So far, three theories have a great deal of 'numerical accuracy' going out to over 10 significant figures, or even 16.  Let me group the Standard Model to include QM, QED, QCD, and everything quantum.  SR/GR is another such group.  And some would say that EM (various forms of Maxwell's Eqns) belong in QM, or GR, but could be considered the third theory, adding to this group the unification of EM with the weak force.  The strong force belongs with quantum.  I think that covers it.  Nothing of greater significance, with super strong experimental data to 'prove' these theories.

And yet I have more to say.  Which was invented/discovered first?  Nature or math?  So, math must be assumed to be based on Nature.  So, my opinion is mathematics is intrinsically based on Nature, and extended by man's imagination to other realms. That said, it implies there is no evidence for the original question's premise.  Say what?

As these three 'math' theories have yet to be fully integrated (EM is arguable already integrated), it can not be yet determined/proven that Nature is intrinsically mathematical.  Exploring the 'extremes' breaks every theory, to date.  Thus, today's math is not up to snuff to be how Nature operates.  This point does not directly or indirectly address your question. 

It states math as known today is a failure for representing 'all' of known Nature.  And a corollary to Godel's Law states mankind will never have all the equations for all Nature observations.  So, the Nature's math 'model' will always be incomplete.  Therefore, the question's premise will remain an unprovable theory.

Galileo said that the language of Nature is mathematics, but it was at the time of Galileo.  Since then, different mathematical systems have been used, and we know that Euclidean geometry is not the language, but one of them in its own domain of validity.  Since the goden age of mathematics, there are so many different structures and systems that they can't all be used in physics. Mathematics is thus a separate discipline and is merely an elaboration of logics.  Theories must be logically consistent so that they can't yield contradictory predictions, that's why they are expressed in mathematical form.

If nature is the collection of phenomenon and each phenomenon is described through mathematics, then there is no phenomenon in nature which is non mathematical in behavior. Take any phenomenon from nature (= a consistent and logical structure) we come to identify, know it and give it a name because of its behaviors and properties we come to observe and describe through mathematics. Nature : space, matter and energy, separately or together are objects that can be observed and studied through the vernacular of mathematics using the principles of logic and deduction. If we can find one thing in nature whose behavior is not described in mathematics , then one can tempted to say nature is not mathematical, but what we can still say is the existing mathematics is not big enough to describe that phenomenon, and a new mathematical structure will be developed that describes it, although we do not have that counter example. 

I would urge for caution here. It is dangerous to assume that nature is intrinsically mathematical when, at root, we know it is quantum mechanical and we lack a mathematical understanding either of wavefunction collapse or distinct, unique observations (in the case of those eschewing collapse).

A way to approach this would be to examine whether there are any instances of either physics apparently falsifying math, or math falsifying physics.

There is not a single recorded instance of physics apparently falsifying math, whereas there are a number of instances of math seemingly falsifying physics (Bell's inequalities, for instance)

Those main applicable parts of mathematics used to describe physics ( i.e. nature ) appears as a language which in conjunction with some physical postulates ( e.g. Newton laws) interpret and anticipate quantitative and even qualitative features of natural phenomena .Actually Mathematics is an invented language by human mind like verbal languages used to name and label the nature ingredients. Number by itself is not an inherent concept of nature, rather it was invented as  the constituents words of mathematics.  Number theory as the pioneer of mathematics languages has not attained the position of differential calculus which revolutionized the natural phenomena explanations via the differential equations. Differential equations succeeded in anticipation of  solar planet orbits, diffusion of heat energy, space-time curvature, wave properties of particles and light and too many applications in the realm of physics (i.e.nature) with this assumption that dynamical phenomena change continuously. Obviously discovery of quantum nature  of matter  limited this approach to macroscopic properties of physical events.Mathematics resembles the other verbal languages with specific grammar  and vocabulary in order to describe concepts and things. This language is not an absolute tool and has been based on human logic. We should not forget that the mind and imagination are parts of nature and consequently the invented concepts of human mind are not the nature, but serve as a language to describe it. 

Manouchehr, these ( Quote "Mathematics is an invented language by human mind like verbal languages used to name and label the nature ingredients" Unquote) are affirmations without proof. 

Very respectable scientists with impeccable credentials say the exact opposite, and this debate - the Aristotelian vs. the Platonic views of reality - has been going on for some time and is still ongoing, with solid arguments on both sides.

Mere unbuttressed affirmations , however, do not contribute anything to the debate.

H Chris, your statement is also a proposition, please prove it, but before asserting take a look at the topic question " Assuming the premise ...."  it use "assuming" and "premise" . I would like to see your proof.

I would like to suggest, that in the context of questions about the relationship between maths and physics, there are two meanings of the word mathematics.  There is mathematics as an invented language which humans apply.  Then there is mathematical behaviour which is independent of language. An example of this is the arithmetical bahaviour of stones, apples, grains of sand or sodium and chlorine atoms in a crystal of salt -- a certain number of which would add to a certain length.  This is different from the arithmetical behaviour of water or 3-space.  Do you not agree that these different arithmetics are there, without us modelling them with mathematical language.

I think, it will be worthy to repeat a note that I already wrote in other thread of SG discussions. In macrocosm (at least), we can never verify the validity of a force law and inertia law each separately. We can only verify the equation of motion containing both force and inertia laws.

The equation of motion can be written in the dimensionless form - I did so in the trivial case of Newton and Coulomb laws in combination with the Newton inertia law (in article on arXiv: 1206.0405). In the dimensionless form the physical constants as gravitational constant, permitivity of vacuum, and Planck constant disappear. (The equation of motion still contains the speed of light which can however be regarded as an etalon to evaluate a change.)

The dimensionless equation of motion says that the nature can, probably, be described using only a pure mathematics. The physical description occurs, likely, due to our (of human beings) establishment of some artificial physical quantities as, e.g., mass or electric charge. The nature may not to "know" these quantities. It seems, if we used only the "natural" quantities, the physical description of reality could be done with exclusively mathematical equations. (According to my opinion, based on the above mentioned circumstances, the physical constants are merely the transformation constants between our, human, artificial physical quantities and the natural quantities.)

