Philadelphia, PA

Dear Messing, 

Many thanks for your answer at length. Let me reply, at first, to just the opening of your posting. You've added other points, which I think may be more interesting for the present question, but your opening is also worth some attention. 

You wrote:

You have to recall that Einstein was in many respects the last of a dying breed, doggedly pursuing criticisms of the emerging "new physics" he himself helped found. EPR (his 1935 paper co-authored with Podolsky & Rosen) was his most thorough, scathing attack yet on quantum mechanics. It is one of the most cited papers in physics, and cited because Einstein was correct: QM entailed non-locality.

--pause---

I believe that what you say here is correct, as history. Yet, I am doubtful of the relevancy of these points to the question at hand. It is no doubt, generally important to know that Einstein consistently resisted QM --especially as understood by Bohr, Heisenberg and Born. The famous EPR paper was crafted as a criticism of the claim for completeness of QM with its uncertainty relations. Most physicists now believe that the paper failed to achieve its stated aim, but as you note, it is a very important paper in spite of that. Given the kind of thought experiment proposed in EPR, contemporary physics draws the conclusion of non-locality. But though we suppose that Einstein was wrong about non-locality, we might draw the conclusion from what has been said here so far, that Einstein was quite a physicist after all, and that he knew what he was talking about--specifically when  claiming (in 1938) that "Fundamental ideas play the most essential role in forming a physical theory."  Your comments above certainly do not rule this out. 

You continue:

However, Einstein sought to use this argument (that the logic of this theory allowed for "spooky-action-at-a-distance") to show QM to be incomplete/flawed. Instead, his work inspired and laid the groundwork for Bell's inequality which in turn gave Aspect the capacity to empirically in 1982 test what Einstein probably wouldn't have considered "ein Gedankenexperiment" rather than clear indications of QM's now clearly revealed flaws. Your quote dates 3 years after that most devastating attack yet, but despite his vast achievements in physics the only paper so cited it's referred to using an acronym is the paper showing his argument correct but his conclusions (and ideas) wrong.

---end quotation

Here, as it seems to me you get closer to the question. But whether or not his ideas on this particular topic were wrong--it seems that some were, other not--, this would not show that Einstein and Infield were also wrong in 1938 to claim that  "Fundamental ideas play the most essential role in forming a physical theory."  A physical theory, after all, is not quite the same thing as a true physical theory.  For example we currently have many competing proposals for new physics (as with "string theory, LQG, etc.) , which, I think you'd agree, can't all be right, but they are all physical theories. If we accept the claim in my quotation from Einstein and Infield, this does not imply that the ideas or concept must be correct in order to "play the most essential role in forming a physical theory." 

I conclude that your opening paragraph doesn't really address the question here as much as it addresses Einstein's relationship to QM. But if we look to Bohr, Heisenberg and Born instead, then I think that Einstein would have made much the same point: "thought and ideas, not formulae, are the beginning of every physical theory."  The thought in QM seemed to have been that the statistical character of predictions and the uncertainty relation are basic, and there is no incompleteness about it, nothing missing, nothing else to know. Einstein thought that doubtful, though he didn't dispute the empirical correctness of QM. The fact that Einstein was disputing Bohr, Heisenberg and Born, tells us nothing in answer to the present question. 

I hope you will not be too disappointed that I am yet to address the rest of your lengthy reply--for which I am grateful! But it seemed wise to get this first paragraph out of the way before proceeding to the rest of your thoughtful remarks. Part of what is of interest here is the question of a general understanding of physical theory which extends beyond the devotees of any given theory or approach. 

H.G. Callaway

You have to recall that Einstein was in many respects the last of a dying breed, doggedly pursuing criticisms of the emerging "new physics" he himself helped found. EPR (his 1935 paper co-authored with Podolsky & Rosen) was his most thorough, scathing attack yet on quantum mechanics. It is one of the most cited papers in physics, and cited because Einstein was correct: QM entailed non-locality. However, Einstein sought to use this argument (that the logic of this theory allowed for "spooky-action-at-a-distance") to show QM to be incomplete/flawed. Instead, his work inspired and laid the groundwork for Bell's inequality which in turn gave Aspect the capacity to empirically in 1982 test what Einstein probably wouldn't have considered "ein Gedankenexperiment" rather than clear indications of QM's now clearly revealed flaws. Your quote dates 3 years after that most devastating attack yet, but despite his vast achievements in physics the only paper so cited it's referred to using an acronym is the paper showing his argument correct but his conclusions (and ideas) wrong.

That said, there are two issues here. The first is that even mathematicians rely on hunches, speculation, thoughts, ideas, and odd sources of inspiration that (if they are lucky) become neat, tidy proofs that show nothing of the inspirations or ideas behind them. Many a physical theory was formulated by accident or via a somewhat desperate play (Planck's constant, X-rays, etc.), and there can be little in the way of physical theory without some reason to investigate a phenomenon and some formal (e.g., logico-deductive) framework used not only for inquiry but analysis.

The second issue, however, is modern physics. Quantum mechanics divided the mathematical representations of physical systems and their properties in a fundamental way. In classical physics, one represents an "observable" with the value that is observed (the momentum represented by some value of momentum, speed is a value of speed and mass is of mass, etc.). Quantum mechanics doesn't do this. Instead, "observables" are mathematical functions (operators), not values. There is no direct (or known) correspondence between the "observables" and the properties of a physical system. That was bad enough, but in order to obtain a relativistic quantum mechanics and/or a quantum version of electrodynamics (QED), much of the formulation of the theory came from taking mathematical expressions and equations as classically expressed and making them fit into the mathematics of QM.

CERN announced in 2012 that they'd found the Higgs of the standard model. A few months ago I read a paper that detailed the work since then investigating what was found, and concluding that what was found was very likely what they said they found ~2 years ago. After so much time, how can you not know whether or not what you have found is what you predicted you'd find? Because there exists an at best vague divide between particles, the representations such systems, and the formal framework demanded to investigate what these systems "are" (by "formal framework" I mean that part of QM, QFT, and so forth isn't simply the statistical or mathematical structure but demands that we prepare a "system" in a particular way such that we can transcribe these preparations into the necessary mathematical notations required such that upon measurements of specified kinds we can relate the outcomes via the formalism to say something about the state of a system that exists so far as we know in a purely mathematical realm).

One of the reasons we have so many different formulations of field theories and so many interpretations of experiments (or of theories! nobody spent decades debating the interpretation of classical mechanics) is because so much of it is based upon mathematics. One QM divorced the formal representation of a system and its properties for that system, all extensions of it (field theories, particle physics, etc.) became far less empirical. As I think the Higgs example demonstrates quite well, it wasn't just a matter of taking classical theories and incorporating them into the mathematics of quantum physics, it was also a matter of depending so much on the mathematical structure that once something our models predicted is found we have to do a significant amount of work figuring out what we actually "found".

The first "real" multiverse theory was Everett's approach to QM: follow the math. The reception of his theory was chilly at best, and he had no choice but to leave academia to lead a hard life as a multimillionaire. Later, DeWitt & Deutsch reincarnated his work and it was renamed the "many-worlds interpretation", one of the class of relative state interpretations of QM. Around that time, Viliken and others were investigating the mathematical structure of the standard model and developed a multiverse theory based on an inflationary cosmology. As Bernard Carr pointed out in the volume Universe or Multiverse? many physicists are attracted to multiverse theories for the same reason others are so leery of them: they are based more on mathematical beauty/elegance than anything else and are currently untestable. Yet there are many multiverse theories.

More and more of physics is information-theoretic, and experiments that have nothing to do with quantum computing still refer to qubits. Mathematics used to be a tool (albeit an essential one) for physicists. Now, for many physicists the kind of "thoughts and ideas" essential to solving problems in physics are similar to those motivating mathematicians.

Thoughts and ideas will always come first. Nobody will start doing a Sudoku puzzle and just happen to solve the Riemann hypothesis (in fact, the work on it or about it has been motivated by many ideas, from the zeta function to quantum computing). However, I think that in many ways the nature of the ideas that drives a lot of physics (at least physics in fields like particle physics or those who must use QM/QT) has changed from those that drove Einstein and the last of the physicists to see Lord Kelvin's two clouds grow to become the raging storms that swept up Laplacian determinism, rent it, and scattered its pieces among the waves and sands of time.

Philadelphia, PA

Dear Messing, 

Many thanks for your answer at length. Let me reply, at first, to just the opening of your posting. You've added other points, which I think may be more interesting for the present question, but your opening is also worth some attention. 

You wrote:

You have to recall that Einstein was in many respects the last of a dying breed, doggedly pursuing criticisms of the emerging "new physics" he himself helped found. EPR (his 1935 paper co-authored with Podolsky & Rosen) was his most thorough, scathing attack yet on quantum mechanics. It is one of the most cited papers in physics, and cited because Einstein was correct: QM entailed non-locality.

--pause---

I believe that what you say here is correct, as history. Yet, I am doubtful of the relevancy of these points to the question at hand. It is no doubt, generally important to know that Einstein consistently resisted QM --especially as understood by Bohr, Heisenberg and Born. The famous EPR paper was crafted as a criticism of the claim for completeness of QM with its uncertainty relations. Most physicists now believe that the paper failed to achieve its stated aim, but as you note, it is a very important paper in spite of that. Given the kind of thought experiment proposed in EPR, contemporary physics draws the conclusion of non-locality. But though we suppose that Einstein was wrong about non-locality, we might draw the conclusion from what has been said here so far, that Einstein was quite a physicist after all, and that he knew what he was talking about--specifically when  claiming (in 1938) that "Fundamental ideas play the most essential role in forming a physical theory."  Your comments above certainly do not rule this out. 

You continue:

However, Einstein sought to use this argument (that the logic of this theory allowed for "spooky-action-at-a-distance") to show QM to be incomplete/flawed. Instead, his work inspired and laid the groundwork for Bell's inequality which in turn gave Aspect the capacity to empirically in 1982 test what Einstein probably wouldn't have considered "ein Gedankenexperiment" rather than clear indications of QM's now clearly revealed flaws. Your quote dates 3 years after that most devastating attack yet, but despite his vast achievements in physics the only paper so cited it's referred to using an acronym is the paper showing his argument correct but his conclusions (and ideas) wrong.

---end quotation

Here, as it seems to me you get closer to the question. But whether or not his ideas on this particular topic were wrong--it seems that some were, other not--, this would not show that Einstein and Infield were also wrong in 1938 to claim that  "Fundamental ideas play the most essential role in forming a physical theory."  A physical theory, after all, is not quite the same thing as a true physical theory.  For example we currently have many competing proposals for new physics (as with "string theory, LQG, etc.) , which, I think you'd agree, can't all be right, but they are all physical theories. If we accept the claim in my quotation from Einstein and Infield, this does not imply that the ideas or concept must be correct in order to "play the most essential role in forming a physical theory." 

I conclude that your opening paragraph doesn't really address the question here as much as it addresses Einstein's relationship to QM. But if we look to Bohr, Heisenberg and Born instead, then I think that Einstein would have made much the same point: "thought and ideas, not formulae, are the beginning of every physical theory."  The thought in QM seemed to have been that the statistical character of predictions and the uncertainty relation are basic, and there is no incompleteness about it, nothing missing, nothing else to know. Einstein thought that doubtful, though he didn't dispute the empirical correctness of QM. The fact that Einstein was disputing Bohr, Heisenberg and Born, tells us nothing in answer to the present question. 

I hope you will not be too disappointed that I am yet to address the rest of your lengthy reply--for which I am grateful! But it seemed wise to get this first paragraph out of the way before proceeding to the rest of your thoughtful remarks. Part of what is of interest here is the question of a general understanding of physical theory which extends beyond the devotees of any given theory or approach. 

H.G. Callaway

Physics, much like anything else in history, has followed its share of fads and fashions over the ages. Every important age of physical discovery has donated a certain emphasis and vocabulary to its descriptions of the natural world. What has never changed, or rather, has remained invariant; is what Albert Einstein refers to as the concept of measurement:

"The ideas must later take the mathematical form of quantitative theory, to make possible the comparison with experiment"

What is a "quantitative theory" has at times been up for debate over the last 500 years. However irrefutable at any time is that what you can decidedly measure must be fundamentally consistent with your ability predict it using whatever framework - and note that I don't say understand - in order that it remains of value to the human intellect.

At Einstein's time scientific speculation was rife about the very nature of matter, space and time as we have come to understand it today. Many bright minds of his generation expended themselves in the search for consistent and elegant mathematical "suitcases" to capture universally applicable laws about how the universe worked. Many of them, including Einstein himself at times, were to be bitterly disappointed. Nonetheless, being theoreticians, we look inward and outward to gauge the depth and fabric of physical reality. We do it today and we did it then in the form of Poisson's or Mach's great minds. However back then there were symbols and mathematics to help describe things, today there are computers as well. Soon there might be quantum computers .. things change, people change, the universe ultimately changes .. so can its descriptions and the thus the ideas that lead to the descriptions.

Applied mathematics (as opposed to the pure art which is representative to a greater extent of abstract symbolic notions) is the language of measurement and pattern. Just like musical notation is the representation of sound with symbols, from basic notations for pitch, duration, and timing, to more advanced descriptions of expression, timbre, and even special effects; just as this notation can describe what we hear, mathematics is the language that describes (brilliantly) what we have perceived and measured through instruments since the dawn of time. The reason it does that is because we can count and thus measure in the sense of counting and there doesn't seem anything more special to it than that. Another race might not necessarily perceive reality the same way, but the physics either experiences would have to be the same.

Abstractly speaking, in both the case of music or scientific measurement however, note that the musical notation is not music, and the music is never the notation, the two are effective "representations" of one another. The same can be said to be true of a physical theory; reality is quite distinct from what is actually used to describe it because what is used to describe it may not be able to describe every facet of the phenomenon satisfactorily; which begs the question: can we really perceive and give description to every phenomenon in the universe ? in reverse : reality is what we can measure and that thought can lead to any number of logical messes, the notable one being that there animals that can hear certain tunes we can't. The mathematical description used for both sets of tunes is actually the same. However for humans one set of tunes just doesn't exist ! So in that sense Einstein speaks of these fundamental ideas connected to theories - the math is then a way of discovering or extending to what we don't know about the universe that is as real and responsible for discovery as using a measuring stick.

To simplify a little, as long as we can make some kind of prediction about a phenomenon in a measurable and repeatable way, anywhere we should reasonably like to choose, then we have some right to think that we might know something about it. However, there are degrees of this implicit in the very definition: Einstein's famous legacy : the Theory of Everything is an extreme push towards a fundamental elegance, a point philosophy if you like, and that is where his "ideas are fundamental" bit actually comes from. It is almost as if we wanted to say every piece of music could be described by just combinations of one simple note (I exaggerate and simplify quite grossly). It might be, but the problem is to do profoundly with scale and reverse engineering and calculating the rules for a universe backward is the realm of (presently impossible) computing.

An idea  like the speed of light is everywhere constant succeeded in giving us a profound bit of physics, it is romantic and if you can solve a parabola, you are probably overqualified to do basic computations in it .. but not all physics in the stellar "patchwork" is so easily captured. Therefore, by today's high standards, both mathematics and ideas are equally important to theories (in my opinion) - they are just ways of understanding the reality of the universe and there could indeed be many more that may yet be found (e.g. inverse computing algorithms); so to answer, no Einstein (or Feynman for that matter) were not saying math is less important than physics per se, they were saying math is used as a tool in physics but that nature could very well write her own music (the ubiquitous measurable phenomena) with the tools left to describe what is actually there .. the degree to which any idea will fit nature's scheme depends upon exactly what nature actually is and does as we go beyond the realm of everyday experience; and to date nobody has been able to answer that in a unified or satisfactory way. I hope that sheds a little light on what is an enormous philosophical subject (Quantum Computing, Shannon's, Turing's and von Neumann's work give more insight into the relationship of math to the fabric of reality which the latest fad says is actually **information**).

Dear Dr. Callaway,

Mathematics might be seen an abstraction of patterns in nature through a very particular formal language, one which aims at being universal.  As a human construct, it can be also viewed with the lens of intuition as long as those abstractions pose an interplay between induction and abduction. Abduction in the expectation that physical events, once described by mathematics, are exemplars of regularities in classes of phenomena, therefore the relation of effects to causes becomes plausible  though their mathematical description (ideally sound and complete). It is also inductive from the experimentalists view: as statistical significance grows (as well as the existence of mounting evidence for stating causal relations from causes to effects) the expectation is for the law to work. When it does not in general cases, the theory can be easily dismissed as incorrect. But when it does not work for particular (corner) cases, those classes of phenomena often point to new directions of the physical concepts. But, and this is the important issue, the directions are heavily informed by the mathematics.

There are some avenues for thinking along this line.

1. Assume there exists a Platonic mathematical reality correspondent (strictly isomorphic) with physical reality. Therefore, as we discover the grounds for physical phenomena, the discovery extends to the mathematical realm. It is, nonetheless unsatisfactory and contrary to the evidence in the evolution of languages.

2. Mathematical idioms are developed to aid explanation, but are only tools. Utilitarian views such as this one seem closer to experimentalists, but then it becomes necessary to provide an explanation on how intuition builds upon them in cases such as "guessing" the right equation (e.g. Dirac) for a given phenomenon, in particular in cases where common intuition fails due to apparent counterfactuals.