The platonic view isn't that Nature is made of numbers, but that mathematics only describe the shadows, and the underlying reality casting them isn't accessible to us.  There is no support at all that reality is but a mathematical network.  Concepts are neural knots called chunks, and it is a big step to think that their relationship with the structure of Nature is anything more than an analogy.  Focusing on physics gives the illusion that the correpondance is as perfect as an isomorphism, but phenomena like consciousness and others, even if subjective, point to the suspicion that there is something more. That's where the Aristotelian view comes in, and there is no evidence at all that these higher level phenomena can be reduced to physics, although it is an ingrained belief among many scientists. However, we know that it is impossible to acertain through brute force.  If a theory of everything (meaning only on the low level of physics) is ever found, it would be a hint in favor of reductionism, but it is only probable that will never happen, no more that at the turn of the 20th century.  Keep your mind open.

Example: the concepts of wave and corpuscle.  For a long time it has been though that penomena in Nature were identical either to a wave or to a corpuscle, with the corresponding mathematical descriptions, differential equations and Lagrange equations. Quantum mechanics posed the question: whether a wave or a corpuscle? It attemped to fudge these two concepts through an axiomatic formulation and the answer: both. But this axiomatic system is logically inconsistent, and predictions of quantum mechanics rely on ad hoc inputs. Actually, the answer is: none, wave and corpuscle are but human mathematical concepts that aren't found in Nature. Such rebuttals of assumed mathematical structures of reality is unrare in the history of physics.

I should add a comment about the Mathematical behavior. In order to clarify this term i recall the first steps of application of math in physics by Galilei (father of  physics). He observed that all heavy things when falling  from a tower ( i.e.ignoring the air resistance ) represent almost the same behavior, then he derived a physical law that all things falling down with the same  acceleration and used (probably for the first time) the mathematical symbols for his notion. This history reveals all requiring concept regarding the relation of mathematics and nature  for this initial steps. First he  observed (or perceived ) the phenomena with visual, touch and other senses, he compared these data with trained concepts such as length, mass (measure of matter) time etc. Since then he explained the notion by using the symbols substituted for length,mass etc. He discovered an arithmetic relation between these quantities (i.e. the cinematic of falling bodies). Without the definition of mass , dimensions ;spatial and temporal ; and gravity these equations do not indicate anything unless the relation of some numbers. When we write 1+1= 2 this is the mathematics language but when we use this equation to count the nucleons in an atom we label to these symbols the concepts hiring from physical (nature) concepts. In this sense any physical laws could be described by some equations. Consequently i agree that governing physical laws, get  the universe to obey its own behavior, by inspecting these universal laws, human being invent a language to describe it in details. For the final example i refer to different verbal languages invented by human, irrespective of different symbols and grammar that can be used for the common concepts such as small,big etc. these languages can transfer the concepts and meanings, and the final result (description of concepts) are the same.

Thank you sincerely L. Neslušan.

I think, what you say is important.  I have been working myself, using dimensionless mathematics.  I use wave-number and position, rather than momentum and position.  If one uses this approach, eventually, as one proceeds, in deriving theory, one comes to a point of difficulty, where we can either adopt a dimentioned constant, or we can use that difficulty as a guide toward better understanding.  Research I would especially like to spend time on is the question: Where does Planck's Constant come from? 

Thank you.

Thank you Stefan.  I think your formal approach is the way forward.

an interesting indication that both micro and macro phenomena are inherently mathematical could also be observed using the Fibonacci Sequence, which is said to occur in smaller organic things as well as the bigger things like galaxies.

The premisse is that there is nothing else than mathematics in Nature.  What does that mean?  Suppose we make a computer simulation that describes every detail of the physical world.  Is its output Nature itself?  I don't think so, we'll get some kind of representation such as graphics, figures etc.  Nature isn't graphics or figures.  Similarly, with a mathematical model, all we get is equations etc., and an interpretation is needed.  For example, what represent that number?  Let's take the Planck constant.  In fine, it is the deviation of a hand with respect to a dial that is a part of a well defined setup, and the constant is the result of an equally well defined numerical calculation. It has a meaning only in a well defined system of unit based on comparision procedures.  If mathematically the Planck constant seems to be a mere number existing in Nature, scientifically it is much more complicated.  It is sort of a pattern that appears identical to itself in a great variety of situations. A physical theory has always two parts: the mathematical formulation, and the interpretation that makes the link between the mathematical concepts and reality. This interpretation, according to the premisse, belongs to Nature, but it isn't mathematical.  Interpretation is what is needed to define the experimental setup and the system of unit, in sum, the whole scientific procedures.

http://www.mathsisfun.com/numbers/nature-golden-ratio-fibonacci.html

The best evidence we have that nature is intrinsically mathematical is its predictive power. Mathematical models are ultimately judged by their ability not just to explain current and past observations made of the physical world but by their ability to successfully predict future observations before they are made. As a physicist I often hear people argue that science is a belief system that ultimately hinges on faith in the same way any theological or philosophical belief system does. I do not even attempt to deny that. All belief systems, at their core, are methodologies for divination, for predicting the future. I choose to use mathematics for my prognostications of the future as it has significantly better predictive power that using tea leaves, tarot cards, any of the various books ascribed to a Deity or deities, astrology, intelligent design, etc...

Now the question of whether Nature must operate by mathematical structures is difficult to test, as Stefan nicely pointed out. Here is when we run into the problem of self-reference. The most successful methodology for testing whether a theory about the physical world is correct (or to be more precise is not incorrect) is to make predictions about the future based on the methodology and then test whether those predictions are accurate. If we assume the negative, that the natural world is not intrinsically mathematical, I do not see any testable predictions that could be used to test the soundness of such a theory. In the end, this is the best evidence we have that the natural world is intrinsically mathematical. Divination is ultimately a pragmatic practice rather than a dogmatic one. Is your assumption about the structure of nature able to lead to successful predictions about future observed phenomena? So far the assumption that nature is intrinsically mathematical has the most successful track record for divining the future (and by an astonishingly significant degree) over any other assumption about the intrinsic nature of the universe. There will be many who will point out that there are many things that a mathematical approach are currently unable to successfully predict, and that in the past specific mathematical approaches were ultimately found wanting in their predictive power and supplanted by a new system. However, I have yet to find a case in which a mathematical theory which was eventually found lacking in certain areas of predictive scope was supplanted by a more successfully predictive theory that did not follow the premise that nature is intrinsically mathematical.

If one were to be found, I would have no problem denouncing my devotion to the cult of mathematics in favor of a more reliable system for divination. Call me a bandwagoner, a pragmatist, a damned soul, or a warlock, but my ultimate goal is to predict the future, and as it stands mathematics has a track record that would make Nostradamus jealous.

Thank you Claude. I hope you don't mind. I am not expecting responses in this thread that argue against my premise.  They tend to steer debate away from the subject I'm interested in. Instead, I'm looking for views (with which many might disagree) that might contribute toward an argument in favour of what I believe.   

Thank you Lawrence,

I would like to extract something you said, and highlight words you use that are actually profound.

"The most successful methodology for testing whether a theory about the physical world is correct (or to be more precise is not incorrect) is to make predictions about the future based on the methodology and then test whether those predictions are accurate. If we assume the negative, that the natural world is not intrinsically mathematical, I do not see any testable predictions that could be used to test the soundness of such a theory."

So if I can find an example where Nature cannot get along without some help that is exclusively mathematical, I might have a good argument??

Exactly, Steve. I think proof by negation (reductio ad absurdum for the latin geeks) would be an interesting way to approach the question. So the question then becomes:

If you begin with the premise that mathematics are not an intrinsic property of nature can you deduce an absurd conclusion?

OK.  Good.  Thank you Lawrence, I think we might be at an interesting moment in the debate.

Do you think the paper below is evidence of a certain experiment whose progress is reliant on information that is strictly mathematical?  I am attaching a page of my own that might provide a little more.  I regret not writing it directly in here, but that would have been easy.

Article Logical independence and quantum randomness

Steve, the better way to reinforce your belief (your own word) is to try and refute it by all available ways, and fail.  I miss an argument showing that an interpretation isn't part of a theory, and/or that an interpretation is mathematical.  Moreover, is a prediction necessarily mathematical, or just a generalization based on observation?  Galileo didn't need any mathematics (nor divination) to predict that every object dropped from the Pise Tour will fall on earth.  Does the fact that the behavior of any object is the same entails that Nature is mathematical?

The possibility of a prediction is called determinism, not mathematics.  But now, quantum mechanics is not deterministic.  Yet, a mathematical description is still possible through probabilities.  Should we give an ontological status to probability theory, as it allows for example to derive exact laws like in thermodynamics in very general terms?

A book of records sponsored by a well-known brewery has an entertaining entry concerning the "most inaccurate value of Pi". Apparently, the Indiana Legislature almost passed a bill in 1897 that would have legally established the value of Pi as 4 (a 'fact' that would then have been subject to copyright law):

http://www.pleacher.com/mp/mfacts/pifacts2.html

One might ask what items of evidence were presented in favour of this view... and what evidence to the contrary was being ignored.

Why are the equations of physics mathematical? This presupposes that we have a complete understanding of nature - which we don't. The equations we do have are inevitably mathematical. Topics which aren't taught in physics (unified theories of all four forces, consciousness, wavefunction collapse etc) are ignored precisely because nobody knows how to express them mathematically. That does not mean they don't exist - nor that they can be dismissed as unimportant or some figment of the imagination!

No mathematical physicist has yet presented an equation that uniquely fixes the value of any physical constant. That may have been one of the main goals of string theory originally, but that ambitious dream has now fallen by the wayside. By now we hear talk of the string landscape, the anthropic principle, myriad clues of cosmic fine-tuning and speculations concerning some multiverse. Such discussions stem from an appreciation of just how improbable it is that nature can give rise to complex life if the constants of physics are fixed with no regard for biology. This of course implies that the physical constants are not completely inflexible.

Before proceeding with this question, it seems to me that we should at least consider the evidence that Pi is not equal to 4.

Claude,

A prediction is not necessarily mathematical, but in order to predict the future in a reliable way (and by reliable I mean orders of magnitude better than a guess or any other method) it is necessary to use mathematics. Whatever interpretation is given to the ability of a mathematical law to predict the future says nothing about the mathematics. It is just a metaphor to make it easier to learn. The mathematics stand on their own entirely based on how well they predict the future (and by prediction I don't mean saying that if I let go of an object on the top of a building it will fall to earth, but that if I drop an object I will tell you accurately where and when it will touch the ground).

I find it interesting that there a lot of comments about quantum mechanics being a non deterministic  theory. That is only one interpretation of the theory, which has nothing to do with its predictive power. There is no physical model for quantum mechanics which seems to bother a lot of people. Who cares? The physical model again is a metaphor that can be put in a language other than mathematics. The fact that the ability of the mathematics of Quantum Electrodynamics to predict the future is many orders of magnitude better than any theory that the human race has ever developed is the only meaningful way to judge it.

Stefan,

I'm glad you stay on topic and examine the actual question. I do disagree, though, that P is not falsifiable by observation. If a new conceptional language were developed which provided provided predictions for the future which were not correctly predicted by any mathematics then I would say that the mathematical description of nature has been disproven in favor on this new more accurate system. For example, if someone comes forward with the hypothesis that the actual intrinsic structure of nature is not mathematics but is divine. That in fact, there is Deity which has both created and has complete control over the workings of nature and this Deity has for some reason communicated through a human being such that they are able to predict the future better than any mathematical model then I would call it a day and look for another job. So far such a thing has not occurred, which does not prove that mathematics is intrinsic to the natural world but has at least increased it's odds as it has successfully recovered for countless attempts to disprove it since it was developed. The mathematics of the most current unification theories are incredibly complex. Imagine the jump from counting to differential calculus. Now take that jump and increase it by 3 orders of magnitude. The pace at which it is developing is staggering compared to just 5-10 years ago, most probably due to the recent developments in experimental measurements.

That's all for now. Please, everyone who comments here take the time to read all the comments made before to be sure you are not repeating what someone else has said, or are bringing up ideas which are not germane to the question.

Lawrence, precision isn't an exclusive propriety of mathematics.  I can predict the length of the meter standard in the future just by measuring it precisely.  The law is, objects compare always the same, and comparing is a well defined procedure that can't be expressed in pure mathematical terms.  In other words, it is an interpretation of the mathematical concept of length.  We can't fish numbers in Nature.  We have to make a correspondance between a variable, and a procedure that yield a number.  The end result is not a number, but a length compared to another length or something similar.  The very objects that are compared is an integral part of the theory.

Claude, not sure what you mean by 'we can't fish numbers in nature' ? If a pride of lions has 20 individuals, then irrespective of how you label twenty - tyve, yirmi, vingt, zwanzig, twintig, or ashinrin da, the reality is that the number 20 has arisen from nature. The ways one can fish numbers from nature - including, as Robin notes, π , whose value is imposed by nature and not by the Indiana legislature, is just endless.