3. Mathematical idioms are cognitive constructs that, along with non-deterministic (i.e. intuitionistic) mechanisms are able to work in a heuristic self-refining fashion.  Although it may seem mysterious, intuition as heuristics is a powerful way of overcoming the limitations of finding the appropriate starting point for the mathematical grammar to work. As a heuristic, it may be an asymptotic process but it remains a product of evolution by natural selection in the end.

Regards

"Pure mathematics is, in its way, the poetry of logical ideas." Albert Einstein

The difficulty level of Einstein's computations in his epocal 1905 articles is even less than elementary. However, the articles designed a new way to understand the reality, a completely new model of the world. The moral is that is important what do you compute, and not how difficult the computations are. Einstein's genius was to formulate his new postulates (axioms) and to believe in their consistency - that is, in the consistency of the new theory. The consistency was later proven by experiments, which demonstrated the truth of some predictions made by the new theory. Of course, if the new postulates are consistent and sound, one can make as much and as complicated mathematics over them, as one needs. But this is a property of mathematics (its soundness) and not of physics. As to physics, I believe that most theories will remain only partial descriptions of the reality, despite of their unboundedly complicated mathematics, which is (only internally! - I mean, inside the theory) sound. We will assist to a continuous evolution of our understanding of the real world, without ever touching a final complete theory. [But this last idea is only a belief of myself - I cannot prove it completely.]

Are "thought and ideas" central and essential and the mathematics secondary and more important for experimental results?

Dear Dr Callaway,

In my opinion…thoughts and Ideas are central of physics or I would say understaning the 'Nature'... “The Apple” gave the idea to Newton which he formulated…Every “Eureka” event started from an idea…

But at the same time I also feel…having exceptional mathematical mind is a bless of God to some very special people…Same idea that clicked to Newton might have clicked to so many people earlier…people those were not mathematician…may be farmers, singers, actors etc…but when a great mathematician observe the same event in nature…he is having the capability of formulating and quantifying the thing that everyone was feeling…and explain it to others through a mathematical theory…understanding this mathematical theory is easier than formulating it…and now this theory will be like a brick on foundation of science on which buildings can be build…

I have no doubt that from the day one on earth mankind will be knowing that things fall down….but how much time it will take, what will be its velocity etc etc can be told by Newtons only…

So “hen or egg”…I don’t know...but I feel IDEA first..because we should not forget that we do not respect a mathematical theory till it matches with real world...Real world: that gives ideas to everyone...

Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories".

The rigorous, abstract and advanced re-formulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in the so-called analytical mechanics. It leads, for instance, to discover the deep interplay of the notion of symmetry and that of conserved quantities during the dynamical evolution, stated within the most elementary formulation of Noether's theorem. These approaches and ideas can be and, in fact, have been extended to other areas of physics as statistical mechanics, continuum mechanics, classical field theory and quantum field theory. Moreover they have provided several examples and basic ideas in differential geometry (e.g. the theory of vector bundles and several notions in symplectic geometry).

Ideas and imagination are fundamental in navigating the unbounded space of truth  of nature and developing  physical theories. Ideas are also reasons of developing abstract mathematical theories as well.  To extrapolate an existing theory (mathematical ) to a larger one, first an idea has to be developed within in.

The Newtonian laws of physics were painstakingly developed via learnt  ideas and observing nature.  Einstein used to say how uncomfortable in the mathematics he was using but strong in  imagination and ideas of different breeds to see space time as curved and developed his mathematical theories to describe and materialize his observation. The equations were written based on his ideas of space time, not the converse that space time was found curved from the equations. 

Maxwell published his second paper in 1862 On Physical Lines of Force which was mainly concerned with constructing a model for the medium (known as the ether) that would account for the electric and magnetic effects. Using this model he considered the electromagnetic wave and found that its speed would be equal to the ratio of the value for an electric current measured in electrostatic units to the value of the same current measured in electromagnetic units:

This ratio had been calculated to be the speed of light and so Maxwell drew the conclusion that:

Light consists in transverse undulations of the same medium which is the cause of electric and magnetic oscillations.

I think, in this case, mathematics was essentials in forming the physical theory of light.

From responses thus far, it seems clear that the bookkeeping aspect of mathematics is essential, but that additionally many feel that ideas are at least somewhat mathematical in composition. Those seem to be the themes here, but there is at least one more, perhaps somewhat different consideration, that goes somewhat further: What if Pythagorus was right, and "All is numbers!" Consider the golden ratio and pi. This perhaps reaches into the realm of metaphysics, but appears to me to have a connection to this question.  

Philadelphia, PA

Dear all, 

Many thanks for the wealth of contributions to this thread. So far, I see some sympathy with the initial quotation from Einstein, and also some resistance to it. Part of the difficulty is Einstein's contrast between "ideas" and mathematics. I suspect that just about any ideas we might care to consider from physics could be given a more mathematical or less mathematical treatment. The inclination to focus on the more mathematical or less mathematical treatments may differ from person to person, partly in accordance with personal mathematical accomplishments and inclinations. I think it worth considering, too, that those using physics to solve particular problems come to intensive practical experience with relevant mathematics, and this sets up strong habits of turning to the familiar formulas. 

Are physical ideas intrinsically mathematical? Well, in the quotation, Einstein thought there was a useful contrast between formative theoretical ideas and mathematical formulations. He was surely thinking of his own work. Regarding special relativity the idea is to adjust classical physics on the supposition of the constancy of the speed of light as measured from any frame of reference moving with uniform velocity. Notice that this doesn't require a formula to state it. In GR, it is often said that the basic idea is the equivalence of gravitational and inertial mass. Its a "generalization" (the "G" of "GR") because it also takes in accelerating frames of reference. Einstein came to reject Newtonian space and time and the supposition of a single universal frame of reference (the Newtonian "container.") 

Notice, though, that no matter how flat-footedly conceptual our approach to physical theories, one might reformulate the same point in set-theoretic terms. You could map the conceptual relations involved with set-theoretic relations; and further, it is often held that all of mathematics might be reduced to logic plus set-theory.  A reformulation of physical insights in set-theoretic terms seems to trivialize the claim that physics is essentially mathematical. (By the same kind of argument, everything else would be essentially mathematical, too.) Whatever the higher mathematics involved in quantitative physical theory, this could, then, be reformulated set-theoretically. But the point I'm inclined to take from these considerations is simply that we can expect to understand a great deal of physics in conceptual terms without relying on higher mathematics, and without attempting to solve problems in physics or do calculations. I would emphasize, too, that this is not thought of as a way of doing physics. It is instead a matter of understanding what the physicists are doing, for purposes of the more general educated public. 

Einstein emphasizes, in the quotation above, that mathematics and quantitative theory are needed for purposes of  experimental testing. This point makes perfect sense, I think, since given the experimental manipulation or variation of variables of quantitative theory, one will expect to get a corresponding range of quantitative predictions for testing. (The points of empirical contact are multiplied.) The point is that the quantitative aspect facilitates empirical testing. But if the mathematical aspect of a particular physical theory ceases to facilitate testing --in the sense that no testable predictions are forthcoming from particular complex and mathematically inspired theories, then the appeal of the mathematical will be much diminished. Critics are increasingly making such complaints about the on-going proposals for new physics arising. e.g., from string theory. 

We are also hearing more "Pythagorian" (or, I'd prefer to say, Platonic) themes of late which emphasizes a sense of the self-certifying character of mathematical beauty and elegance. But I think that excessive emphasis in this direction would represent a considerable danger to empirical science in general. Let me suggest a paraphrase of a well known quotation. The arc of mathematical theory is long (as we might say), but it bends toward empirical prediction. Why? Because the historical evidence is that lacking this, pure theory will tend to produce self-stultifying conceptual catastrophe.   

H.G. Callaway

Philadelphia, PA

Dear Messing, 

I've now had a chance to come back to your long posting --which came in near the start of this thread, and I want to briefly address the following paragraph. 

You wrote:

That said, there are two issues here. The first is that even mathematicians rely on hunches, speculation, thoughts, ideas, and odd sources of inspiration that (if they are lucky) become neat, tidy proofs that show nothing of the inspirations or ideas behind them. Many a physical theory was formulated by accident or via a somewhat desperate play (Planck's constant, X-rays, etc.), and there can be little in the way of physical theory without some reason to investigate a phenomenon and some formal (e.g., logico-deductive) framework used not only for inquiry but analysis.

---end quotation

Here I am pretty sympathetic to what you have to say, and in tendency, it seems that you are also expressing some agreement with the Einstein and Infield quotation reproduced in at the start of this thread. 

Einstein's point, I take it, is not to discourage more precise and quantitative formulations of physical insight, but instead to focus attention upon the importance of physical insight or concepts and "fundamental ideas."  The proofs and calculations which follow in the wake of this, may, as you say, "show nothing of the inspirations or ideas behind them." Yet Einstein places this emphasis on formative or fundamental ideas in spite of that. I think we can agree that "there can be little in the way of physical theory without some reason to investigate a phenomenon and some formal (e.g., logico-deductive) framework used not only for inquiry but analysis." 

I think that no one can dispute the importance of mathematics in physical theory. This is not to suggest that it can never become a kind of baroque overgrowth on the body of physical theory, though. Again, I have no doubts about the importance of mathematical physics. Perhaps the most important example of the importance of this came with the work of Beckenstein and Hawking on black holes --where gravitation and QM are equally important, and much useful theory has proceeded without the possibility of experimentation, beyond the crucial thought experiments, as with the question of what happens with entropy when a disorderly physical system is lowered in a container into a black hole. How is the physical development to be understood  in relation to the second law of thermodynamics? Out of this came the thermodynamics of black holes and eventually the hypothesis of Hawking radiation. 

But even such results of mathematical physics can be understood in more conceptual terms, and I submit, it is worthwhile trying to do so. This will not make physicists of us, but it can inform the philosophy of science and public attention to developments in physics. 

I notice that you have some further comments on the role of mathematics in quantum theory. Perhaps you would care to develop your related points. 

H.G. Callaway

Darrigol summarizes:

''Most of the components of Einstein's paper appeared in others' anterior works on the electrodynamics of moving bodies. Poincaré and Alfred Bucherer had the relativity principle. Lorentz and Larmor had most of the Lorentz transformations, Poincaré had them all. Cohn and Bucherer rejected the ether. Poincaré, Cohn, and Abraham had a physical interpretation of Lorentz's local time. Larmor and Cohn alluded to the dilation of time. Lorentz and Poincaré had the relativistic dynamics of the electron. None of these authors, however, dared to reform the concepts of space and time. None of them imagined a new kinematics based on two postulates. None of them derived the Lorentz transformations on this basis. None of them fully understood the physical implications of these transformations. It all was Einstein's unique feat.''

http://en.wikipedia.org/wiki/Relativity_priority_dispute

So according to Darrigol Eintein's main contribution was not mathematical but  what these equations fundamentally mean for physics: with Einstein we have not new mathematics or new fact about the world but a new world we live in, a totally new perspective on the same fact and equations.

Philadelphia, PA

Dear Abbas,

You wrote, regarding Maxwell's work: 

This ratio had been calculated to be the speed of light and so Maxwell drew the conclusion that:

Light consists in transverse undulations of the same medium which is the cause of electric and magnetic oscillations.

I think, in this case, mathematics was essentials in forming the physical theory of light.

---end quote

I think you suggest a valuable argument here. A prediction regarding the speed of light, as I understand the matter, falls out of Maxwell's quantified theory of electromagnetic phenomena. It thus became a test of Maxwell's theory that the measured value for the speed of light should have turned out in agreement. No doubt about it, I think, the mathematics was essential to Maxwell's theory and its confirmation. 

So, let's take another look at the quotation from Einstein and Infield:

Fundamental ideas play the most essential role in forming a physical theory. Books on physics are full of complicated mathematical formulae. But thought and ideas, not formulae, are the beginning of every physical theory. The ideas must later take the mathematical form of quantitative theory, to make possible the comparison with experiment. Einstein and Infield 1938, The Evolution of Physics, p. 277.

--end quotation 

One notices that in the quotation, it is stipulated that the "mathematical form of quantitative theory" is needed "to make possible comparison with experiment." But what would Einstein say about the "fundamental ideas" of Maxwell's physics?  Well, it seems clear that Einstein took from Maxwell the "fundamental" idea of the "field theory."  He saw this idea as basic in over coming the Newtonian idea of (gravitational) action at a distance and the expectation of a purely mechanical or billiard-ball universe. 

I take the following quotation from Einstein's 1940 discussion of Faraday, Maxwell and field theory:

But if the electromagnetic field could exist as a wave independent of the material source, then the electrostatic interaction could not longer be explained as action at a distance. And what was true for electrical action could not be denied for gravitation. Everywhere Newton's actions at a distance gave way to fields spreading with finite velocity.  (Einstein 1940, "Considerations Concerning the Fundaments of Theoretical Physics," Science, 91, no. 2369, p. 489.)

---end quotation

I submit that, though Einstein, consistent with the 1938 quotation could certainly appreciate the importance of a calculation of the speed of light from Maxwell's physics, he though the idea of field physics more basic and important.  This is the kind of "thought and ideas," to use Einstein's words, which stand at, "the beginning of every physical theory." From this perspective, your argument seems to turn of an ambiguity concerning what "forms" physical theory. 

H.G. Callaway

Dear all,

I would prefer to interpret this Einstein quote as a contribution to science theory in general.

It later inspired Poppers idea of critical rationalism. In the light of this more general approach, that is not only relevant in physics, it means to me that thinking has priority compared with mathematics and experiments. It has it's charme because it opens the doors for creative thinking. Therefor it can be understood as an antipositivistic statement.

But the discussions went on and today we are aware, that it is too easy to follow a pure Popperian approach, because science as it is really proceeded is much more complex, experiments do also inspire thinking as well as mathematics does. Poppers conceptualization and with it the mentioned quote has been criticized often and fundamently by Lakatos, Kuhn, Feierabend, Hacking and a lot of other science philosophers.

The quote in my opinion is written in a time when the zeitgeist was positivistic and physics was in a change or crisis and new ideas were necessary.

The methodological implications of the quote in my opinion are, that mathematics and experiments can be understood as tools of thinking. And these tools should be be used to test the results of this process in sophisticated experiments. The problem herein is that the other way is also possible, e.g. that observations from experiments could lead to ideas.

My consequence for the asked question of H. G. Callaway about the relations between basic concepts, mathematical formulations and physical theories is, that mathematics and formulae as well as physical theories are products of our minds and are all necessarily built on basic concepts and preassumptions. There is probably a conceptual hierarchy in the used categories, because physical theories usually contain mathematical formulations and physical theories, as well as mathematics necessarily include basic ideas (e.g. truth).

Regards Thomas

IMHO ( with apologies where needed).There are absorbing valuable  insights in this thread. From an engineering point of view simplistically put,  it seems basic concepts are a must ,mathematics becomes a tool to solve and show the relationships, patterns, cause/ effect which help in rationalizing logically the concepts . It is hard to imagine an equation generating a physical concept,mostly it is the other way around. Mathematics is a beautiful subject with enormous power.  It seems to be based on abstracting patterns from "reality" and also perhaps from Qualia (totally unrelated to known phenomena) which later becomes recognized as useful tools to explain new discoveries in "reality".Examples are to be found in Number theory (cryptography). Linear algebra which easily extend idea of cosine to multidimensional systems.

 Cheers

Hello everyone,

I would like to add further from my previous post.

I would like to categorize the ideas here…First one is the very basic idea…Ideas that starts a new science…a new way of doing/understanding things….Every science begins from an Idea and keeps on maturing…this maturing process is never ending…In this maturing process so many further ideas (second typd) come…mathematics, experiments, logics etc come in path of coming further ideas…so it’s a infinite loop…

After the very first idea the role of mathematics comes in picture…even a mathematical expression can give birth to so many new ideas…and more importantly when we proceed from science to engineering i.e. implementing it, mathematics is very much important…’Metals expand as they are heated and when this expansion is constrained thermal stresses develop’ is whole basic behind thermal stresses, which itself is a vast topic…the ideas are beginning and very very important also…but it will die at birth itself or will not become a mature subject of science if it will not get breath of mathematics….

Nature is very vast and so physics too…As we mature…Nature opens new and new treasures of knowledge to us…It (Nature) waits till we are enough mature to digest that knowledge…and at the same time “everything in nature is going towards equilibrium (state of least energy) from the path of least resistance” and we Human are not exception to it…Mathematics is not a new subject…In my opinion it (mathematics) was a natural selection by lazy human brain to understand the nature/physics because it was the easiest way…Today science has advanced too much so explaining it can not be simple even with this simplest possible tool...that's why we find big big complex expression in the books of physics:-)...Both are that much intermingled that seeing them independently can be discussed only but can not be concluded…

Regards

Some thoughts about the discussion so far:

(A1) Do we-as-physicists agree that there is something spooky about “basic CONCEPTS”? (Even many psychologists are a bit hazy about this so it would not be surprising). Would it affect the discussion if, instead, we had a clear idea of the physiological/biochemical mechanisms whereby these nontrivial basic concepts are (#) acquired, (#) stored, (#) organized, and (#) retrieved within the brain? --- [There are some plausible theories about this, based on the ideas of J.Piaget (1896-1980), and W.R.Ashby’s “Design for a Brain” --- both leaning tacitly on Darwinian-like ideas].

(A2) Do we agree that there is also something spooky about MATHS, at least in the way it comes to us? We don’t see mathematical symbols when we look down a microscope, and yet somehow they find their way into the curriculum. [Here at least we can perhaps trace the painstaking investigations of Frege and suchlike others --- but there is still some uneasiness about connections to reality].