What is a belief?  A belief is not the truth, it is something that we heard, read, or even thinked a large numbers of times, and that has become a habit.  It is activated unconsciously and used in our reasoning as a premisse that is never challenged.  Because of that, a belief need not a flawless demonstration to be reinforced, a sophism or even a mere hint works as well. We ignore the evidences against it because they contradict what we have taken for granted, that thus seems to disprove them.

The mathematical intrinsicness of Nature is one of these beliefs.  Because since Galileo, the advances, and even successes in physics, have been made possible by mathematics.  But there have never been the shadow of a trace of a epistemological theory backing it.  It seems beside that Kant in his "Kritik der Rein Vernunft" tried and disprove it.

But our theories of today aren't in agreement with that belief, especially quantum mechanics.  The axioms of QM aren't logically consistent.  The axiom of unitary evolution is obviously incompatible with the one of projection.  They both predict two different wave functions in the future.  They aren't used in the same conditions, but there is no axioms defining them.  We only know impirically that in such or such setup, the collapse happens so or so.  It is clearly not a rigourous mathematical foundation.  But the key point is that nevertheless, QM works, then it describes Nature anyway.

Despite the random nature of QM, we can restore an approximate determinism for very simple systems, described by fundamental physics, and persist in the illusion that physics is mathematics.  But in very complex systems, where the collapse happens may yield a genuine impredictable behavior, as that may have macroscopic consequences like with the Schrödinger's cat.  The human behavior for example may be completely impredictable, and not because is is impossible to model completely the brain, because there are so many collapses.

Mathematics, and also computer science, made a huge progress when it was realized that logics can be expressed in mathematical form, through 0's and 1's.  It is a way to keep consistent, since 0 is not 1 and conversly.  It is a trick used by our brain to model its environment and making sense of it, while keeping consistent : a neuron fires or not.  But Nature doesn't need such a trick, it doesn't model itself, it is and that's all.  Nature "knows" when there is a collapse or not.

Perhaps in the future, there will be a theory that treats this problem in a consistent way.  But perhaps not, and perhaps this theory will suffer from other inconsistencies.  Presently, it seems that we must conclude that Nature isn't intrinsically mahematical.

Claude,

I'm not sure what you trying to say. The issue is not whether the mathematics used in science are logical or satisfy your human need for meaning. Again I will say this for the third time and hope it is understood. If I can predict an event with an accuracy (not precision!) of better than 1 part in a trillion with a measurement which has never been done before on an event that has never been observed before based exclusively on the use of a mathematical model then it becomes very hard to claim that you can disprove that nature is not intrinsically mathematical.

I won't go into the excrutiatingly painful details of all the strange misconceptions about quantum mechanics. Never have I seen a scientific theory so thoroughly misunderstood yet painfully argued by people who have extremely deep rooted beliefs about a theory which they clearly have never studied to the level of a professional in the field.

If you haven't studied quantum theory at the graduate level, please don't make statements that sound like they were formulated by someone who just read a lot of post-modern philosophy and musings on the implications of quantum theory which fill the physics section or barnes and noble and then had a bong hit.

Lawrence, sorry but you always repeat the same error.  The meter standard has never been measured on the 20 april of 2056, but the result will be 1 m.  Physical theories are just more complex, but fundamentally they are no more than that.  They encode regularities that are already in Nature, they don't create them.  Then they works out the logical consequences.  The precision or accuracy is in the procedure, not in mathematics.  We have one part in a trillion because we have atoms with very stable states to compare with.  It is clearly a physical object, not a formula.

Chris, counting is also a comparision procedure.  Pi in not imposed by Nature, for the simple reason that if you measure it, you won't get pi because of the gravitational field.  Pi is in a theory that is based on the axioms of Euclidean geometry.  Even in the absence of a gravitational field, it could happen that the circle ratio is not exactly pi, but very near.  That's what fooled Galileo, as in his time it was thought that Euclidean geometry was an absolute truth.

Claude, sorry, but no. Irrespective of the latter day structuring into a set of axioms of arithmetic by Peano et al, numbers did exist before this latter-day structuring. The structuring is pure taxonomy, nothing more. 

Pi by the way is certainly not a theory, it is a number, to wit the value of a ratio found in nature irrespective of Euclidean geometry : it is the ratio of girth to diameter in a flat plane, and the value exists irrespective of the existence or not of Euclides or Peano. Flat planes and curved planes, whether convex or concave, exist independently of taxonomy.

The examples from nature are endless - Boticelli, etc.: the so-called golden ratio exists independently of us.

The only thing that is human-related in numbers is our habit of using a base 10 - rooted in our number of fingers - but this is no more fundamental than calling one 'one' or 'ichi' - it's content-free, labelling semantics.

Always putting man at the center of the universe and positing that numbers would not exist without mankind seems to me totally untenable, something perhaps akin to Medieval cosmology which insisted that the Earth was the center of the unverse ....

In a gravitational field, Pi remains pi if it's defined as the ratio in a flat plane - the definition is independent of the environment. Now if you want to use that ratio to actually measure something, rather than in abstract math, then it becomes engineering or physics and no longer arithmetic - and then of course the value adapts to the material conditions at hand, but it does not affect the value of the number if defined for a flat environment. 

As for QM - there is no such thing as collapse - only instances of coherence and decoherence, which are pure reflections of the set of variables that become encompassed within, or leave, the valid Schrödinger Equation describing the system at all times. The evolution of the Schrödinger equation over time mandates which variables become included by timelike coherence, and which decohere - it's a mathematically describable phenomenon.

Because we're generally not very good at math, we are unable to work out and resolve the wave function in any but the very simplest cases - however, it is easy to prove that wave functions exist and have some mathematical formulation in all cases. We're just not very good at working out the math, is all: e.g., the easily, straightfowardly expressed Millennium problems we can't even solve, for instance. Perhaps if we had 20-liter brainboxes containing 100 trillion neurons we'd be better - in any event the fact that we cannot work out the external-reality math further underprops the notion that math exists outside us. If it were man made, we'd probably work it out.

Chris, the question is, is there flat planes in Nature.  That is, flat plane in the sense of Euclidean geometry, since there is no other way to define it.  The answer is no, because of the gravitation field, but also because of dark energy, and because the Universe started as a point singularity.  If space-time is not a manifold, the very notion of a surface would be meaningless, and pi with it.

Decoherence is formulated with the projection postulate.  It is the idea that collapse happens all the time, and what we observe is sort of a classical thermodynamical average.  But an electron beam can be deflected by an electric field, and still exhibit an interference pattern in an adequate setup.  An electric field is a macroscopic system with a large number of photons.  That's though a very simple system with no amplification of small quantum fluctuations.

On the subject of comparing lengths of rods, so one can measure one against the other.  You will find the ratio of their lengths to be rational, never irrational.  Is this a mathematical fact about Nature?

Can I make this argument?

Nature complies with conservation laws:  momentum, energy etc.  By Noether's theorem, these conservation laws each stems from a symmetry.  Symmetries are mathematical entities.  They are any entity complying with some group of transformations.  A transformation is mathematical information.  Hence, Nature complies with mathematical information.

Steve: Thank you for clarifying the original question, and editing the original clarification, I read your posted paper, 3 paragraphs, and scanned the paper at the posted URL.  I must ask a clarification of "unitarity is redundant." Seems I am reading that the opposite of what you intend, as the next sentence, I would disagree with.

I provide a reworded premise in a below paragraph, towards perhaps a stronger position using what is called a "properly worded problem statement." At least to me it states where my stated "position" is coming from. Even though, I still believe in my first posted position of believing Nature is not intrinsically mathematical. I can still put myself in the other guy's shoes, even if I disagree with the premise.  I hope to provide some value for you, and even more for me, as exposure to RG posts has enlarged my vision.

Below, I coin the term "equation set" to be 'the set' of mathematical formulas behind Nature.

I hope all future posts explicitly state in the very first sentence, their overview topic, of evidence for Nature to follow mathematical/equations/formulas predictions, and even stronger than mere prediction, that Nature 'consists' of 'only' those equations, whether known by man or not ever to be known, but still an "equation set" not only models Nature, but controls Nature.

Having read all posts, I will clarify my first post with the insights I have gained.  I will restate my set of 'evidence' in different terms below.

The evidence I 'see' for your premise is the "accuracy" of the equation set predictions.

If that accuracy can be shown, experimental data or proven theory, to extend to not just 10 to 16 significant digits, but to an infinite number of significant digits, I would call that very hard evidence for that "equation set" being what is 'behind' Nature.  Thus, a way to define M, the "Mathematical-ness variable" of Nature, might be established as approaching the limit of infinite significant digits.  Heisenberg Uncertainty Principle need not apply if one is not examining, at the same time, two complementary quantities.

Additional clarification of my answer is the limitation inherent to addressing only the 'quantitative' Nature, and not the 'qualitative' Nature.  Being 'mathematical' means that qualities are strictly derived from quantities.  Thus, another measure of evidence would be to show there are no stand-alone qualities.  That all qualities are cases of quantities.  Yes, I am not sure I even know what this means. 

As much of science is done first via discussion of qualities, and then deriving the quantities, if Nature is intrinsically mathematically, then qualities would take a back seat, in my mind.  And as human minds are not directly number crunching entities, dealing with qualities as the primary method of living and development, it might be very, very hard for such a creature to flip flop to seeing Nature as strictly quantities, with qualities that degenerate to distinct quantities.

The "holographic" view, that 3D space is a mapping from a 2D sphere, is a strong statement that Nature is mathematical, in my view.  If the holographic view is 'proven', then what happens on the 2D sphere must now be questioned if it strictly obeys, in all ways, another, second "equation set" for only 2D.

There is the limit of the "equation set" completeness to take into consideration.  Must the equation set be 'fully' known, or can a theoretical analysis 'prove' that in the limit the set is complete?  My next step for your premise would be set up this limit and prove it.  Again, I am not sure what I mean.  If someone were to do this, then I would agree it's a good approach.

Where the above paragraph comes from my thinking about the paper whose URL you posted.  To be more precise, I use the wording (a quote with two bracketed phrases of mine) from the paper below.

Whenever a proposition [your premise] and a given set of axioms [Nature or Mathematics or both] together contain more information than the axioms themselves, the proposition can neither be proved nor disproved from the axioms - it is logically independent (or mathematically undecidable). If a proposition is independent of the axioms, neither the proposition itself nor its negation creates an inconsistency together with the axiomatic system.

So, I read this to mean that Nature and Mathematics must be defined as a formal system based on axioms.  Then, to prove your premise/proposition, must show there is no logical independence.  There is no outcome randomness.

My opinion of this approach, perhaps it's not one you intended to be perceived, appears to be valid. 

However, both the paper and this approach is limited by a severe hampering assumption, that is common to many mathematically approaches to physics.  The overall, comprehensive problem is simplified, in order to derive modelling equations.  It's arguable that no simplification has been made.  I would say limiting discussion to quantum physics is a simplification, as it leaves out GR.

I feel, it is a way to get traction on your premise.

Thank you Peter for your contribution, and especially or taking my question seriously even though you disagree with my premise.

I'll try and clarify a couple of points you ask about.

"unitarity is redundant."

The paper I posted is just the beginning of the introduction of a larger article I shall post, hopefully quite soon.  By unitarity is redundant, I mean that when a quantum system is prepared in a certain state, and then measured, not for any complimentary state, but for that same state, then you will get that state for certain.  That is the circumstance when unitarity is redundant. Another way to put this is to say, that the self adjoint-operator is a member of a sub-algebra of some larger algebra. For all measurements the sub-algebra is necessary, except in the case of when a measurement is made on a system already prepared in that state.  For that kind of measurement, you don't need Hilbert space.

I like your point that the evidence we have for the quality of Nature, is quantitative.  I think this might be a statement of representation through isomorphism

Your point on incompleteness  --- If elementary algebra, on which physical theory rests, is treated an a formal axiomatised theory, then it can be proved that that algebra is incomplete.  You don't need to look further into the physical formulae.

"Whenever a proposition [your premise] and a given set of axioms [Nature or Mathematics or both] together contain more information than the axioms themselves, the proposition can neither be proved nor disproved from the axioms - it is logically independent (or mathematically undecidable). If a proposition is independent of the axioms, neither the proposition itself nor its negation creates an inconsistency together with the axiomatic system."

The Paterek research shows there are experiments whose results correlate random outcomes with logically independent propositions.  And predicable outcomes with logical dependence.