(A3) So is Callaway’s question ultimately trying to get these two spooks to support each other thus producing something more tangible and productive? --- If so, that would seem to be sensible, at least as a step in the right direction.

(A4) As an overview, we might construe the situation as fruitfully reconciling two epistemological domains: The inner “thought” domain within the individual, AND the Separate-But-Comparable maths/logic/computer domain within Science/Society-as-Such. --- http://iopscience.iop.org/1742-6596/329/1/012018   &/or   www.ondwelle.com/MolecularScheme.ppt

----==ooOoo==----

(B1) These are basic problems which seem generally applicable --- not just the province of physics. Indeed if we are really trying to understand the process, we should surely start with much simpler cases. In fact it might be seen as misleading to dive into the complexities of EPR or Maxwell when (e.g.) Piaget offers us comparable ELEMENTARY-SCIENCE problems faced by young infants.

(B2) E.g. Instead of EPR --- consider how we gain the concept of a circle --- or, more basic still, a CLOSED LOOP (allowing for squares etc as well). Piagetian theory offers an account of how the mental concept is assembled (based on encoded verb-like ACTIONS) — and we could apply some basic TOPOLOGY-theory to deal with any parallel mathematical requirements in the external world of Science/Society.

(B3) If we can work through that, to whatever standard of rigour we choose, then we might want to work our way toward wilder complexities (such as how EPR-concepts are processed mentally) --- but probably not in one giant step!

Philadelphia, PA

Dear Traill and contributors, 

It strikes me that what you say above may be somewhat off topic, Dr. Traill. Perhaps others will see the matter differently, though. 

I think the present thread is not so much about the origin of basic physical concepts. It is more about the relationships of basic concepts to mathematical expression and development. However, it has been suggested, along the way, that the basic concepts which Einstein emphasizes, may sometimes have their origin in mathematical theory.

If I may offer an analogy, the difference I'm pointing to is like that between the question of the origin of life and the question of the evolution of living things or species. We can have a pretty good idea of evolution, even in regard to the evolutionary relationships of particular species, say, the relationship of Darwin's island-dwelling finches to ancestral mainland birds, without every broaching the question of the origin of life. In a similar way, it seems to me that we can ask and answer questions concerning the relationships of concepts and theories, without every looking into the question of the psychological origin of either.  

Basically, since you ask, I don't regard either concepts or theories as "spooky," and the question here has much more to do with the relative importance of concept-based exposition/explanation and mathematical theory in our philosophical and public understanding of physics--and of what it is that physicists do.

The work of Piaget is of undoubted importance and broad interest for many fields of study and scholarship, but I simply do not see that it would be very helpful here to take up the psychological origin of concepts. Nonetheless, I want to thank you for your interest and for your kind offer to inform us on Piaget's work. 

As I say, others may see the matter differently. 

H.G. Callaway

Dear H.G. Callaway,

Could it be that the answer to your question (Are "thought and ideas" central and essential and the mathematics secondary and more important for experimental results?) can be found in the works of F.S.C. Northrop? What follows are his ideas and words mainly.

Science, according to Northrop “… proceeds in two opposite directions from its many technical discoveries. It moves forward with the aid of exact mathematical formulation to new applications, and backward with the aid of careful logical analysis to first principles. The fruit of the first movement is applied science, that of the second theoretical science. When this movement toward theoretical science is carried through for all branches of science we come to first principles and have philosophy” (Northrop 1931).

Right at the beginning of his 1916 paper Einstein writes that it was Ernst Mach who suggested that Einstein’s own Special Theory of Relativity (1905) as much as the classical mechanics of Galilei and Newton had a epistemological flaw: objects were moving relative to one another with constant rectilinear motions. While obviously everyone looking at natural phenomena can observe that objects are usually moving with changing velocities and non-rectilinear. Einstein writes in his paper of 1916 that this defect necessitated the discovery of the new and more general first principles (basic concepts).

Einstein’s solution of this problem was an easy one. The defect being epistemological, its removal must be epistemological also. He writes: “The Laws of Physics must be so constituted that they hold for any conceivable system of moving objects whatever. This way we go to generalization of the Relativity Postulate.” The other speculative discovery of the second first principle needed for his theory was “The general laws of Nature must be so constructed through equations which hold for all coordinate systems, that is, that are covariant for any conceivable substitutions”.

Einstein then (p.769) proceeds to tell us with what pure mathematical and mathematical physical aids he will rigorously deduce from these two basic concepts his equations.

The consequence of Einstein’s example is according to Northrop (1985) that “physics can only remove its own defects by making the science of epistemology more basic than crucially experimentally verified-by-others theoretical mathematical physics”.

Willem Beekhuizen

The following paper:

Leibniz's place in the history of physics by Joseph Agassi.

http://www.tau.ac.il/~agass/joseph-papers/Leibniz.pdf

is very interesting. It is an historical account of the debate debate between Leibniz and Newton and the later influence the Lebiztian relational conception of space and time on later philosphers , geometers and scientist up to Einstein.  We can see in this historical account the long ancestry of ideas on space and time that lead the the type of physics that Einstein developed.  

Philadelphia, PA

Dear all, 

It strikes me, reading through the recent additions to this thread, that some might be interested to have a look at Einstein's 1940 paper " Considerations concerning the fundaments of Theoretical Physics" 

A pdf is available at the following address:

http://promo.aaas.org/kn_marketing/pdfs/Science_1940_0524.pdf?origin=publication_detail

See paragraphs 4-5 in particular which compares fairly directly with the passage from Beekhuizen's posting: 

Science, according to Northrop “… proceeds in two opposite directions from its many technical discoveries. It moves forward with the aid of exact mathematical formulation to new applications, and backward with the aid of careful logical analysis to first principles. The fruit of the first movement is applied science, that of the second theoretical science. When this movement toward theoretical science is carried through for all branches of science we come to first principles and have philosophy” (Northrop 1931).

---end quotation

That said, I would add that I have my doubts on Northrop's take on epistemology, where he says “physics can only remove its own defects by making the science of epistemology more basic than crucially experimentally verified-by-others theoretical mathematical physics”. This is to suggest the view that epistemology is no "queen of the sciences" but instead corrigible in light of empirical discoveries. More on that theme later, perhaps.  There seems to be much attractive in his work, though. 

I also want to briefly recommend the paper by Agassi, recommended by Bassard. The debate concerning relational accounts of space and time is no doubt quite important to the development of physics. 

What do others think of the recent contributions? More later. 

H.G. Callaway

Dear Callaway,

Thank you for the link to Einstein's 1940 paper.

Northrop’s first meeting with Einstein was in 1927 and later they had annual private meetings in his Princeton Residence.

About “Northrop's take on epistemology” – that seems to match Einstein’s ideas about it rather well. Northrop was invited to write a chapter (“Einstein’s Conception of Science”) in the famous Schilpp series: The Library of Living Philosophers, Volume VII: Albert Einstein: Philosopher-Scientist (Cambridge University Press, 1949). In the last chapter, “Einstein's Reply to Criticisms” Einstein discussed Northrop’s contribution, amongst other things, as follows:

“The essays by Lenzen and Northrop both aim to treat my occasional utterances of epistemological content systematically. From those utterances Lenzen constructs a synoptic total picture, in which what is missing in the utterances is carefully and with delicacy of feeling supplied. Everything said therein appears to me convincing and correct. Northrop uses these utterances as point of departure for a comparative critique of the major epistemological systems. I see in this critique a masterpiece of unbiased thinking and concise discussion, which nowhere permits itself to be diverted from the essential.

The reciprocal relationship of epistemology and science is of noteworthy kind. They are dependent upon each other. Epistemology without contact with science becomes an empty scheme. Science without epistemology is — insofar as it is thinkable at all — primitive and muddled. However, no sooner has the epistemologist, who is seeking a clear system, fought his way through to such a system, than he is inclined to interpret the thought-content of science in the sense of his system and to reject whatever does not fit into his system. The scientist, however, cannot afford to carry his striving for epistemological systematic that far. He accepts gratefully the epistemological conceptual analysis; but the external conditions, which are set for him by the facts of experience, do not permit him to let himself be too much restricted in the construction of his conceptual world by the adherence to an epistemological system. He therefore must appear to the systematic epistemologist as a type of unscrupulous opportunist: he appears as realist insofar as he seeks to describe a world independent of the acts of perception; as idealist insofar as he looks upon the concepts and theories as the free inventions of the human spirit (not logically derivable from what is empirically given); as positivist insofar as he considers his concepts and theories justified only to the extent to which they furnish a logical representation of relations among sensory experiences. He may even appear as Platonist or Pythagorean insofar as he considers the viewpoint of logical simplicity as an indispensable and effective tool of his research.

All of this is splendidly elucidated in Lenzen’s and Northrop’s essays.”

---end quotation

Here is a link to Einstein’s comments: https://www.marxists.org/reference/archive/einstein/works/1940s/reply.htm

Willem Beekhuizen

Philadelphia, PA

Dear Beekhuizen, 

Many thanks for your thoughtful and very useful reply. Much more could be said, but I will restrict myself, for now, to emphasizing the following passage quoted from Einstein's reply, in your posting: 

The reciprocal relationship of epistemology and science is of noteworthy kind. They are dependent upon each other. Epistemology without contact with science becomes an empty scheme. Science without epistemology is — insofar as it is thinkable at all — primitive and muddled. However, no sooner has the epistemologist, who is seeking a clear system, fought his way through to such a system, than he is inclined to interpret the thought-content of science in the sense of his system and to reject whatever does not fit into his system. The scientist, however, cannot afford to carry his striving for epistemological systematic that far. He accepts gratefully the epistemological conceptual analysis; but the external conditions, which are set for him by the facts of experience, do not permit him to let himself be too much restricted in the construction of his conceptual world by the adherence to an epistemological system. He therefore must appear to the systematic epistemologist as a type of unscrupulous opportunist: he appears as realist insofar as he seeks to describe a world independent of the acts of perception; as idealist insofar as he looks upon the concepts and theories as the free inventions of the human spirit (not logically derivable from what is empirically given); as positivist insofar as he considers his concepts and theories justified only to the extent to which they furnish a logical representation of relations among sensory experiences. He may even appear as Platonist or Pythagorean insofar as he considers the viewpoint of logical simplicity as an indispensable and effective tool of his research.

---end quotation

This is an excellent little summary, which could do with detailed study and commentary --though it stands as it is, in my view of the matter. As I read it, it seems to me clear that Einstein avoids the high apriori conception of epistemology with mastery; at the same time, he "saves the appearances," of viewpoints he avoids.

I hope that others reading along will appreciate the fine-tuned precision of your short posting. and download the full Einstein text you recommend. 

H.G. Callaway

"Brownien motion is transport phenomenon named after the botanist Robert Brown. In 1827, while looking through a microscope at particles found in pollen grains in water, he noted that the particles moved through the water but was not able to determine the mechanisms that caused this motion. Atoms and molecules had long been theorized as the constituents of matter, and many decades later, Albert Einstein published a paper in 1905 that explained in precise detail how the motion that Brown had observed was a result of the pollen being moved by individual water molecules. This explanation of Brownian motion served as definitive confirmation that atoms and molecules actually exist, and was further verified experimentally by Jean Perrin in 1908." (see link below).

From this example, it seems to me that the thought of atoms and molecule existence was known before Einstein's paper. The way by which Einstein handled this thought mathematically is more essential than the thought it self.

http://en.wikipedia.org/wiki/Brownian_motion

I think they both have the same epistemological status with respect to foundation, because the genesis of the basic mathematical conceps themselves has an ideatic nature. Such concepts - even abstract - embed essential empirical features of the reality. Mathematics creators design concepts and axiomatic systems with an eye (mind) back on physical reality and another forward on the possible development of their theories, including application in other theories. This is one of the reasons mathematics is "unreasonably" effective in physical theories and reality.

Philadelphia, PA

Dear Abbas, 

Einstein's theory and explanation of Brownian motion was no doubt quite important and especially for physics. Still, as I understand the history of the matter, the chemists had long had little doubt of the reality of atoms and molecules, so that Einstein's paper chiefly served to bring the physicists along. I think of the chemists' related convictions as based in the long development of techniques of quantitative analysis and the long history of the identification of the elements --culminating in the periodic table and the prediction of unknown elements and their properties. Physics seems in this case to have "stood on the shoulders of giants" of chemistry.

H.G. Callaway

Philadelphia, PA

Dear Quintana, 

Many thanks for your contribution. I am myself somewhat skeptical of claims for Einstein as a Kantian, though he no doubt knew the German classics. What I tend to see is something more like a striving for "thematic unity," in science and in physics, which is exactly what engenders his emphasis on "thought and ideas." I suspect he is more likely to quote Goethe than Kant. The stress on "thematic unity," in Einstein culminates in his search for a "unified field theory." As a good empiricist, he is well aware of the contrary developments in physics. 

I notice that you have an interest in Max Tegmark's work and the thesis of the Mathematical Universe. That development is certainly of interest here, though I find myself more skeptical about this, and Tegmark's views certainly seem to contrast with what we find in the opening quotation from Einstein and Infield, at the start of the present thread. Perhaps you would do us the honor of commenting on the opening quotation --at the start of this thread of discussion and exchanges.

See the short video of one of Tegmark's talks on "Our Mathematical Universe." 

https://www.youtube.com/watch?v=rtT5itRX_Rc

Tegmark seems to stand at the extreme Platonizing end of the contemporary philosophy of physics. This development seems to have arisen from the lack of evidence relevant to contemporary theories of quantum gravity & etc.. Would you agree? Einstein makes room for the concern with mathematical elegance and beauty, yet only within limits of empirical viability--as I see the matter.

H.G. Callaway 

Philadelphia, PA

Dear all,

Next we come around to the theme of quantum mechanics and its developments. In many ways, this is the most interesting case. The long debates concerning the interpretation of quantum mechanics are exactly what makes QM extremely interesting in relation to Einstein’s emphasis on “thought and ideas”—and the place of mathematics. Just because the debates on interpretation of QM have been so protracted, there is considerable tendency to escape into more purely mathematical approaches to QM. I notice that Messing devoted much of his early posting to this theme, and that there have been no replies to his views—neither supporting arguments or criticism. It strikes me that this lack of response is a result of the obscurity of his argument. Readers will perhaps correct me, if I have missed something. Debates on the interpretation of QM are all well and good, I believe, though again, there is an obscurantist tendency to glory in the dust that may be raised and then to want to escape responsibility for the debate by flight into complex mathematics.

Philosophical doctrines of categories arose, originally, in Aristotle, from a classification of predicates (read “ideas”). Thinking about the role of “thoughts and ideas” in contemporary physics, it is worthwhile considering the contemporary standard model’s categorical system. The conceptual achievement is quite significant—in comparison to the prior “zoo of particles.” There is a central distinction made between fermions, including six species of quarks, three kinds of leptons (electron-like particles) and three varieties of neutrinos; the fermions contrast with the bosons, or force-carrying quanta: the photon, three varieties of weak-force carriers, and eight gluons to carry the strong force. The hypothesis of the graviton, the supposed boson of gravitation, has not yet been confirmed and is quite problematic. This is so, in spite of the fact that gravitational waves are confidently predicted. (Great sums have been spent on gravitational wave detectors.) One might think that this would render the graviton equally unproblematic, yet the graviton remains more problematic, in spite of the general wave-particle duality of traditional interpretations of QM (as I am informed), because gravitation is non-re-normalizable. All in all, in spite of the anomaly of the graviton, the classifications of the standard model of particle physics looks like a fairly neat categorical system –with some remaining unruly threads of considerable interest.

The standard model also predicted the Higgs boson and the Higgs field, in order to explain mass in the standard model. The Higgs boson is the quantum of the Higgs field, and early efforts with the LHC at Geneva, have been devoted to detection of the Higgs boson, taken as a sure sign of the Higgs field. It is the Higgs field which is regarded as accounting for the mass of particles. One can understand how the Higgs field gives mass to particles by analogy with other field phenomena—as I have seen Susskind explain the point. Any field will have an influence upon the movement of relevant particles through it. But such influence amounts to a change of energy, and by Einstein’s equation, energy is equivalent to mass.

The confirmation of the detection of the Higgs boson depended upon statistical evaluation of a great masses of data. The statistical test or standard of confirmation was placed pretty high, and a great mass of data had to be compared to abstract statistical standards. That scientists should still have been working on this some years after the confirmation was announced seems to me a matter of empirical and mathematical checking of prior results. There is no reason in this to think that the related physics is “far less empirical.”

The various theories of the “multiverse,” may just prove to be the Achilles heel of many developments in contemporary mathematical physics—those centered around Everett’s interpretation, and the “anthropic principle,” in particular. Alternative universes to which we have no empirical access are prima facie very problematic as physics, and the high frequency of such math-based ideas has been a cause for alarm in many quarters. Much of this seems to be based upon a simple refusal to accept the irreversibility of quantum measurement and decoherence phenomena. This is closely connected with the tendency to valorize the mathematical elegance of the uniform development of the Schrodinger wave equation. I tend to become more skeptical when I hear talk of the wave equation of the universe as a whole, and recall at this point that the mathematics of QM and QFT must function to take in the empirical phenomena and to facilitate (statistical) predictions.