Claude,

You wrote the following:

"Chris, the question is, is there flat planes in Nature. That is, flat plane in the sense of Euclidean geometry, since there is no other way to define it. The answer is no, because of the gravitation field, but also because of dark energy, and because the Universe started as a point singularity. If space-time is not a manifold, the very notion of a surface would be meaningless, and pi with it."

The existence of a gravitational field does not have any influence over the question of whether the universe is euclidean or not. In fact recent experiments that measured the microwave background radiation has shown that the universe is most likely (with a .4% margin of error) euclidean in geometry.

http://map.gsfc.nasa.gov/universe/uni_shape.html

Many people confuse the idea that in a gravitational field the path of light is bent with the global geometry of space-time. These are two very different concepts.

In a pride of lion, there is the number 40, and the number 10, and the number 10^5, and the number whatever.  40 legs, 10 males, ~10^5 stands of mane.  We get the number 20 only associated with the definition of an individual lion, and this definition isn't mathematical.  We can't add apples and oranges, but we can add fruits.  However, I had exactly the same in my basket.  Everything we observe in Nature doesn't necessarily obey arithmetics.  Velocities don't add, but rapidities do. The number of photons evolves in time.

In quantum mechanics, if we count the number of particles in a box, each time we'll get a different number because the collapse can occur either in the box or outside it. Counting doesn't make sense at all without a well defined physical procedure, and in the case of QM, this procedure disturbs the system.  There is no interpretation of quantum mechanics that gives a real meaning to the number of particles in a box before a measurement without running into logical contradictions.

So it is only the choice of concepts we make that allows a mathematical formulation. We are actually imposing mathematics upon Nature.  But it isn't, and probably can't be, described by only one mathematical model.  Quantum gravity escape any unified theory, two incompatible mathematical structures are necessary.  It is only by restricting the domain of validity in an ad hoc manner that we get the illusion it is mathematical.

Velocities do add. You can add vectors as well as scalers.

In QM, a particle in a box is confined to the box. It has a zero probability of being outside the box (hence the term particle in a box). The size of the box determines the possible energy states of the particle in the same way that the length of a string determines the possible harmonics it can produce.

Lawrence, I won't answer you.  Putting down my proficency is neither correct nor respectful.  If I have made some mistakes, -- that's quite possible -- point them out and explain why, but you can't impose your own beliefs on me for whatever reason.  There is no referee here. Also, read more carefully what I wrote, I wrote it carefully myself.

Claude, I would say your use of the phrase "an individual lion", is implicitly mathematical. Lions are countable.  Even if they are never counted by anybody, by any beings or the lions themselves, they are countable. Countability is a mathematical distinction.  If countability is not mathematical, then what is it?

Steve, countability is obviously mathematical, but not the definition of a lion. (Individual is the term of biology, the last taxinomic division, not of mathematics.)  In mathematics, only the elements of a set can be counted, and some sets can be defined mathematically.  But in physics, there aren't.

Now what is countability?  In arithmetics, the natural numbers are defined by the axioms of Peano in an inductive way. (That Induction is what leads to Gödel's theorem and paradoxes in the foundation of mathematics.)  Then, we can bijectively associate each elements of a set to one of the smallest numbers.  There is a theorem showing that, in whatever order this association is done, the biggest number is always the same.  Therefore the cardinal of a set is well defined.  But what happens when it is a set of undistinguishable particles?  Have we ever a mathematical system able to deal with undistinguishable objects?

In the old arithmetics of Galileo's time, this problem didn't show, but in the axiomatic approach of the 19th century, everything, even the most intuitive statements, must be proven.  Euclide couldn't prove his postulate, but admitted it anyway.  Today we know that it was an error.

So you are agreeing that countability is a mathematical distiction?  Would you say, that the set of all lions is a countable set?

Claude.

If I had seen this question, as a responder, and had disagreed with the premise, I would have steered clear of the question and left it alone altogether.  

Life is not that simple in that your question deserves a more detailed answer, also modifying the question. See elsewhere, e.g., the attachment.

Leo.

Claude , quoting : 'Euclides couldn't prove his postulate, but admitted it anyway. Today we know that it was an error.' - that is not correct at all.

Euclidean geometry is correct and self-contained as it is, the same way as Riemannian geometry is, and the same way Lobatchevsky geometry is. All three of them are absolutely not incorrect - they just have different scopes and areas of applicability, but as mathematical constructs they are individually perfect, and perfectly consistent and self-contained .

The fact that we live in a curved universe has no bearing on the internal consistency of Euclidean geometry. Moreover, the fact that axioms may not be universal and may not be applicable in certain circumstances or environments is, if anything, a great plus - it pushes the envelope of knowledge and of possibilities and leads us to look for other environments where a different, yet equally self-consistent, set of axioms may apply. Many folks - Blencowe, Duff, Monastyrsky et al. - have commented on the fact that whenever a self-consistent and self-contained set of axioms works, then it may well actually map onto a reality, somewhere - the very definition of a mathematical universe, or mutiverse 

Indeed - nowhere is there a self-consistent set of axioms that says, for instance, that 1+1= 3.542 or the color blue: it just does not work, and Peano's axioms are severely constrained by external reality. Outside Riemann's, Lobatchevsky's, and Euclides's, there does not seem to exist room for a further 3D geometry: you can't build a consistent set of axioms in 3D different from these three.

I'm not sure why you persist in considering the physical environment and/or conditions where you live as determining truth: you might well live in a severely warped universe, and that would not in the slightest impugn on the self-consistency and validity of Euclidean geometry - these are wholly separate things.

Thank you Chris.  All very interesting,  I am particularly interested in this remark: 

"...Many folks - Blencowe, Duff, Monastyrsky et al. - have commented on the fact that whenever a self-consistent and self-contained set of axioms works, then it may well actually map onto a reality, somewhere..."

Thanks.

Assuming the premise, you need no evidence. Is, then, the premise right? Maybe not as formulted, the trouble being in the word "intrinsically." I've attached an answer.

Leo.

Decoherence addresses the relationship between quantum and classical mechanics.  It is sort of an ensemble average, it doesn't describe the actual system but derives classical probabilities.  Decoherence doesn't replace completely QM, like thermodynamics doesn't implies that the Newton laws are false.  In some situations,the initial formulation is necessary.  It isn't possible at all to explain entanglement whithout both the unitary evolution and the projection axioms.  In a EPR type experiment, we even observe the correlation although the photons pass through a cristal before their polarization is measured.  The multiple random absorption-emission processes, described by the same decoherence, don't alter the outcome.