At the end of this posting, I want to come back to a point of agreement with Messing, who finished up his long posting as follows: 

Thoughts and ideas will always come first. Nobody will start doing a Sudoku puzzle and just happen to solve the Riemann hypothesis (in fact, the work on it or about it has been motivated by many ideas, from the zeta function to quantum computing). However, I think that in many ways the nature of the ideas that drives a lot of physics (at least physics in fields like particle physics or those who must use QM/QT) has changed from those that drove Einstein and the last of the physicists to see Lord Kelvin's two clouds grow to become the raging storms that swept up Laplacian determinism, rent it, and scattered its pieces among the waves and sands of time.

--end quotation

The point of agreement comes with the tendency to hold that "thoughts and ideas will always come first."  We have witnessed a quite grand development of mathematical physics of in recent years, and I would not dare to say that nothing will come of this. That would be hubris, for the curious layman. But even mathematical physics, as with "thoughts and ideas" or physical insight,  must eventually come down to the matter of empirical testing of the theories developed.  

I agree, too, that the character of the ideas involved in the development of physics have changed, since Einstein's times. But that is far from saying that his emphasis on "thought and ideas" has lost its importance and relevancy.

H.G. Callaway 

A mathematical expression of a part of physics had to be generated from a physicist and has to be understood by physicists.  What is understood through reading of mathematical expression is not any more in the realm of mathematic but in a realm of idea that is not necessarily unlike the realm of ideas evokes by philosophical discussion.   Although the two outer realm of expressions are different, the inner realm of understanding might be one.   Mathematical expression of ideas are precise while narrative expression of ideas are not as precise although philosophical discussion are much more precise than causual discussion on general subject.  

The mathematical tools that were available to Einstein cames from a long history in mathematic ideas that were influenced by Kant's philosophy.  

''The need for a thoroughgoing epistemological elucidation of the foundation of geometry induced Riemann, about the middle of the century just closed, to propound the question of the nature of space; the attention of Gauss, Lobachevski, and bolyai having before been drawn to the empirically hypothetical character of certain of the fundamental assumptions of geometry.   ...  Like the fundamental assumptions of every natural science, so also , on Riemann's theory, the fundamental assumptions of geometry. to which experience has led us, are merely idealizations of experience.

In this physical conception of geometry, Riemann takes his stand on the same ground as his master Guass, who once expressed the conviction that is was impossible to establish the foundation of geometry entirely a priori.

'  Mach 1906

When Einstein or any physicist of his time learned the geometry of Riemann, they might not be consciously aware of the philosophical ideas underlying this geometry but this geometry was inspired by such ideas and these become implicitly present in the mind of the physicists that learn these mathematical tools.  Ideas propages explicitly but also in conceal ways.  Only a long historical investigation of the evolution of ideas allow to make explicit the multiple mode of diffusion of ideas.

Philadelphia, PA

Dear Quintana, 

Many thanks for your kind words. I'm much inclined to say that anything Woit writes about contemporary physics in its popular manifestations, is well worth reading and taking seriously. Here you point out his review of Tegmark:

Peter Woit (Author of “Not Even Wrong”) in his blog at: http://www.math.columbia.edu/~woit/wordpress/?p=6551

--end quote

Many thanks for sharing the link. 

H.G. Callaway

Philadelphia, PA

Dear Brassard, 

Your quotation from Mach, and brief discussion of the historical import of Riemannian geometry, are definitely of interest here. 

You wrote: 

When Einstein or any physicist of his time learned the geometry of Riemann, they might not be consciously aware of the philosophical ideas underlying this geometry but this geometry was inspired by such ideas and these become implicitly present in the mind of the physicists that learn these mathematical tools. Ideas propogate explicitly but also in conceal ways. Only a long historical investigation of the evolution of ideas allow to make explicit the multiple mode of diffusion of ideas.

---end quotation

More generally, non-Euclidean geometry underwent very significant development in the 19th century and was substantially available to physics and to relativity theory, when it was needed. It is worth noting, in this connection, that it also provided grounds to question the a priori conception of space and time--which in Kant's epistemology had been "a priori forms of intuition."  

Basically, the mathematicians had been attempting to reduce the axioms and postulates of Euclidean geometry to the minimum needed to deduce the theorems, and testing the postulate of parallels in particular, they negated it, combined it with the other axioms and postulates, and attempted to deduce a contradiction --to show that the postulate of parallels was not needed as a separate assumption in generating the Euclidean system. But no contradiction was forthcoming, and in consequence, systems of non-Euclidean geometry were developed. 

The postulate of parallels states that, given a line and a point not on that line, there is only one other line through the point which is parallel to the first line. This can be denied in two basic ways: assuming there is no line parallel to the first, or that there are many. Thus developed systems of non-Euclidean geometry. 

But to bring this back to the theme of "thought and ideas," as science developed, in relativity theory, and non-Euclidean geometry was found useful --and non-contradictory--this brought about a critique of ideas from Kantian epistemology. Physical space is not Euclidean, and the question of its exact character has an empirical element to it. So much the worse, then for Kant's "a prior forms of intuition." But if this could fall to empirical science, then it certainly puts in question the entire project of epistemology as an a priori  theory, superior to empirical science. Epistemology also has an empirical character, however general. I would think that Einstein's skepticism of a priori elements of experience has a similar basis. 

One may argue, further, that an a priori basis of epistemology and categories is misleading, even as a starting point, since it tends to an overly rigid treatment of existing systems of categories and concepts, even when the a priori is sometimes surrendered under questioning. My point is that though we are sometimes justified in being more conservative about particular concepts and laws, this is a kind of thing which requires argumentative support against opposition and cannot be taken for granted. Of course, people do get quite attached to their own particular modes of thought, but our sympathy with such attachment should not extend to the absence of critical or skeptical perspectives--as needed from case to case. 

H.G. Callaway

You said: Fundamental ideas play the most essential role in forming a physical theory.

In the case of Einstein, it could be true.  Apart from Riemann, Lorentz and other scientists, the most important influence on Einstein was Ernst Mach's critique to Newton's universe.   And Mach was, more than other, an anomalous philosopher of science, inspired by the "pure" philosopher Richard Avenarius.  In touch with this incontrovertible, deep influence, we can say that Einstein was inspired by a fundamental idea.    

Einstein's ideas or thoughts in his own words were most influenced by the experience of the compass his father handed him always pointing north independently of any kind of movement, in actuality 'pointing' incontrovertibly to an underlying global versus local phenomenon, his general relativity or any other school of thought to this day has yet adequately to describe.

Philadelphia, PA

Dear Tozzi, Nev et al, 

Thanks for the new contributions above and for your interest in this thread of discussion. 

I tend to agree that Einstein was inspired by (several) basic ideas, and we have been exploring the relationship of this to mathematical formulations in physics.  I would not likely have put this particular question up except that the quotation from Einstein and Infield struck me as important and worthy of examination and consideration. One related question has been whether something similar is true, say, of Bohr, Heisenberg and Born. I do believe that philosophers of science can be an important influence on scientific thought and development, though I expect that the physicists will have the last word in physics. It is not surprising that many in physics show little interest in philosophy of science.   

I had not heard this particular story about Einstein's father giving him a compass. However, I have heard similar stories about great thinkers carrying a compass and regarding it symbolically. The general idea seems to be that study of nature may provide a kind of orientation to one's thoughts and a kind of guidance. What seems to me beyond reasonable doubt is that it is worthwhile exploring physical theory as a paradigm of human thought, exploration and accomplishment. On the other hand, I tend to be skeptical of the notion that human beings will ever arrive at a complete "theory of everything" --though I don't regret it, since I think that would take much the zest out of the scientific life.

H.G. Callaway  

Dear H.G. Callaway,

''physicists will have the last word in physics.''   Physicists will always have the last word in physics but it is not to say that a physicist is not simultaneously a philosopher.  There are different type of physicists, a small minority of them is concern in unifying, expanding, understanding physics in a new way, to reformulate it from other basis or point of view. All of them have to know the language of physics which is mathematics, and those belonging to that minority need to be particularly good at mathematic.  Among this minority, there is a small minority of what I would call: philosopher-scientists.  There are not philosopher in the sense of being trained in philosophy (some are but most are not) but are philosopher is the sense of seeing the relevance to physics of some important philosophical ideas, the ideas that gave (and are still giving) rise to physics.

Carlo Rovelli wrote:

"It's sort of the fashion today to discard philosophy, to say now we have science, we don't need philosophy. I find this attitude very naïve for two reasons. One is historical. Just look back. Heisenberg would have never done quantum mechanics without being full of philosophy. Einstein would have never done relativity without having read all the philosophers and have a head full of philosophy. Galileo would never have done what he had done without having a head full of Plato. Newton thought of himself as a philosopher, and started by discussing this with Descartes, and had strong philosophical ideas.

But even Maxwell, Boltzmann, I mean, all the major steps of science in the past were done by people who were very aware of methodological, fundamental, even metaphysical questions being posed. When Heisenberg does quantum mechanics, he is in a completely philosophical mind. He says in classical mechanics there's something philosophically wrong, there's not enough emphasis on empiricism. It is exactly this philosophical reading of him that allows him to construct this fantastically new physical theory, scientific theory, which is quantum mechanics.

The divorce between this strict dialogue between philosophers and scientists is very recent, and somehow it's after the war, in the second half of the 20th century. It has worked because in the first half of the 20thcentury, people were so smart. Einstein and Heisenberg and Dirac and company put together relativity and quantum theory and did all the conceptual work. The physics of the second half of the century has been, in a sense, a physics of application of the great ideas of the people of the '30s, of the Einsteins and the Heisenbergs.

When you want to apply thes ideas, when you do atomic physics, you need less conceptual thinking. But now we are back to the basics, in a sense. When we do quantum gravity it's not just application. I think that the scientists who say I don't care about philosophy, it's not true they don't care about philosophy, because they have a philosophy. They are using a philosophy of science. They are applying a methodology. They have a head full of ideas about what is the philosophy they're using; just they're not aware of them, and they take them for granted, as if this was obvious and clear. When it's far from obvious and clear. They are just taking a position without knowing that there are many other possibilities around that might work much better, and might be more interesting for them.

I think there is narrow-mindedness, if I might say so, in many of my colleague scientists that don't want to learn what is being said in the philosophy of science. There is also a narrow-mindedness in a lot of probably areas of philosophy and the humanities in which they don't want to learn about science, which is even more narrow-minded. Somehow cultures reach, enlarge. I'm throwing down an open door if I say it here, but restricting our vision of reality today on just the core content of science or the core content of humanities is just being blind to the complexity of reality that we can grasp from a number of points of view, which talk to one another enormously, and which I believe can teach one another enormously. "

Philadelphia, PA

Dear Qintana & Blassard, et al.

You both raise some very interesting questions, and I will want to come back to your contributions in detail. I hope others will also respond. But for now, I thought to make use of the following posting, which I made yesterday on another thread. This concerns the relationship of philosophy and science. 

I'm not much inclined to think of science as part of philosophy, but it is not entirely separate either. Philosophy has its distinctive, traditional and more recent, sub-disciplines and themes: metaphysics, epistemology, logic, ethics, political philosophy, etc., and also includes philosophy of science. In consequence a specialized knowledge of philosophy, which usually includes its history and periods, is quite different from a specialized knowledge and competence in science or in particular sciences. As academic disciplines and categories of library books and journals, philosophy and science are surely different; and I don't think this broad strokes differentiation is unimportant. It reflects differences in the basic competencies of people in the distinct fields.

Philosophy of science belongs to philosophy, though many scientists are quite interested in it. Likewise, philosophers, when interested in the nature of knowledge or of the sciences, will naturally take up the history of science and even developments in contemporary science. As has become evident in recent years, history of science is of considerable interest both to philosophers and to scientists. Partly in consequence of that fact, philosophers and scientists often find themselves engaged with the same questions, though their approaches may be different.

When people in science are inclined to say, about some topic or discussion or claim "that's not science its philosophy," I find myself sometimes sympathetic sometimes not. I find myself most sympathetic when the scientific claims are excessively speculative and without much chance of empirical testing. Its not that I think that something distant from the prospect of empirical testing could not be science; and it is, in general terms, a difficult matter to say exactly what should count as "excessively speculative." Still the complaints are worth hearing out --though sometimes mixed with a bit of the experimentalists' schadenfreude over the disappointment of theoreticians' large-scale or over-sized ambitions. These complaints may encourage alternative theoretical developments or greater concentration on theoretical developments which facilitate empirical testing.

I am least sympathetic to the charge that something is "not science but philosophy" if this is used to suggest that philosophy is simply a repository of unanswerable questions and metaphysical puzzles. On the contrary, I think that philosophy is at its best, in relation to science, when it takes scientific results and developments seriously and does not attempt to impose some a priori or skeptical scheme upon the sciences. Philosophy may include a catalog of unanswered questions and puzzles, historical or contemporary, but it is not at its best if it simply glories in our ignorance and quandaries.

On the contrary, philosophy of science may aim to systematize what is best in the sciences and detect its basic norms and cognitive processes: "This is what science at its best is doing and how it does it." Subsequently, the sciences may introduce new variations, and it is always interesting to try to understand the significance of this. But I suppose that neither philosophy nor epistemology in particular can give any absolute laws to the sciences. It is more a matter of weighting and balance of judgment. If someone proposes a pure a prior method for the natural sciences, then I think the philosophers should be shocked. But this is basically in light of the past success of science in insisting on empirical testing.

H.G. Callaway

Philadelphia, PA

Dear Bassard, 

you wrote:

''physicists will have the last word in physics.'' Physicists will always have the last word in physics but it is not to say that a physicist is not simultaneously a philosopher.

--end quotation

Basically, I agree with you and almost all of what you say in your last posting on this thread, including the quotation from Carlo Rovelli. Moreover, in the unlikely eventuality that some recognized philosopher would have the last word on a question in physics, I think that would just mean that the fellow would thereafter count as a physicist, too. 

Still, in the present context, I am inclined to emphasize, following the opening quotation from Einstein and  Infield, the dependence of development in physics, and science generally, upon empirical testing, experiment, measurement and the prospect of such (which is sometimes sketched in mathematical physics). Certainly, I think that philosophy will provide no escape from this dependence. To say so, of course, is to take a philosophical position.  Physics may borrow, for a time, upon the wealth of past theoretical accomplishments, empirically tested and confirmed, in projecting plausibilities of new physics. But eventually, the physical theoretician's borrowed plausibilities have to face up to the physical experimentalist's demanding conscience. 

H.G. Callaway

Philadelphia, PA

Dear Quintana,

You wrote:

But notice that Einstein’s quote is not saying we must trust our intuitions: “But thought and ideas, not formulae, are the beginning of every physical theory.” In other words, mathematics alone will not allow us to make progress. It may be that scientists like Tegmark have come to trust mathematics too much and now make the astonishing claim that “all is mathematics”.

--end quotation

I quite agree here, and generally, I thought that what you had to say about "intuition" is helpful. I'd tend to agree also with the notion that something like Euclidean geometry is built-in to the mammalian visual system--good for hunting and gathering, apparently. In any case, it is very difficult for us to visualize non-Euclidean space or 4-dimensional space-time relations, etc. Einstein's "thought and ideas" is more a kin to a go at (or hypothesis of) thematic and explanatory unity in relation to a range of known and problematic phenomena. 

Reading further, I came across an on-line pdf of the famous EPR paper of 1935.

Can Quantum-Mechanical Description of Physical Reality Be considered Complete?

http://www.drchinese.com/David/EPR.pdf

The pdf is based on a photocopy of the Physical Review article. 

On the first page, the authors, including Einstein, of course, state:

The elements of the physical reality cannot be determined by a prior philosophical considerations, but must be found by appeal to results of experiments and measurements. A comprehensive definition of reality is, however, unnecessary for our purposes. 

--end quote

Surely, Einstein would say much the same about the role of a priori mathematical considerations. Mathematical physics basically details and quantifies the physicists' "thought and ideas" (or hypotheses), and may draw out implicit consequences of existing or accepted theory --and sketch the way to experimentation.  

Though most doubt of Einstein's skepticism on non-locality in this very influential paper, the EPR paper is well worth reading in the original. It is often said to be the place of origin of the idea of entanglement. 

H.G. Callaway

Dear H.G. Callaway,

Am not qualified to follow many of the complexities woven into this string. But your question twigs thoughts. 

Einstein and Infield refer to "beginning" and "later" - but your question refers to "central" & "essential", and "secondary". Einstein appears to be referring to seriality while you are invoking different criteria. You may be editing. (Or trying to interpret?)

Having said that, the major thought your question twigs with me is that I perceive a broader pattern in which Einsteinian-type "thoughts and ideas" are followed by both mathematics and experiments, producing a crop of mathematicians and experimenters who then become arbiters regarding such questions as "Does God exist?" I find this unsettling. (I think Max Tegmark is playing both sides of this fence and I admire him for his efforts - an intellectual bomb-thrower not unlike the political Newt Gingrich in the 1990's.) There is a disconnect between this process and the critical seriality Einstein describes. It's not necessarily so, but it can be so - and I believe it exists and is a problem

Einstein categorically believed in "objects", and said so. He believed in "motion", as did Newton. I do not. Regardless who is correct, we must start somewhere - these are our assumptions. But the assumptions we start with drive our mathematics and our experiments. (For example, I believe we can look for "motion sensors" in the eye and brain complex of animals forever and never find them. They won't exist if motion does not exist.)

If a datum are a system, the assumptions (i.e. thoughts and ideas) underlying the datum are what Einstein & Infield were referring to. It is difficult for me to understand how where one begins does not affect where one goes - mathematically and experimentally. Mind you, Art Winfree (and Riemann?) seems to stand as one of many mathematician counterexamples in going from mathematics to experiment to "thoughts & ideas" as a methodology.

Thanks for the question. It caused me to put my LCC (i.e. Large Concept Collider (i.e. the human brain)) in gear.

David Skidd

Philadelphia, PA

Dear Skidd, 

I think you pose a useful sort of question, and you are right that there is a bit of tension between the opening quotation from Einstein and Infield and the question I raised about it. But let me try to go more directly at your question. 

You wrote:

If a datum are a system, the assumptions (i.e. thoughts and ideas) underlying the datum are what Einstein & Infield were referring to. It is difficult for me to understand how where one begins does not affect where one goes - mathematically and experimentally. Mind you, Art Winfree (and Riemann?) seems to stand as one of many mathematician counterexamples in going from mathematics to experiment to "thoughts & ideas" as a methodology.

---end quotation

The distinction between "assumptions" and "conclusions" seems to be relevant here. Different theoretical physicists, say those concerned with the proposed "new physics" of quantum gravity, may surely start out with different "thought and ideas" (or hypotheses) regarding how the outstanding questions of this area of inquiry might best be answered. Aiming for a thorough development of their (provisional) assumptions, they might well develop quite different systems in mathematical physics. The mathematical elegance and beauty of these developments is certainly of interest, and they may add considerably to the appeal of a particular approach. Most appealing of all, would be if they can provide means of empirical testing of the developed system. So, as you say, "where one begins" will "affect where one goes - mathematically and experimentally." Yet the point may be misleading. 

I think that quantum gravity is particularly interesting in this connection because there is very little available by way of direct evidence or proposed experimental implications. Yet, if the various developments or proposals in quantum gravity never come to any experimental test, then none of them will be acceptable as physics. They would be, instead, various speculative developments of physics. In consequence, "where you start," the initial "thought and ideas" may take you nowhere as regards answer to question and problems in physics.  

But this is not always how things go. Consider instead Einstein's theory of general relativity. The basic "thought and ideas" are centered on "the equivalence principle," say, or the concept that space-time is dynamic. "Matter tells space-time how to curve and space-time tells matter how to move." Einstein saw the need to extend special relativity to gravitational phenomena. His basic ideas were then formulated as a system of equations which allows one to make calculations and predictions. Einstein's theory had to pretty much agree with the confirmed predictions of Newton's theory of gravitation, but also allowed for observational testing of distinctive predictions. (Subsequently, there have also been experimental tests as well.) 

Famously, Einstein predicted the curving of the paths of star light in the vicinity of a massive object, such as the sun. In 1919, observations were made by Arthur Eddington, among others, which were taken, world-wide, to confirm Einstein's predictions. The theory remained controversial for a long time, but from the time of its confirmation, by observing the relative positions of stars in the vicinity of the sun, during a total eclipse. it was clear that this was the first new theory of gravity since Newton. (Einstein's Nobel prize, though, was for his earlier work on the photoelectric effect, contributing to the early development of QM)

It is important to notice in this connection that defenders of Newton's theory could make the very same observations. It was Einstein's GR which suggested the importance or significance of these observations, but scientists didn't have to share Einstein's approach in order to test its implications. It is in such cases that the original "thought and ideas" are eventually accepted into the body of physical theory. Those following the contrary (Newtonian) "thought and ideas" eventually gave them up in the face of the evidence supporting GR. So, we see that "where one begins" does not always determine where on ends up.  

H.G. Callaway

Philadelphia, PA

I've come across an interest quotation from Paul Dirac, relevant to the present question; and I think it well worth taking a look. The quotation comes from the Preface of Dirac's Principles of Quantum Mechanics, in the 4th edition, Oxford University Press, 1958, and the text is available on line:

https://isush4u1.files.wordpress.com/2012/01/dirac-principles-of-quantum-mechanics.pdf

Dirac, who is more positive on the role of mathematics writes,

Mathematics is the tool specifically suited for dealing with abstract concepts of any kind and there is no limit to its power in this field. For this reason a book on the new physics, if not purely descriptive of experimental work, must be essentially mathematical. All the same the mathematics is only a tool and one should learn to hold the physical ideas in one's mind without reference to the mathematical form. In this book, I have tried to keep the physics to the forefront, ...examining the physical meaning underlying the formalism wherever possible. (p. viii)

---end quotation

It seems clear that there is some contrast with the opening quotation from Einstein and Infield here, and Dirac can also be quoted as favoring more purely mathematical approaches. Still, he also holds that (it is possible) to "learn to hold the physical ideas in one's mind without reference to the mathematical form." He thinks it important, and worthy of his own efforts to "examine" the "physical meaning underlying the formalism wherever possible."

Though the commitment here to the importance of "thought and ideas" is weaker than what we have seen in Einstein, a commitment to understanding the basic physical ideas is still quite evident, though he thinks a book in the "new physics" (that is, relativity and QM) "must be essentially mathematical. Mathematical genius that he no doubt was, it is difficult not to sympathize with Dirac's commitment to the essential role of mathematics in physics. Yet, he thinks it is not quite so "essential" as to rule out his own efforts "to hold the physical ideas in one's mind without reference to the mathematical form." 

Perhaps this argument from Dirac will help bring out the friends of greater emphasis on mathematics. But it seems to me that however great the need for mathematics within physics, (given Einstein's contrary emphasis), there is still considerable room for examining physics in terms of its basic concepts --leastwise for those of us who are not professional physicists and who do not hope to actually do physics. Surely, logic and analysis can take us quite a way. 

H.G. Callaway

Dear H.G. Callaway,

Thank you for your thoughtful response. Your comments:

"It was Einstein's GR which suggested the importance or significance of these observations, but scientists didn't have to share Einstein's approach in order to test its implications. It is in such cases that the original "thought and ideas" are eventually accepted into the body of physical theory. Those following the contrary (Newtonian) "thought and ideas" eventually gave them up in the face of the evidence supporting GR. So, we see that "where one begins" does not always determine where on ends up."

points to the potential difference between considering your original question from the perspective of an individual's thought processes (e.g. Einstein or Dirac) vs. the accumulated/compounded processes of a collection of minds (e.g. the science of physics).

I have an unsettling  sense that when humans achieve consensus explanations supported by mathematics and experiment we somehow create a de facto "kitty-bar-the-door" consideration of premises underlying the explanations. If so, perhaps that is why generational turnover may be so helpful.

As an aside, it strikes me one "actually do(es) physics" by posing questions such as yours. Aristotle might think so, as well.

David Skidd

I'we been following with much interest this thread, as it puts forth and discusses important ideas, very well moderated by prof. Callaway. I chime in as a sympathizing outsider.

It seems to me that the way concepts are associated with mathematics, in a physical theory, is rather arbitrary (and, at any rate, phenomena are different entities than their theorizations). Hilbert tried to pin it as problem #6 in his 1900 list. Yet, even accepting his stance as a definition of physical theory, which works for quantum mechanics, relativity theory and successive ones, it seems to me it would be anachronistic to apply it to prior theories.

Galileo perceived physics as written in mathematical language. Today, the relationship between mathematical invention and physical discovery can be obfuscated for the opposite reason; that is, because people deem that the language of physics is just human, and physics is what physicists do.

In “The psychology of invention in the mathematical field”, J. Hadamard took a sort of survey about that, and reports Einstein's reply in appendix II. In my opinion, there is no ultimate relationship between fundamental concepts and the language used to express them, and it's not at all futile that people with different areas of expertise review this topic once more.

Philadelphia, PA

Dear Vesely, 

Thank  you for your kind words and interest in this thread of discussion. It strikes me, from your mention of Hadamard, that you are interested in the psychology of creativity. I've always thought this a laudable and significant interest and topic, though it is not exactly the topic here.

In the present thread, Einstein's talk of "thought and ideas" has been interchanged pretty freely and generally with "concepts" or talk of a conceptional approach to physics --in contrast with mathematical formulas and formalisms. The mention of "ideas" will perhaps invite Hadamard's theme of thinking in images, while the talk of a conceptual approach might invite the contrary notion of thinking in words. While I do not myself doubt that people sometimes think in images (and thinking silently in words might even be regarded as a matter of auditory images, from a psychological perspective), I do not believe that this point or contrast is crucial for the distinction intended in Einstein, and now, as we find in Dirac. Our question has more to do with the understanding of physics which is possible without going into a great deal of mathematics.

Hilbert's problem #6 is the problem of an axiomatic system for physics --in mathematical formulas, as I understand the matter. So, that project does not seem to contrast with your point from Galileo, though one might, it seems, invoke Galileo to argue against the opening distinction in the quotation from Einstein and Infield. It does not seem to me, though, that this is an argument or approach you want to make in your remarks. 

Instead, if I've understood you correctly, you bring up two contrasting considerations about a conceptual approach to physics. The first point, I take it, is that there is some danger of obfuscation if physics is made more accessible, to the educated layman,  because "people deem that the language of physics is just human, and physics is what physicists do."  It seems that your comments on an "arbitrary" relation between physical concepts and mathematics belong here. However, I doubt that what we are doing on this thread is physics proper or that understanding physics in the fashion proposed here will qualify anyone to do physics themselves. But if there is any real danger of this sort, then I would point to half a library of contemporary books on physics, written by physicists and science writers alike. It is not generally though that these many books present such a danger. Instead they are generally taken to stimulate interest in contemporary physics and its history. This is all to the good, I take it. 

Secondly, you remark that "it's not at all futile that people with different areas of expertise review this topic once more." Here, I agree, of course. But I would emphasize that the conceptual system of physics is a highly specialized and selected one, and not something at all very easy to master. But, in spite of that, I think it a worthy topic of study and analysis. One might, of course develop some critical or skeptical perspectives, but that is quite another thing from developing the capability to productively contribute to the development of physics. It is best to leave that to the physicists themselves who have the requisite specialized education, training and experience in the field.  

Again, as I said above, I think the topic of the psychology of creativity a very interesting one, but it is not the question here. In consequence, I do not see that it is directly relevant that some people describe their own creative processes in one way or another. Some of this may be quite plausible --e.g., Hawking's talent for geometrical images, perhaps. Some other examples might be less plausible, as an account of creative processes and of the psychology of creativity. (What creative people actually do need not agree with their own descriptions of it.) Once we set aside the generalized question of the psychology of creativity, for present purposes, then it seems to me that what the physicists say about the relationship between physical concepts and mathematics will remain the center of interest. 

H.G. Callaway

We cannot separate reasoning using ideas expressed in natural language and reasoning using mathematical formulae. In thought experiments that are usually expressed in natural languages are also used for doing mathematical reasoning or used to prepare an actual experiment.

Quotes from http://en.wikipedia.org/wiki/Thought_experiment:

"A thought experiment is a device with which one performs an intentional, structured process of intellectual deliberation in order to speculate, within a specifiable problem domain, about potential consequents (or antecedents) for a designated antecedent (or consequent)" (Yeates, 2004, p. 150).

In philosophy, thought experiment have been used at least since classical antiquity, some pre-dating Socrates. In law, they were well-known to Roman lawyers quoted in the Digest.[6] In physics and other sciences, notable thought experiments date from the 19th and especially the 20th century

The ancient Greek δείκνυμι (transl.: deiknymi), or thought experiment, "was the most ancient pattern of mathematical proof", and existed before Euclidean mathematics,[2] where the emphasis was on the conceptual, rather than on the experimental part of a thought-experiment.

Perhaps the key experiment in the history of modern science is Galileo's demonstration that falling objects must fall at the same rate regardless of their masses. This is widely thought[3] to have been a straightforward physical demonstration, involving climbing up the Leaning Tower of Pisa and dropping two heavy weights off it, whereas in fact, it was a logical demonstration, using the 'thought experiment' technique.

The 'experiment' is described by Galileo in Discorsi e dimostrazioni matematiche (1638) (literally, 'Discourses and Mathematical Demonstrations') thus:

Salviati. If then we take two bodies whose natural speeds are different, it is clear that on uniting the two, the more rapid one will be partly retarded by the slower, and the slower will be somewhat hastened by the swifter. Do you not agree with me in this opinion?

Simplicio. You are unquestionably right.

Salviati. But if this is true, and if a large stone moves with a speed of, say, eight while a smaller moves with a speed of four, then when they are united, the system will move with a speed less than eight; but the two stones when tied together make a stone larger than that which before moved with a speed of eight. Hence the heavier body moves with less speed than the lighter; an effect which is contrary to your supposition. Thus you see how, from your assumption that the heavier body moves more rapidly than ' the lighter one, I infer that the heavier body moves more slowly.[4]

Ernst Mach always argued that these gedankenexperiments were "a necessary precondition for physical experiment"). In these cases, the result of the "proxy" experiment will often be so clear that there will be no need to conduct a physical experiment at all.

Philadelphia, PA

Dear Brassard,

Try this slight paraphrase:

Salviati. If then we take two bodies whose natural speeds are different, it is clear that on uniting the two, the more rapid one will be partly retarded by the slower, and the slower will be somewhat hastened by the swifter. Do you not agree with me in this opinion?

Simplicio. You are unquestionably right.

Salviati. But if this is true, and if a large stone moves with a higher speed  while a smaller moves with a slower speed, then when they are united, the system will move with a speed less than the higher speed; but the two stones when tied together make a stone larger than that which before moved with a high speed. Hence the heavier body moves with less speed than the lighter; an effect which is contrary to your supposition. Thus you see how, from your assumption that the heavier body moves more rapidly than ' the lighter one, I infer that the heavier body moves more slowly.

--end of paraphrase

The initial argument proceeds on the supposition that stones of different weights fall at different speeds. If so, then, combining a heavier and a lighter stone, we ought to get a medium speed. That argument is accepted. But, then, on the contrary, it is pointed out that by combining the two stones we get one heavier than either. So, on the original supposition it ought to fall faster than either stone alone. Thus the combination of the two stones must both fall faster than the heavier component and also more slowly: a contradiction. 

The argument is a reductio ad absurdum of the original supposition. 

My point is that there is a style of (vague or comparative) "quantification" short of mathematics which belongs to logic and analysis. There is nothing in that reflection against the practice of thought experiments, though. 

H.G. Callaway

Wise reply.

Dear H.G. Callaway,

first let me thank you for your explanations. I just wish to try and clarify my point. Hadamard's interest for psychology is not pertinent here. It reflects lively and far reaching discussions about the relation between mathematics and sciences, which were trendy before the two world wars. What could be of interest here is the high level of those discussions. E. Mach, F. Enriques and others took part in similar meta-physical discussions, too. As such questions had no clear-cut answer, sometimes people, educated laymen inclusive, look at them just as a loss of time. My opinion is that such discussions – also the present one – promote further advancement in science, and are defensible as such.

Coming back to Hadamard's essay, I was a little bit surprised to read the following sentences at the beginning of the introduction. He says: "We speak of invention: it would be more correct to speak of discovery. The distinction between these two words is well known: discovery concerns a phenomenon, a law, a being which already existed, but had not been perceived." He gives an example of each, invention and discovery, then goes on: "Such a distinction has proved less evident than appears at first glance. [...]." I think it is worth paying attention to that sentence.

I agree with you that there are highly specialized and selected conceptual systems of physics. And Einstein's physical concepts are certainly of a different character than skillfully written exercise books in mathematics. But, if the difference between discovery and invention is relinquished, what becomes of the relationship between physical concepts and mathematics?

Philadelphia, PA

Dear Vesely, 

Before the two world wars, and in the late nineteenth century, much in philosophy, at least, was somewhat psychologized. I think of that, in part, as a matter of people seeking common ground as the great empires divided the Western world. In any case, it appears that Hadamard's work was taken up in that tendency. The question about "discovery" and "invention" is an interesting one, as is the question about the psychology of creativity. But it strikes me that the question about discovery and invention concerns both conceptualization and mathematics.

I wonder, however, why we should suppose that the "difference between discovery and invention is [should be?] relinquished" or given up. That would seem to require an argument or some explanation, since you have introduced the claim. 

I take it that, say, Maxwell invented, or in any case, first formulated, "Maxwell's equations," and in doing so, one normally supposes, that he discovered certain laws of electromagnetic phenomena. Or we might look instead to Faraday, whose work was a precursor to Maxwell's. Between them, in any case, it seems that they invented the physical concept of the "field," --in contrast to the long tradition of thinking that interaction was all a matter of contact of particles (if not Newtonian "action at a distance"). This eventually inspired Einstein's work, in which gravitational phenomena are looked at in terms of the Einstein "field equations"; and in his later work, Einstein sought a "unified field theory" to unite the theories of gravitation and electromagnetism.  However, though he invented several systems of equations, he failed to discover any adequate unified field theory.

It may be that there are particular contexts in which the distinction between discovery and invention becomes problematic. The matter is something worth thinking about in its own terms. But we normally think of "invention" in relation to technical artifacts, say the light bulb or television, and there are debates, say about whether Marconi invented radio. I certainly tend to think of conceptual systems as invented--and branches of mathematics, too (given Godel's incompleteness proof). Didn't Euclid invent or first formulate Euclidean geometry? I suppose that first formulation of a system of mathematical symbols and rules is close enough to "invention" that the difference needn't worry us here. As a question, "Who invented non-Euclidean geometry?" makes sense. Once such a system is invented, it will then make sense to speak of further discoveries regarding its theoretic consequences,

Yet what is invented may be useful and illuminating, and capable of fruitful, systematic development --in relation to potentialities of the natural world --or less useful, sometimes dead-ends perhaps. (Consider "phlogiston theory," e.g., in comparison to the theory of combustion based on the concept of oxygen or oxygenation. There is better and worse among inventions, and regarding conceptual systems, we certainly do evaluate them in relation to their use in prediction and experimentation. But the laws of nature, I take it, are not technical artifacts. Regarding them, we naturally speak of discovery. 