No interpretation nor formulation like decoherence has succeded in removing the logical inconsistency in QM.  At best, like in a sleight of hand, they distract from the real problem.

"Decoherence addresses the relationship between quantum and classical mechanics" is one issue out of many in a well-defined mathematically framed context. So why bother about this here? Seems to me a self-fulfilling prophecy.

Best wishes,

Leo.

In the time of Euclid, mathematics was an idealisation of physics.  At the time of Galileo, it was the same.  But today, mathematics is an independant discipline, and is constructed as an axiomatic system.  Many different axiomatic systems are possible.  If Nature is intrinsically mathematical, only one such system should describe it.

Euclid seeked a proof for his postulate, but couldn't find any.  According to the philosophy of then, it wasn't possible to simply take it as an axiom. So he said the following: I can't prove this proposition, but I am convinced that it is ""true" anyway, so I "postulate" it.  In this context, "postulate" has not the meaning of "hypothesis," but of "assumed theorem."  It was an error, the postulate of Euclid isn't "true", it is only one of the possible axioms for a complete and consistent system.

Galileo took the postulate of Euclid for granted, as if it was an intrinsic property of Nature, and his observations were compatible with it, so it kept the status of "assumed theorem." But he couldn't do better at his time, since although he studied the effect of gravitation, it was in a domain where general relativity isn't necessary to have a very good approximation. 

Even if the Universe is flat on average, we can't say that pi is intrinsic to Nature, since a physical circle ratio isn't and can't be pi.  It isn't according to the best theory, and he can't because it is impossible to falsify its value, since it is impossible to have an infinitely precise measurement.  Even if only the tausendth or the billionth decimal place is wrong, it isn't pi since it is defined with no error bar.

That's where idealisation should be considered. It works through the introduction of the concept of infinity.  Mathematics is in substance the study of infinity.  But there is none in Nature, and even if there were, anyway we can study it with only a finite number of experiments.  Also for the natural numbers we can't do without infinity, since their definition involves it automatically, which leads to paradoxes.  We can count lions, but as there can't be an infinite number of them, (we can't add 1 to 10 bucks and get 11 bucks, and so on,)  there is no perfect correspondence between mathematics and Nature.

Now this ludicrous pretension that made much too many very good minds to stray.  Supersymmetry is self-contained and self-consistent.  Most of the theorists thought that it was so beautiful that God necessarily used it.  Only little petty problem: none of its predictions are verified.  Same thing for superstrings and many more theoretical musings.  (Perhaps God hates mathematics, hates to play dices, or even doesn't exist at all.)  It isn't possible to do physics inside mathematics alone, physics is an experimental science, period.  If every system works somewhere, it is still worst since that means Nature is described by incompatible axioms.

Assuming the premisse that Nature is intrinsically mathematical, we arrive at an absudity. Have a good day.

Maybe we should first define what it precisely means when we state that "Nature is intrinsically mathematical." There is no way out in that you must define a context. In that of theoretical physics, the statement is a -- trivially! -- open door to intense discussion regarding, e.g., string theory. In that theoretical-physics context you can also discuss MUH. Most biologists do not understand a word of the statement that "Nature is intrinsically mathematical." That is, what is "Nature" in this discussion?

No folks, sorry for that but from my point of view the worm is in Stefan Gruner's statement below,

"Therefore you would have to design and conduct an observation or experiment by means of which could measure Nature's own Mathematical-ness."

I'm pretty sure you can't since we then quarrel about how to define M = "Mathematical-ness" for the rest of the time. Let me therefore define how I see the issue: Mathematical-ness is very simply our -- yes, our -- ability or, if you want, lack of ability, to describe natural phenomena by means of mathematics. What type of natural phenomena is the next question, which we cannot, and need not, answer yet. After all, mathematics is the only way of quantifying what happens in nature. Yes, sorting algorithms and formal logic may well belong to the tool kit but that is bothering about the math and not our question.

In summary, statements such as "Nature is intrinsically mathematical" are, if I may say so, intrinsically ill-defined. We first need to specify what such a statement means. I have seen no signs yet that you want to do so. Nevertheless, I have already published an answer to this fundamental question. The previous paragraph hints at the basis of the argument. If you want to continue with answering an ill-defined question, please go ahead. I wish you a lot of fun. In my opinion, that's what we all need every now and then ;-) Enjoy!

Although the question dates back quite some time, I take yesterday's gathering with a couple of other PhD students as an motivation to expound on the question myself because yesterday's discussion revolved around a similar issue.

First of all, what do you define as "nature" and what do you define as "intrinsically mathematical"? For me, not necessarily because I am a non-native speaker, these notions are in need of a definition. Since the OP tagged the question to belong to theoretical physics, it may serve as a good start to define nature as physical reality and thus only require the notion of reality to be defined assuming that physical denotes the attribute to the noun physics. Reality can be defined as the set of all objective situations and processes, I. E., as what we can observe and measure without individual altering due to perceptive bias etc. By doing so, we exclude subjective perception of mental states for instance but also several social phenomena. The latter makes the definition restrictive but we do not attempt perfection at the present.

If we agree on a definition akin to the one proposed in the previous paragraph, we run into the well-discussed situation of Wigner's article on the effectiveness of mathematics as a tool and medium in physics. So, from the practical perspective, one may use Wigner's article as a reference and thus as an evidence in the sense of scientific support of your hypothesis to get published.

However, if we take a step backwards, we may question whether it is necessary to obtain a deep answer on the question. If we think of physics as the study of mathematical models aiming to describe physical reality by experimental as well as mathematical and computer-science tools, we see that the definition allows us to employ mathematics as a tool as well. On the other hand, the result obtained theoretically can be checked by other methods as well. The result of this process is that math is objective and valid as well as reliable if one uses the correct physical concepts for the initial models and does no mistakes in the pen and paper work ;). From this perspective one may say that physics is the endeavor to examine your hypothesis by working through a lot examples. Although examples are no rigorous proof in the sense of mathematics, the established and well-tested theories of physics such as mechanics, electrodynamics, thermodynamics etc. should provide good enough evidence for your hypothesis.

I am personally skeptical that the philosophical question you posed can be answered in general. It is too general and one may attempt answering more specialized questions and clarifying the involved notions first.

Best regards

David

PS: I apologize for mistakes on the part of my Autocorrect.