To sum up, I think you may owe us an explanation of why you say or suppose that "the difference between discovery and invention" is to be "relinquished," and how you think this is connected with the guiding question here, drawn from Einstein and Infield, regarding the relationship between "thought and ideas" on  the one hand and mathematical formulations of physics on the other.  Perhaps, instead of contributing to the present thread, you would do better to ask your own question?

H.G. Callaway

Dear H.G. Callaway,

My point in bringing the cases of thought experiments is to emphasize that the realm of ideas and the realm of mathematics are not clearly separated.  The topic of the thread assumes  such separation.  Most people not trained in mathematic nor logic will be able to understand Galileo's thought experiement making use of some natural common sense mathematical intuition we all have without being trained.  

Philadelphia, PA

Dear Brassard, 

I thought your example of the argument from Galileo was a quite useful one. The topic of the thread assumes a difference between "fundamental ideas" or "thought and ideas" which "play the most essential role in forming a physical theory," and  books full of "complicated mathematical formulae," in the words of Einstein and Infield. 

This seems to me different from saying that the "realm of ideas and the realm of mathematics are ... clearly separated." What little mathematics was contained in the argument from Galileo was easily removed without disrupting the argument.

The point is that it is possible to go a good way in understanding physics without recourse to a great deal of mathematics. In spite of that fact, philosophical discussions of physics often contain a great deal of mathematics.

Stipulating, as you do, that "most people" are able to understand Galileo's thought experiment on a "common-sense" basis. I think you support the point from Einstein and Infield.  Still, people do tend to differ on related issues regarding the technical distinction between logic and mathematics. I simply shifted the argument from Galileo to show its logic sufficient to the purpose.  

H.G. Callaway

Philadelphia, PA

Dear all, 

I've come across an additional quotation, from physicist Richard Feynman, on the relationship between basic physical concepts and mathematics, which fits in pretty well with the quotations above from Einstein and Dirac. This comes from The Feynman Lectures on Physics, Desktop Edition, Vol. II, p. 2-1 (California Institute of Technology, 1964). 

Feynman writes: 

"Mathematicians, or people who have very mathematical minds, are often led astray when 'studying' physics, because they lose sight of the physics. They say: 

'Look, these differential equations --the Maxwell equations--are all there is to electrodynamics; it is admitted that there is nothing which is not contained in the equations. The equations are complicated, but after all they are only mathematical equations and if I understand them mathematically inside out, I will understand physics inside out.' Only it does not work that way. 

Feynman continues: "A physical understanding is a completely un-mathematical imprecise and inexact thing, but absolutely necessary for a physicist." 

--end quotation

I've separated things out a bit here to clearly distinguish what Feynman is himself claiming from what he is attributing to "people who have very mathematical minds." I think he makes his point very clearly, and it is pretty much in accord with what we have found from Einstein and Infield at the start of this thread. 

Now, I expect we might well find some contrary quotations, too. Still, who is willing to take on three winners of the Nobel prize in physics on this question? Is there a contrary argument which is yet to be considered in this thread? 

H.G. Callaway

Albert Einstein Creative Thinking

"All great achievements of science must start from intuitive knowledge. I believe in intuition and inspiration.... At times I feel certain I am right while not knowing the reason." Thus, his famous statement that, for creative work in science, "Imagination is more important than knowledge" (Calaprice, 2000, 22, 287, 10). (see link)

https://www.psychologytoday.com/blog/imagine/201003/einstein-creative-thinking-music-and-the-intuitive-art-scientific-imagination

Philadelphia, PA

Dear Abbas, 

Many thanks for your thoughts and for Einstein's. In many ways, imagination is more important than knowledge, though I find it difficult to quite picture imagination not arising from pre-existing knowledge and justified belief. Imagination is involved in the growth of knowledge. That is precisely the difference between imagination and fantasy. Inspiration and intuition we tend to judge of ex post facto --as with the empirical success of a hypothesis. The imagination proposes, and empirical prediction has the last word. 

Ultimately, in science, the products of imagination must be brought to empirical tests. But that is not to say that we cannot judge of better and worse regarding yet untested scientific hypotheses. When social enthusiasm and the great, tragic political bandwagons and fanaticisms of history are held up to this standard, such as Einstein could see the difference. Social-political contrivance parading as imagination, in contrast, is mere travesty. 

H.G. Callaway 

Dear Abbas,

You can understand the geometry of Euclid both visually and through logic.  The same with most mathematics.  I studied Hilbert space,  and and most of my understanding was visual but at the same time totally algebriaic and axiomatic.  But althought the later is used to give expression to the mathematic , it is the visual part that I  used  in my imagination.  Although Hibert space are used to represent functions as infinite dimensional vectors, all the reasoning can be done imagining surface and line and point in 3D in our head.  

Knowledge is expanded through imagination.  Imagination is not simply our capacity imaging capacity that is a kind of controled fainted waking dream but is mostly our creative part and this one is mostly unconscious although everyone has his/her own routines and practice forstering it.   Formal representation of physical knowledge through mathematical expression is a pre-requise for a physicists who will have to first to acquire physical knowledge and if he/she manage to come with new results through its imagination will have to communicate it in that form.  It is like an english fiction writer.  Knowledge of english , litterature and creativity are all necessary for creating good english fictions.  

Music is intimatly related with body movement as dance is the most evident testimony.  Since the most ancient time it has been recognized that music is filled with mathematical orders and so it is natural to think that musical composers , musical imagination has to be close to the mathematical imagination and to the generation of body movement by our nervous system.  A lot of empirical studies in different sciences have confirmed this.  It is certainly not a superficial and peripheral connections but a fundamental one.  Many such as MIthens have proposed that a dancing musical language first evolved in our primate ancestors prior to sign and oral language.   All the forms in space, all movements, all complex mathematical structure can be seen as a form transformation. If animal nervous system are essentially form transformation instrument then a musician is simply a extension of our nervous system that incorporate an exterior instrument to its body.  Becoming a musician is like extending our body.  Human being are animals with a shape shifter body nervous system.  Creativity/learning/growth is the putting in place of body extension interaction mechanisms. The whole body is layer upon layer of such extension put in place from conception.

Dear Ramon,

Although the mathematical language is totally formal by opposition to our other languages, it is nonetheless a language humans created since they began to concentrate in cities.   It was created by us but we did not created and are still not creating it arbitrarily.   It originally was build by formalizing accounting practices, engineering measuring and building practices of a many kinds, of borrowing logic from sophisticated language practices etc.  Although we disguised these borrowing by formal axiom definitions and avoid any undefined element to enter the language, the language rules and the defined structures and everything in it came from operations in the real  human societal world of interaction with nature.   So the basis of mathematics are a kind of physics of the world as experience by human in their multiple practices.  The fact that the mathematical world has isolated itself by making no explicit reference to the outside world like all our other langguages do does not remove the fact that almost everything into it came from our practices in the world.   It is not an a priori science of the world but an a postiory science of the world.  Von Newman predicted that mathematic creation would decline if from time to time new empirical import are not introduced into the mathematical world. 

From my humble point of view, it is not possible to clearly separate mathematics of ideas or thoughts on a given issue. Mathematics are not only a language with its inner rules, but it is also a set of idealized translations of the reality. For instance, a circumference or one sphere are objects translated easily in formula which keeps the distance constant for a set of points. The first way of seeing the circumference as an sphere in two dimensions or a sphere as a circumference in thee dimension might bring us to the mind the idea of trying to see what is a sphere in four or in five dimensions and try to imagine them. The answer is that it turns quite difficult imagine an sphere in dimensions higher than three, while their formulae is straightforward: just put the distance in such dimensions equal to a constant.

Thus when we employ formulae in fact we are making economy of thought or ideas trying to determine the properties that we want to determine or know. The principle of equivalence was one of the most interesting ideas of Einstein ( as he said) and it is quite intuitive if we think in the lift, but what it result useful is when you realize that it allows to transform a Riemann geometry in a local Minkowskian one. But in any case we are employing physical ideas or thoughts that allow us to calculate properties or results as the perihelium of Mercury.

Philadelphia, PA

Dear Baldomir (& contributors), 

Many thanks for your recent contribution on this thread, which seems to sum up a recent line of thought and postings.  However, you also keep more strictly to the question, as I see it. The comments on creativity and the sources of ideas and hypotheses, though interesting as a topic, seem to me to be off the path a bit. 

But, to clarify, I am wondering if you intend your posting to explicitly disagree with what we have seen from Einstein and Infield at the start, viz.,

Fundamental ideas play the most essential role in forming a physical theory. Books on physics are full of complicated mathematical formulae. But thought and ideas, not formulae, are the beginning of every physical theory. The ideas must later take the mathematical form of quantitative theory, to make possible the comparison with experiment. --Einstein and Infield, 1938, The Evolution of Physics, p. 277. 

---end quotation.

It is not that Einstein --or Dirac or Feynman-- could possibly be accused of thinking that mathematics is not important to physics. Again, it seems clear to me that people engaged in mathematical physics may benefit greatly from purely mathematical developments in their work --developments arising directly from the related mathematical formulations of interest. Yet, mathematical physics is one thing and physics something else again. So, regarding any more mathematical development, it seems, it should always make sense to ask "Well, yes, I see your point. But is this physics?" 

Again, I do not imagine that anyone thinks they can do physics without a great deal of mathematics. But the interest of the question posed by the Einstein quotation, and the others, concerns the room for purely conceptual explorations and analysis of what the physicists may be up to at any given time. If the "thought and ideas" really have the role which Einstein took them to have, then it seems the determined layman and the philosophers ought to be in a fairly good position to follow along. 

You remark:

The principle of equivalence was one of the most interesting ideas of Einstein ( as he said) and it is quite intuitive if we think in the lift, but what it result useful is when you realize that it allows to transform a Riemann geometry in a local Minkowskian one.

---end quotation

You emphasize here, one of Einstein's crucial ideas, the principle of equivalence, central in GR. You also point to the mathematics arising from it. But, Einstein himself, wrote, in the quotation above, that "The ideas must later take the mathematical form of quantitative theory, to make possible the comparison with experiment." So, do you really disagree with him --or with the quotations in this thread from Dirac and Feynman? 

H.G. Callaway

Dear Callaway,

Your question is very interesting and you are right to base it in the paragraph of the book of Einstein-Infeld (by the way a book that Einstein collaborate for getting financial support for Infeld and which is not to be proud). In 1905 Einstein published four fantastic papers and one of them was special relativity, which a minimum of mathematical level. Later on his old teacher Minkowski was who formalized it in a proper form into a pseudo-Euclidean metric. And the same happened with general relativity published in 1916, that a mathematician as Hilbert formalized it also. Therefore it seems that in the case of Einstein he can separate formal mathematics from physical ideas or concepts. In the case of Feynman (creator of a mathematical technique as the path integrals) or Dirac ( applying topology to physics for first time, Clifford algebras for spinors or introducing the distribution functions) I think that it is not possible to say the same.

But speaking in general on this interesting question I think that Einstein was too influenced of his problems with Hilbert for distinguishing on these concepts: mathematics and concepts in physics. From my humble point of view, both things are not different if you employ them within Physics. In fact a physicist employs mathematics as a child when is counting the money for paying the bus, i.e. in a natural form. These kind of mathematics are necessary for having a physical idea or thought.

Another different kind of mathematics is the formal one. Lagrange formalized the Newtonian mechanics in a very elegant way without changing the results in any form, although this form of understanding mechanics is the logic one without any kind of doubt. Minkowski wrote special relativity in four dimensions in a beautiful form and much more deeper than Einstein's one, but without adding new physics at all. And in quantum mechanics there are several methods to explain it which are all of them equivalent from the physical point of view, i.e, no new physics is in any method but formal mathematics which makes it to appear only different at a first look.

Summarizing, any physical thought needs mathematics because it is the only form to apply it that I know, while formal mathematics are only necessary to understand properly the relationship of different thoughts or ideas of physics in a simple or with a minimum of principles assumed. If you think in a theory as a whole Einstein was wrong but if you think in certain physical phenomenon he was right. In fact when he wrote his article "On the Electrodynamics of the moving bodies" he was far of what we understand for the special relativity theory although the main physics was there.

Hello H.G. Callaway,

In "What is Life?" Erwin Schrodinger (1944) writes:

"This little book arose from a course of public lectures, delivered by a theoretical physicist to an audience of about four hundred which did not substantially dwindle, though warned at the outset that the subject-matter was a difficult one and that the lectures could not be termed popular, even though the physicist’s most dreaded

weapon, mathematical deduction, would hardly be utilized. The reason for this was not that the subject was simple enough to be explained without mathematics, but rather that it was much too involved to be fully accessible to mathematics." (emphasis added)

I believe this reinforces Einstein & Infield, but I wonder if you believe it addresses your question? For those thinking the subject matter at issue is not physics, Schrodinger observes:

"How can the events in space and time which take place within the spatial boundary of a living organism be accounted for by physics and chemistry? The preliminary answer which this little book will endeavor to expound and establish can be summarized as follows: The obvious inability of present-day physics and chemistry to account for such events is no reason at all for doubting that they can be accounted for by those sciences."

Regards,

David Skidd

    Dear Callaway,

    One of the dangerous things, associated to try to separate ideas or thoughts from mathematics within Physics, is that it seems possible to go to a room for thinking deeply and then to have an idea ready to just apply pure mathematics. This process seems to me quite lazy and wrong way to make research. Usually the physicist must have a mathematical-physical problem more than an idea, and after that just helps (in both fields of knowledge) to joint the hidden links for its solution.

    Nowadays there are several ideas or concepts that most of the physicist know but nobody can find, for the moment at least,  the mathematical formalism to solve them. One is quantum gravitation, high critical temperature superconductivity, etc. The problem is to find the mathematical-physical solution and this is only using mathematics properly, besides perhaps other ingredients more as to imagine the consequences of the solution before solved it with a certain method.

   Clearly Einstein knew that the aether and electromagnetism were not compatible as Lorentz and Poincare said previously defining the problem. What was revolutionary is to accept that time could depend of the state of motion or space, as the mathematics prove. Therefore, from this point of view, he was the highest believer in the mathematical results as the mean to discover reality( in opposition to what would appear from the words in the book with Infeld). He didn't feared to change his concepts or ideas for accepting the results of his calculations. But previously he needed to solve the problem with clear mathematics as can be seen in his papers.

I am adding below a quotation from an article by Miles Mathis [More on the Golden Ratio and the Fibonacci series". It is worth a read and quite up your street I think ;)]

".....As usual, it is because they have too much math and too little mechanics. Instead of trying to visualize this as field mechanics, most people have been analyzing it as pure math. Most of the current and historical math not only doesn't help us see the field mechanics, it blocks it. And this example stands as a near-perfect indictment of modern physics, which has been hampered by a lack of visualization and physicality for at least two centuries. Since the Copenhagen Interpretation in the 1920's, it has been even worse, since visualization was no longer simply a rarity (due to the normal or average abilities of physicists); beginning with Bohr and Heisenberg, it was outlawed. Banned, verboten, förbjudet."

Ideas need imagination, and may be mathematics is needed to bring them to reality. You cant say which is more important, especially mechanical devices. I think excess of one precludes the other in anyone.

Narayanan

Mainz, Germany

Dear Bhattathiri, 

Thanks for your comment and for pointing in the direction of the scientific imagination. 

I'm generally inclined to think that there is a good deal of continuity between imagination and conceptualization. Consider that some of the first written languages came in the form of hieroglyphics. This is suggestive of my point. Again, I'm inclined to think that when we attempt to draw an image of a 4-D object, say, then the result turns out so highly ambiguous that the results almost acts as a word instead of an image. Or, looking in the opposite direction, the written mathematical formulas used to represent a QM superposition strike me as almost graphic in style. Conceptualization seems to involve a regimented or strictly purposeful kind of imagination--which is weeded or selected by reference to the purposes of theory and its empirical tests. 

So, I´m not inclined to think that conceptualization or even abstract mathematics eliminates imagination. Written or spoken language, or stories told, can be literal and descriptive or alternatively they can be very imaginative in character. Mathematicians are also quite imaginative within their own domain. 

But in sympathy with some of your comment, and the quotation from Einstein at the start of this thread, none of this is to say that there cannot be too much mathematics and too little physics in our physics. 

H.G. Callaway

The trouble with this statement is that we can know things we cannot understand. I have lately become interested in the field of cognitive load, which attempts to situate the construction of knowledge in an evolutionary context. One key feature of evolution is that the 'organism' must survive evolutionary changes, which limits the number of mutations that are feasible at only one change. 

It seems to me that they are right in applying this principle to the ability of our knowledge to accommodate (in the Piagetian sense) to new ideas. Anyone who has worked in a very different culture will be aware of the impossibility of explaining some concepts because of the sheer number of points of misfit between the interlocutors' world view.

Physics has indeed brought us to the point where we can know things that we cannot understand, because any understanding would entail abandoning concepts that are central to our functioning in everyday life. But we have also the example of 'big data' analytics, where we can make decisions based on the combination of so many parameters that no human can 'understand' their interactions. 

A curious age to live in…

Right, Ronan. May be we have to abandon supposedly central concepts if we have to make any progress other than technological. And I don't mean religion.

Narayanan

Dear H. G. Callaway:

I barely recall this dialogue but apparently (it seems) I didn't offer the response I should have.

"Einstein's point, I take it, is not to discourage more precise and quantitative formulations of physical insight, but instead to focus attention upon the importance of physical insight or concepts and "fundamental ideas."

Unfortunately, too often great minds are quote-mined out of context and their quotations become divorced either from their intention or form their general research. Einstein absolutely discouraged fundamental physical insight and concepts that he though did not constitute what he regarded to belong to these. He devoted the majority of his life to critiquing in every way possible the foundations of modern physics (the he himself was instrumental in developing). We have, e.g., Pauli's letter's bemoaning Einstein's most quoted and discussed co-authored paper, EPR, and the only reason it is so famous and so important is because it turns out that everything Einstein intended to show by it was absolutely wrong. The thousands of citations of his paper were not his intent, but would have been his nightmare. He ABSOULTELY discouraged such thoughts and devoted the majority of his career to doing so, such that he became to his colleagues something of an relic.

Mainz, Germany

Dear Messing,

It is not my impression that you had earlier left out some needed comment.

You wrote:

Einstein absolutely discouraged fundamental physical insight and concepts that he though did not constitute what he regarded to belong to these. He devoted the majority of his life to critiquing in every way possible the foundations of modern physics (that he himself was instrumental in developing).

---end quotation

However, I think you help make the point of my insisting on the quotations from Einstein, above. It was not my point, of course, that he was always right in the basic concepts which he emphasized, and as you suggest, our contemporaries often believe that he was wrong about QM and in his resistance to it, on conceptual grounds. But the question here is not about whether Einstein or others, such as Dirac, were right or wrong regarding the particular conceptual approaches they adopted. The question concerns "the relationship between basic concepts and mathematics in physical theory."

Many thanks for your contributions, but it seems you have neglected needed attention to the question --and subsequent postings?--in returning to this thread after some lapse?

H.G. Callaway

You are probably correct. I had forgotten ever posting here, and tried to catch up so as to respond with something worthwhile, but appear to have failed to do so. My apologies.

Here is a case of scientific ideas of a fiction writer that surely did not come to them from mathematical considerations.

Eureka (1848) is a non-fiction work by Edgar Allan Poe (1809–1849) which he subtitled "A Prose Poem", though it has also been subtitled as "An Essay on the Material and Spiritual Universe". Eureka describes Poe's intuitive conception of the nature of the universe with no antecedent scientific work done to reach his conclusions.

 The work has its origins in a lecture Poe presented on February 3, 1848, titled "On The Cosmography of the Universe" at the Society Library in New York.[6][7] He had expected an audience of hundreds; only 60 attended and were confused by the topic.[8] Poe had hoped the profits from the lecture would cover expenses for the production of his new journal The Stylus.

Eureka is Poe's attempt at explaining the universe, using his general proposition "Because Nothing was, therefore All Things are"

 "I design to speak of the Physical, Metaphysical and Mathematical – of the Material and Spiritual Universe: of its Essence, its Origin, its Creation, its Present Condition and its Destiny".

 "that space and duration are one"[11] and that matter and spirit are made of the same essence.[12] Poe suggests that people have a natural tendency to believe in themselves as infinite with nothing greater than their soul—such thoughts stem from man's residual feelings from when each shared an original identity with God.[

He postulated that the universe began from a single originating particle or singularity, willed by a "Divine Volition".[1] This "primordial particle", initiated by God,[34] divides into all the particles of the universe. These particles seek one another because of their originating unity (gravity) resulting in the end of the universe as a single particle. Poe also expresses a cosmological theory that anticipated black holes and the Big Crunch theory[3] as well as the first plausible solution to Olbers' paradox (the night sky is dark despite the vast number of stars in the universe)

Ultimately individual consciousnesses will collapse back into a similar single mass, a "final ingathering" where the "myriads of individual Intelligences become blended".[14] Likewise, Poe saw the universe itself as infinitely expanding and collapsing[15] like a divine heartbeat which constantly rejuvenates itself, also implying a sort of deathlessness.[1] In fact, because the soul is a part of this constant throbbing, after dying, all people, in essence, become God.

 Lacking scientific proof, Poe said it was not his goal to prove what he says to be true, but to convince through suggestion.

 Albert Einstein in a letter written in 1934, noted that Eureka was eine schöne Leistung eines ungewöhnlich selbständigen Geistes (a very beautiful achievement of an unusually independent mind).

Source: http://en.wikipedia.org/wiki/Eureka:_A_Prose_Poem

Dear Louis

Thanks for bringing to my attention Poe's book. I just downloded it, but is yet to read it.

I would agree to his concept of material and spiritual universe; but should try to work out something without bringing in God (Who is this fellow to criticize Poe!!?).

Actually that is what I meant in my earlier comment, and the needlessness to bring in religion.

Narayanan

Mainz, Germany

Dear Brassard,

It strikes me that what is curious about Poe and "Eureka," in the present context, is that we are offered no account of the basis on which Poe formed his judgements. That Einstein admired the piece, ex post facto, seems somewhat beside the point of the value of his judgements when he made them. Again, I don't see that Einstein, or other reputed physicists, have been recommending the study of Poe. On the other hand, if we knew that Poe had undertaken detailed study, reading hundreds of books and articles, publishing on the topic in journal and books, and held extensive discussions with well informed colleagues in the relevant field, then the situation would look very different.

That many now reject Einstein's conceptual criticism of QM is important, though few doubt the scientific standing of his judgements at the time he made them, or even their contemporary interest. Yet, recall, too, that when Einstein published his innovative work of 1905, he was virtually unknown and unrecognized. This helps make the point that successful scientific practice may leave behind monumental changes and deep influences on professional acceptance and standards. Accomplishment, and scientific standing, then, are not be be equated with prior recognition. Yet this kind of point hardly suggest that just any lucky guessing should be equated with science. 

H.G. Callaway

Dear Callaway,

Examples of proposition of fundamental scientific ideas that have apparently absolutely no connection with mathematic should tend to reinforce the case for the independence of the process of discovery from mathematic considerations.  Eureka is not a scientific book but a book filled with speculations about everything the author consider important in the cosmos.  No attempt is made by the author to justify the ideas.  In term of acceptable method Poe is everything except a scientist or a philosopher.  In spite of that he proposed some ideas that the best scientific mind discovered much later. Einstein who had posited the basic cosmological equations and had two solutions in front of his eyes: one expanding universe and one static universe and he wrongly choose the later.  Poe had no equations and choose the good answer, not only expansion but cycles of expansion and contraction.  In mathematic, some theorems are enunciated without any proof and it takes hundred of years before a proof.  During that period, other mathematicians cannot use the theorem.  None of the ideas of Poe were usable in science for the same reasons.  Sometime human imagination comes to concepts which that reasons takes century to find the path for connecting it to other concepts.  Einstein was riding a light beam at fifteen noticed that something was totally wrong with the equations of Marxwell.  He took some years for the poor mathematician that he was to fing the reason for his original insight.  Maybe discovery always preceded knowledge.  When the distance from what is discovered from existing knowledge is small, both the discovery and its path seem to come together but bigger is the advance, farther aways the imagination jump and the longer is the period for finding the path and unfortunatly many time the path is never found in those cases. Imagination mostly work in the deep subconscious and is only partially under conscious control.  Poincarre has described how he consciously set up the problem and how ready solutions appeared to him out of nowhere, sometime after a nap, or sometime after month of not even thinking about the problem.  Then reason is necessary to  clean-up the solution and write it down.

http://www.brainpickings.org/2013/08/15/henri-poincare-on-how-creativity-works/

http://www.theguardian.com/science/blog/2012/jul/17/henri-poincare-einstein-picasso

Unconscious imagination operates in the parallel background of our brain while the proof of mathematic and of science where the path to discovery are work out have a narrative linear structure.  The path to discovery is totally different from the path of connecting the discovery to existing concepts. Not only all the person living inculturation experience lived there but all the biological learning of life on earth live there.

Mainz, Germany

Dear Brassard,

Many thanks for your clarification of your reflections on Poe and "Eureka."

You wrote:

Examples of proposition of fundamental scientific ideas that have apparently absolutely no connection with mathematics should tend to reinforce the case for the independence of the process of discovery from mathematical considerations.

---end quotation

Although I think there is reason to see conceptual matters as more basic in scientific discovery, I don't see that the Poe example really helps the point along. I would think to mention here, regarding this quotation, that there are processes of discovery involved in mathematics, too; but we would not expect non- mathematicians to come up with much, in any sophisticated mathematical context. By parity of reasoning, we expected those deeply familiar with the history and development of a particular field to be better positioned to innovate in that field. Many don't, of course. 

I think the Poe story interesting but unconvincing. It strikes me as a backward-looking bias selection. Einstein could admire the ideas retrospectively. But we have to ask how many other similar, but utterly false speculations we are here failing to notice. How many forgotten nineteenth-century theologians, say, set out their elaborate speculations about the physical world?

What falls under the rubric of "imagination" is, no doubt, quite varied. I don't believe that imagination follows quite the same paths in all people and at all times. (Its a bit like dreaming, perhaps. I reflect that some people are able to "program" their dreams on occasion.) But, I think the vagaries of imagination cut little ice as concerns the scientific imagination or regarding normative evaluation of the theoretical products of imaginative solutions to outstanding scientific problems.

Processes in which people suddenly come to an answer or proposed solution, waking up in the middle of the night, or on the next morning, are familiar and convincing--things go on in productive imagination of which we are unaware. But even in these cases, I think to add, "no input, no output." The specific preparation and study of a problem, involving broad and relevant background knowledge of a particular field is a crucial ingredient of the process. So, we don't reasonably expect viable scientific cosmology from a poet, insofar as the person is a poet only.  Some poets have, however, also been cosmologists. Similarly, we do not expect advances in physics coming from philosophers, though some physicists have also been philosophers. 

H.G. Callaway

Mathematical concepts and physical knowledge come from very different places, and it little short of a miracle that they come together in this universe as well as they do. Many scientists have commented on his, starting with Eugene Wigner, who wrote of the "unreasonable effectiveness of mathematics." There is a great deal of mathematics that has no "applications" whatsoever, and one finds quite a number of mathematicians who are inordinately proud of the fact that their subject is "useless". This is a species of vanity with which I have little patience. However, it is fact that, for example, non-Euclidean geometry was developed in the 19th century with no thought of any physical applications, and Relativity was developed in the 20th. Then, incredibly, they needed each other! No-one knows how this process works, and probably never will.

Roy,

In the eithteen centuries natural philosophers were sometime called geometers.  People like Gauss, Rieman who invented non-euclidean geometries knew the revolutionary implications that these inventions would be be physics.  It is only since 100 years that the major mathematicians are not the major physicists. There is absolutly no coincidence if mathematics has applications.  All the roots of mathematic are in engineering, measurement techniques, building techniques, language logical structure, accounting, etc, etc.  The fact that most pure mathematicians are not scientists nor engineers today do not change this fact.  

Roy, Louis

Mathematics and imagination come from different places (?Lt and Rt brains). It is said that mathematics is purely a science of the human mind. But, if mathematics is linked to knowledge of engineering, animals such as bears, birds and ants which build dams and buildings also should be taken as having good knowledge of mathematics. May be all animals are mathematicians. But I think the essential difference is the human beings knowledge of Language or rather his having the 'Faculty of Language' organ (see my presentation uploaded in RG regarding Biolinguistics)  which made him create numbers is what makes human beings unique as mathematicians.

Narayanan

Mainz, Germany

Dear Lisker,

You wrote:

Mathematical concepts and physical knowledge come from very different places, and it little short of a miracle that they come together in this universe as well as they do. Many scientists have commented on his, starting with Eugene Wigner, who wrote of the "unreasonable effectiveness of mathematics."

---end quotation

I am vaguely aware of Wigner's argument, but I wonder if you might be in a good position to develop it here, a bit, for present purposes. It certainly seems to me to be relevant to the original question and the current thread of discussion.

Initially, though, I'm skeptical. To me at least, it seems that the mathematical possibilities are literally limitless and the needs of science much more specific. So, the use of mathematics in science is by selection of patterns which may seem to be exemplified in our conceptual approach to natural processes. From that perspective, mathematics seems to be a kind of pure research into possible conceptual structures or relations, from which selection may be made for scientific purposes.  That the selected mathematics meets the needs of science, then,  seems no great wonder.

Am I neglecting something here which Wigner or others have emphasized? What is Wigner's argument?

H.G. Callaway

Mr. Callaway: I've got lots to say on this subject but it would be difficult for me to put into a logical order without some work and thought. Let me begin with noting several examples of the process I'm describing. One good example is that of Group Theory. This was discovered and developed in the 19th century by Galois, Cayley, Emma Noether Sophus Lie and others. No one thought it could have much application to basic Theoretical Physics until modern elementary particle theory came along. The origins of the modern theory date to 1961, with the work of Weinberg, Glashow, and Salam and the importance of one group in particular SU(2). However, Lie Groups had been found to be useful in the solutions of the basic differential equations of physics.

  The following subjects, which were to deemed to be in the province of pure mathematics only, have been found to have important physical applications: Group Theory, Non-Euclidean Geometry, Topology (notably in Gauge Theory, the theory of the Dirac Monopole, and of course Topological Quantum Field Theory), Tensor Analysic (Elasticity) , Riemannian Geometry (General Relativity),.. Heisenberg was amazed to learn that the mathematics he needed to give a foundation to his version of Quantum Theory was already present in matrices and linear algebra.

   I would also like to point out the importance of the reverse process: "The unreasonable effectiveness of physics in forcing the creation of new mathematics". It has happened repeatedly in the history of these scientists that physicists have come up with mathematical models that make no sense whatsoever, that are completely self-contradictory, yet which give reliable predictions. This compels the mathematicians to discover the underlying conceptual basis, to in a sense, "create new mathematics " to clean up the physics, thus advancing mathematics itself. The most notable example in modern times is String Theory. String Theory was developed to unify the apparently contradictory fields of Relativity and Quantum Theory. It turned out that , however logically satisfying, it has not come up with any predictions. However a new field in mathematics based on String Theory and its associated geometries and abstract spaces, has been enormously successful.

Another  example to be cited is  the Dirac Delta function, which is almost an inherent self-contradiction, but which led to the theory of distributions of Laurent Schwartz, etc.

   There are some branches of mathematics which are not likely ever to have any application in physics, mostly connected with the foundations of logic and set theory: Godels Theorems, Transfinite Number Theory, Axiomatic Set Theory, the Axiom of Choice and so on. But one can never be sure.

Roy,

In the 19th century mathematicians realized that geometry, the ground of physics, was essentially group theory,  invariance under symmetry group of transformation and so began the classification of all new non-Euclidean geometry by Klein Erlangen program (1872,Vergleichende Betrachtungen über neuere geometrische Forschungen).  So the realisation that group was central to physics dated at that time. In Quantum theory , Wigner laid the foundation for the theory of symmetries in quantum mechanics. Wigner's theorem proved ( 1931), specifies how physical symmetries such as rotations, translations, and CPT symmetry are represented on the Hilbert space of states. 

But the group structure before they were formally defined by Galois was already implicit all over in different branches of mathematics.  It took thousand of years of mathematical practices for the first mathematician to recognized this abstract structure as a common underlying structure of hundred of other abstract mathematical structures.  But this abstract structure was already implicitly present into our motor system and many other of our physiological systems as Piaget has shown.  It was implicitly present in many ot the original engineering and measuring practices that were abstracted for the creation of mathematics.  So the empirical world has injected from its foundation most of the abstraction of mathematics.  

"The unrecognized  effectiveness of human practices in the creation of all mathematics".

Narayanan,

Mathematics is abstract but also visually concrete.  Euclidean geometry can be understood through visualisation.  The homo sapiens cortex is much bigger than our primate ancestor and it is in the part related to vision that the bulk of the extra cortex real estated is devoted.  Smell is usefull and I guess that a parfum designer or a good hunter can make great use of it but there is not much about the universe compared to vision that smell provide access.  Vision is intimatly connected to movement and space and so our motor system and our vision are meshed together.

Mathematic is purely a science of the human mind.  Yes and No. It is a science of our practice that the human mind has separated from these practice abstractly.  The human mind is not separated from the human body , it is the mean to control this body and the root of both human practices and of mathematics are built-in in the human body.  The human body did not evolve in a vacuum but in the interaction with the world.  The root of its evolution are the root of the evolution of the cosmos.

Poincaré’s Philosophy of Mathematics

http://www.iep.utm.edu/poi-math/

''Kantian constructivism about mathematics is thus not opposed to scientific realism, provided realism is not taken in a naïve way. For Poincaré, the structural realist hypothesis is that the enduring relations, which we can know, are real, because we have evolved to cut nature at its real joints, or as he once put it its “nodal points” (Science and Method, 287). Mathematics is a sort of by-product of evolution, on this picture. In this way, Poincaré's Kantianism about pure mathematics is supported by a Darwinian conception of human evolution – a picture that enables his philosophy of mathematics to coexist with his diverse views about natural science.''

The platonic world is not as Plato imagined outside the world into a mathematical world but is the evolution of the interacting cosmos and organism evolves a nexus of interaction converging toward the natural nodal points of the interacting world.

Mainz, Germany

Dear Brassard,

An Interesting reply, I think; and I have no doubt, myself, that the human visual system is closely connected with the origin of geometry--as among the first sciences. As you say, it evolved in contact with the world, and we may easily suppose that it has some kinship with what is represented by means of the visual system. That is the plausibility of the Darwinian and quasi-Kantian take on the matter.

On the other hand, however the human visual system facilitates the perception and understanding of geometrical relationships, or the awareness of space and time, modern mathematics surely goes well beyond that. Notice, moreover, that if we suppose that Euclidean geometry is "built-in" to the human visual system (and plausibly also in some of our primate ancestors and evolutionary cousins), then it by no means follows that Euclidean geometry is anything better than a good approximation to physical geometry. There is no justification of the "a priori" claims for geometry in all this. It was precisely the use of non-Euclidean geometry by Einstein which is credited with over-turning the Kantian doctrine of space and time as a priori "forms of intuition."

H.G. Callaway

"Euclidean geometry can be understood through visualization"

Sure. So long as it is under 3-dimensional. In general, it is utterly incomprehensible to most and for those who regularly work in n-dimensional Euclidean spaces, we just get more used to doing so (we are no more able to visualize 10,000thD-space than anybody else.

''modern mathematics surely goes well beyond that''

Yes but could modern mathematics goes further than what a human visual thinking can grasp?  I personnally never went beyond it.   But I know that not all mathematicians think in the same manner.  Some visualizes other apparently do not.  

Research in perception have proposed different type of geometry for the senses, none of them being Euclidean.  Kant assumed an Euclidean geometry because at that time it was assumed to be the only one possible.  In the nineteen century when we realized that there are many geometries and that the choice of a geometry had to be a pragmatic choice based on empirical context.  Mach , Poincarre, Weyl, and others immediatly realized that transcendantal idealism has to be modified appropriatly and at that time Darwinism and biological evolution was accepted and so the a priori could be created gradually by evolution.

You should read Michel Bitbol on how transcendental idealism could be modified a a modern philosophy of physics and science in general.  He has multiple online paper on his website. http://michel.bitbol.pagesperso-orange.fr/

You most probably know 10 x more than me on transcendental idealism.  I have formed an idea of it which if you would challenge it in any details would probably collapse.  But Bitbol seems to be a real scholar of the type I like: solid in philosophy and solid in science.  He is an expert in Quantum Mechanics which he interprets as Kantum Mechanics.

Regards,

Mainz, Germany

Dear Brassard,

I regard "transcendental idealism" as a variety of metaphysics, which like other metaphysical systems, can only be naturalized in degree--so that it attends to scientific standards of evidence and testing. That it could be modified to accommodate later discovery of alternative geometries is suggestive of the ways in which traditional metaphysical systems make themselves immune from falsifying evidence of any sort, though this might also be regarded as a step in the direction of a (non-dogmatic) methodological naturalism. 

You ask:

Yes but could modern mathematics go further than what a human visual thinking can grasp? I personally never went beyond it. But I know that not all mathematicians think in the same manner. Some visualizes other apparently do not.

---end quotation

My prior point, to which you here reply, was that modern mathematics goes beyond Euclidean geometry, but I take it that your question concerning "visualization" is a good one, and relevant to our thread here. (Your point is well taken that some mathematicians visualize more than others.) 

Consider, to start, the two dimensional representation of a three-dimensional object, say a cube, drawn on paper. One of the more interesting aspects of this kind of thing is that it is ambiguous regarding the orientation of the cube. Staring at the representation, one can see it alternatively as oriented in one direction or another. The front and back typically shift in perception. That is worth reflecting on for consideration of the attempts, often made, to represent a four-dimensional object with a three-dimensional shape or figure.  Personally, I find this very difficult, and i suspect that the reason for the difficulty has to do with the multiple ambiguities of orientation of the represented figure. Its seems obvious that if the attempt is made to visually represent ever higher dimensional objects, then the ambiguities will only multiply.

The suggestion I take from this phenomenon of visualization is that the representations of ever high-order figures become increasingly symbolic or conceptual in character. With much effort and attention to detail, we can perhaps satisfy ourselves that we have an adequate 3-D representation of a hyper-cube in 4 dimensions, but whether this should count as full visualization remains doubtful, and increasingly so with every higher-dimensional objects.

Consider in comparison the writing systems based on hieroglyphics. They seem to start with an attempt at visualization, adding also phonetic elements, too, though in the end, the general attempt to see hieroglyphic systems as efforts at visualization of the objects represented becomes beside the point of the system. In the end we are dealing with words and concepts.  So it is, as well, with higher mathematics. In the end, there is so much difficulty with visualization of higher-dimensional and non-Euclidean geometries, that we retreat into a more algebraic style and the rule-governed manipulation of symbols, preserving visualizations and diagrams for special illustrations.

H.G. Callaway

Dear H.G. Callaway,

I cannot visualize a 10 dimensional vector a series of 10 numbers as a arrow in 10D space but I can visualize it a a ten dots in a graph in 2D and sometime this visualization is usefull but sometime not and not appropriate.  In the case of Hilbert Space, space of functions which are defined in infinite dimensional space, it is sometime usefull to reduce the functions of the Hilbert Space to 3D vectors and a lot of intuitive insights become available and help guide us in the maze of the abstract algebra.  When I was studying electricity in the first year of engineering,we were taught the colomb law and all the laws of electro magnetic previous to Marxwell's reduction into its four fied equations.  I did not know about Maxwell's field equations when learning these laws that were discovered previously but I was visualizing them in fields and were seeing the connections that are directly express in Maxwell equations.  And all of Maxwell's equations are really understood by me as simple visual laws.  It was obvious before I learn Einstein relativity that the laws of electricity were invariant under galilean transformations.  I had visualize this fact even before I learned about Maxwell's equation and I remember trying to explain this to my college teachers and he told me the connection about relativity.  The point I am making is that discovering special relativity is very simple and obvious from a visual point of view.  But in my case, visualization is not limited to mathematics, even the most abstract concept such what is transcendental idealism is reduce to a sets of simple drawings or visual concepts and the nice thing about these very simple concepts is that when I tried to find connections between very various fields, I just have to put these little visual concept next to each other and see their structural relations.  I do not only use this type of visualization. I remember my first physic course in grade 11, in the first class the teacher told us about the old Ptolemy astronomical system and how wrong it was and the right way to think was the Copernician model.  I visualized this situation and told the teacher that if I located myself on the earth and observes everything then the Ptolemy system is right and it is only two viewpoints on the same set of observables.  He started argying how wrong I was!!!!!  You see through visualization the two are equivalent at the level of of their observable structures.  I can explain this in terms of philosophy of science now but when I was 17 it was visuallly obvious. Things that take book and book of narrative explanations become 1/10 sec evidences.  I use also a narrative method and tried to locate all that I know into a number of such narratives and tried to fit all of them into mega evolutionary narratives.  From time to time two separate narrative line can merge and fused into a single one.  It is process that goes on for decades.  So it is a work of fiction writer which know that all the plots should go together but is working on each of them separatly and is looking for knowledge out them that help putting them together.   There is a close connection by narrative construction and structures.  All the complex structures had to be constructured through a narrative of complexification and so structural hiearchy IS history.  This is a guiding principle in order to link visualization and narrative construction.  There is finally a third historial guiding aspect to my method.  Ideas emerges in a historical context and there is a relationship between the personal learning history of an individual and the original history of generating these ideas.  I had always been deeply interested in history but at some point I realized a closed connection between the ideas I had with respect with certain concepts I was learning and similar ideas people historically had in history.  I was borned into a religious society which transitted in my teenage years to modernity and I was parrally experiencing in my personal life fight between different conceptions of the world which later I found deep parallel with the historical passage to modernity.  I now make it part of my method to take the insight of the historical process.  THere is too much to say about this but historical consciousness is really really deep and is at the base of the effectiveness of the narrative method.

Before choosing vision as topic of my graduate reseach, I for a while (1984) consider goes in the now developing computer graphic field and try to create a new type of mathematic where the user would work on the visual side and the program would support the algebriaic side.  Since then matlab and mathematica have done a few steps in this program but very very short of the type of ideas I had on this.  Around 1988 while doing a master in vision, I had a visual idea which I developed over a weekend and put in my notebook: basically the idea was what it would be called later as virtual reality and the field of telepresence.  I even had choose ''virtual reality'' as a name describing the idea.  One year later I realized through an article of scientific american that the idea was being developed already for a few years.  The point of mentioning this is not simply to brag but to say that it was not a hard idea to get for someone expose to the few simple ideas virtual reality is based on.  It is true in technology, in mathematics, in physics, in philosophy, in everything.   I really think that eventually we will develop educational method and support tools to train all children in this kind of working methods. It is not a question of being smart, it is a question of using the right method.

Regards,

I agree, Louis. I too feel it comfortable and understandable when I can visualise whatever I read. I think children are great in this, but our teaching makes them lose it. 

Narayanan

Narayanan,

Education , litteracy, training in mathematic although necessary unfortunatly push back the natural children way of learning.  Statistical studies showed that the level of people creativity (I do not the specific method of measurement) goes down from the age of five , the day you step in school.  I do think that it is because school , after Kindergarten, neglect to rely on the arts.  Arts are bodily sense based on learning, learning that are not intellectual.  While litteracy and mathematic and the sciences relies primarily on the intellect.  The intellect is totally creative less, totally dum.  The intellect is the part of our mind that evolved in earliest mammal to control objects while the arts are based on the part of our mind that have evolved in the early mammals to take care of the babies, in the primate to relate to each other and which allowed to create all that is uniquely humans.  Not cultivating the arts while cultivating thing needing the intellect is creating an deep unbalance between the two hemisphere of the brain, it creates a recess of the creative female part of ourself and a dominance of the intellect male part.  All education should be art based and even the science should be art based.  My wife is a Kindergarten teacher for 25 years and she is beginning are Ph.D. in art based education.  It is not a new idea, the romantic were heading that way.  Are you aware of similar movement in other cultures in the past?

Mainz, Germany

Dear Brassard & Bhattathiri,

One of the interesting things about early modern philosophy is that, from its talk of "ideas," one may gain the impression that they believed that they thought in images, (say visual images) or that they believed it better to think in images. It has always struck me, in degree, as an open question whether they actually thought in images, or merely believed that they did so--to the degree their talk might suggest.

I have no doubt that it is possible to think in images, or to do so in some degree, or by preference. There was a normative element involved, since in the empiricist tradition, it was felt that ideas should be traced back to perceptions, or "impressions." An idea that could not be traced back to sense impressions was regarded as doubtful or spurious.

But consider, too, that people might think by speaking --come to express things they had never thought before, --in some lively conversation, say, or, alternatively, think in sound images --of words. In any case, I feel little confidence that how people report their thought practices will be completely accurate. We have to learn to describe our own thoughts, and in some sense, we do so in accord with our own theory of how such things go. The theories we use to describe our own thought processes may differ more than the actual thought processes.

Concerning creativity, I have no doubt that people can be just as creative thinking in words as in any other style or manner of thinking. People who work in higher mathematics can be extremely creative, whether this involves images or visualizations or not. This is equally true of people who work in extremely abstract fields of theory. This is, perhaps more difficult for many to appreciate, if they do not understand the details of the problems under study.

On the other hand, high praise for imagination and creativity can sometimes fall under just suspicion. For some people, "being creative" or "imaginative" will simply mean acting or thinking in an arbitrary and unconventional way. This can imply leaving all needed constraint behind.

Solving a problem generally implies respecting the conditions which contribute to setting the problem. A good example of this is that no one would have taken Einstein seriously, if he has not comprehended, with his revised theories, SR and GR, all that which Newtonian theory has successfully predicted. Conceptual creativity is constrained in this way. Still there is some inclination to think of arbitrary departures from accepted standards or rules as "creative" --especially if they are accepted as such by a social group and help to sustain the group as such--for better or worse. This may easily constitute a formula for corruption. 

I take conceptual creativity as cognitively paradigmatic, whether or not this involves images or visualization.  Arbitrary, especially arbitrary and self-interested departure from existing norms, I take to be paradigmatically doubtful and dubious. No one is perfect, of course, and I am not talking about mistakes or momentary departures.  The question is whether arbitrary, self-interested departures from norms become a regular practice and valued as such. All creation plausibly involves some destruction as well--but not wanton destruction.

H.G. Callaway

Mainz, Germany

Dear Brassard & Bhattathiri,

I hope my brief comments above did not provide too great a difficulty for you. I think that constraint on the scientific imagination is an important theme. It certainly enters in, in physics, in mathematics, and in the relationship of the two. 

Nor should we neglect moral constraint in the organization of scientific research --and the avoidance of corrupt practices. These can hold back scientific progress in ways little less effective than the traditional authoritarian methods of persecution and censorship. Even the persecuting tyrant can be expected to have a suitable chorus of "yes men" hoping to benefit out of the process --by jumping on the moving train, so to speak, or jumping on the bandwagon, to use the corresponding American metaphor. 

I see these two themes of research norms as closely related. Techniques of research" turns out to have a moral core.

H.G. Callaway

Dear H.G. Callaway,

In an earlier posting I did refer to Northrop in relation to Einstein and at this stage of discussion I would like to suggest that he has some clarity to offer again. Northrop discerns two types of concepts depending on the source(s),  from which any scientific or philosophical theory can be constructed. He defines:

A  concepts by intuition is one which denotes, and the complete meaning of which is given by, something which is immediately apprehended. Northrop gives blue “in the sense of the sensed color” as an example of a concept by intuition.

A concept by postulation is one the meaning of which in whole or in part is designated by postulates of the deductive theory in which it occurs. Blue in the sense of the number of a wave length in electromagnetic theory is a concept by postulation.

According to Northrop, these two types of concepts exhaust the available concepts (i.e., providing terms with meanings). Mathematical and physical theories are thus clearly concepts by postulation. It may be inspiring to take note of his classification of them:

(1) Concepts by postulation designating factors which can be neither imagined nor sensed. (a) Monistic, e.g., The space-time continuum of Einstein's field physics. (b) Pluralistic, e.g., Plato's atomic ratios.

(2) Concepts by postulation designating factors which can be imagined but cannot be sensed. (a) Monistic, e.g., The ether concept of classical prerelativistic field physics. (b) Pluralistic, e.g., The atoms and molecules of classical particle physics.

(3)Concepts by postulation designating factors which are in part sensed and in part imagined. (a) Monistic, e.g., The public space of daily life. (b) Pluralistic, e.g., Other persons, tables, chairs, and the spherical moon with its back side which we do not see as well as its presented side which we do see.

(4) Concepts by postulation designating factors, the content of which is given, through the senses or by mere abstraction from the totality of sense awareness, and whose logical universality and immortality are given by postulation. (a) Monistic, e.g., The "Unmoved Mover" in Aristotle's metaphysics. (b) Pluralistic, e.g., Whitehead's "eternal objects", Santayana's "essences" or Aristotle's "ideas."

(4) “provide a natural transition”to the other generic type of concept, i.c. concepts by intuition to be found in F.S.C. Northrop - The Logic of the Sciences and the Humanities, 1947 and in brief in Fred Seddon – An Introduction to etc., 1995).

Willem Beekhuizen

Mainz, Germany

Dear Beekhuizen,

Thanks for your suggestion. I would say, to start, that it matters quite a bit what one may go on to say about concepts in determining how your suggestions fit into the present thread. We want to know about the status and interrelationships of the classification of concepts on offer. If I had thought, earlier on, that you wanted to bring Northrop more fully into this series of exchanges, then I would have perhaps asked you to go into detail on how you think it relevant. Perhaps you think the relevancy obvious. So far, it seems not very obvious.

You wrote:

In an earlier posting I did refer to Northrop in relation to Einstein and at this stage of discussion I would like to suggest that he has some clarity to offer again. Northrop discerns two types of concepts depending on the source(s),  from which any scientific or philosophical theory can be constructed. He defines:

A  concepts by intuition is one which denotes, and the complete meaning of which is given by, something which is immediately apprehended. Northrop gives blue “in the sense of the sensed color” as an example of a concept by intuition.

A concept by postulation is one the meaning of which in whole or in part is designated by postulates of the deductive theory in which it occurs. Blue in the sense of the number of a wave length in electromagnetic theory is a concept by postulation.

---end quotation

I take it that you want to suggest that a "concept by intuition" has its meaning given by something "immediately apprehended." But I doubt there any such concepts. I suspect that the point was made by Wittgenstein in saying "Don't look for the meaning, look for the use" --say, of the word "blue." It was just such "meanings" immediately apprehended which he was criticizing by emphasis on word usage.

The idea of a "concept by postulation" seems more familiar, in some ways. But apparently, the account here leaves room for something "immediately apprehended" is be part of the meaning of "blue" as postulated in theory? If so, the problem affects both sides of the proposed distinction. Generally, I am doubtful on the usefulness of classifying concepts, as suggested, according to their sources or origins.

I suspect you may need to say something about how this distinction is proposed to function in application, if we are to go much further with it. Surely, Northrop made some use of these classifications, and to bring your account into some closer and useful relation to the present thread, we need further details --say, on how he would have answered or replied to the account of the present themes in Einstein and the physicists.

More later, perhaps.

H.G. Callaway