Thank you David.

Here is a very basic question on the matter. If there are three stone pebbles in front of you, no matter how they are counted, they add up to three. Rules of arithmetic, used by humans, give the answer three.

Is it possible that the rules of arithmetic, in Nature, are not the same rules humans use? By the term nature I mean the physical world, the world we can do experiments on.

David is right. Let us return to the original question: Assuming the premise that Nature is intrinsically mathematical, what items of evidence can be argued in support? Leaving aside the question of whether the question as is can be answered (in plain English, makes sense), the venom is in the adverb `intrinsically´. What does that mean? Just a nasty question to start with.

Continuing, why not ask: Assuming the premise that Nature is extrinsically mathematical, what items of evidence can be argued in support? That makes more sense since reality only exists in the eye of its beholder. We can then discuss the problem of whether Nature allows a mathematical description — mind you, description => extrinsic— and I have given a decent answer, which I need not repeat here. I simply refer to Biological Cybernetics 108 (2014) 701-712.

Let us now turn to the conceptual mess we apparently disappear into:

Tegmark's MUH is: Our external physical reality is a mathematical structure.

External is good as it refers to a description of outsiders (insert can’t) but… what does a `is a mathematical structure´ mean? And, more importantly, how can we measure it? After all, physics is a *combination* of experimental reality and mathematical description and only its intense *interaction* makes physics. So, is external physical reality a mathematical structure? We can at best think it is in the sense that it allows a mathematical description, and we can have explicit arguments for that. Even better, can categorize what that means; for me, see above.

Theories that you can’t falsify are no good. You may not like it but you must face physics as is, as an interaction of experimental reality and mathematical description. And you may not like my explicit statements either but, if so, take some time to read a masterpiece and classic in the history of science, E.J. Dijksterhuis, The mechanization of the world picture ( Oxford University Press, 1961; I’ve read its Dutch original, for which he got the Dutch state prize for literature = > reading it was a true pleasure).

To finish, I am quite happy to read Steve Faulkner’s statement of today, `By the term nature I mean the physical world, the world we can do experiments on.´ I happily join him here, but that was not the problem we started with ;-)

Have fun,

Leo.

Thanks for the question, Steve and thanks to Leo for the discussion of the question whether it would not be more appropriate to use the specification "extrinsically" instead of "instrinsically" when referring to a "mathematical" property of nature.

Spoiler: I will address both points.

First of all, I want to modify the example of three peebles of stone and use 42 rocks instead simply because 42 is already a sufficiently large number for my purposes.

Even today, indigeneous tribes are known to exist on Earth which did not develop anthropologically such as to make use of a numerical system as we do as a result of scientific globalization in the majority of the world.

Suppose, a member of a tribe with the above properties finds a spot with 42 stones which block his way. Since the person is incapable of using a numerical system and thus also lacks the ability to perform arihmetic operations, he has two options:

Assuming that the situation has the property that all persons under consideration can carry two stones at once, the spoken--of member of the tribe needs two runs until he has de--blocked the way. In terms of efficiency this is obviously a bad choice if we suppose in addition that our person has quite a tight schedule.

The second option is to run back to the tribe's "headquarters" and communicate the problem. In this case he also runs into the problem that his description of the problem is inaccurate in the sense that he can simply complain about "many stones blocking [his] way".

In short, his linguistic system does not permit a specification how many tribal members are needed at most to carry away the stones in the least possible time. Under the assumption that one person can carry two stones at once, that the tribe consists of enough (>21) persons and that the time needed to get back to the tribe's headquarters and the location of stonesis negligible compared to the time needed to carry away two stones, WE see that the best possible solution is to gather 20 other tribal members.

The problem is: We can find this answer but, by constrction of the setup, the tribal member cannot.

This story shall exemplify that mathematics is a cultural development and basic arithmetic abilities are most likely to have evolved in order to describe the environment and solve quite practical problems in it. The history of geometry may serve as an additional example because it was ultilized in ancient Egyptian geodesy to avoid confusion about estate property as the aftermath of the annual Nile floodings.

A second finding concerns the capability of the tribal member to use arithmetics. From developmental psychology, it is well--known that children lack numerical abilities at very early age. Psychologically, they need to develop the concept that we can store similar things in sets. An alternative but less prominent theory states that the responsible neuronal processes still need to be enhanced by learning; Leo can comment on this better than I can.

This means that the tribal member needs to develop or make use of the handy concept of a set where by concept I do not mean the precise notion of a "set" but rather the general idea it incorporates. Indeed, this approach has the advantage of needing less pieces of information and thus leaving working capacity to the brain to do other things than worrying about stones.

Given these two findings we see that first from quite a practical perspective it was useful to develop the ability to recognize and work with arithmetic structures that can be spotted in the natural surroundings of humans.

The third finding concerns the question whether we need specifically humans for arithemtics whatsoever. Surprisingly, I think the answer is no. Psychologists have performed experiments in apes to test whether they have a sense of fairness. The experimenter handed an inequal number of -- I think -- nuts. Intelligibly for humans, the ape with the lower number of nuts became jealous and complained. Psychologists inferred from this that the hypothesis that apes have a sense of fairness is not too wrong.

We can use the experiment to be more detailed by saying that in order to spot unfairness some intuitive notion of arithmetic concepts needs to be existent also in apes.

If we use this information, we can conclude that also other beings than humans are able to use arithmetic concepts to navigate better in their environment.

This could be regarded as evidence that nature indeed has an intrinsic mathematical property in the sense that not only humans use in a less coarse way than apes of course arithmetic concepts.

On the other hand, we still need to assess what we mean when we speak of "instrinsic" as opposed to "extrinsic". Leo has fortunately done the biggest part of this job but noting correctly that we need an external observer to interact with nature, including simple tasks as counting stones or nuts.

Comparing also the previous discussion between you, Steve, and Leo, I find that there is only a superficial contradiction between your two points:

I'd suggest to think about "instrinsically mathematical" as "having the property (this is intrinsic) to be described by mathematics by an external being (this is the extrinsic part)."

Although the construction is monstrously worded, it is more precise and I think not too bad as compromise between the two points of views: On the one hand, your comments, Steve, seem to emphasize the "having the property to be described by mathematics"--part whereas Leo focuses on the "mathematics by (human) beings".

Feel free to drop a comment.

Best regards

David

PS: I recommend the book by Dijksterhuis suggested by Leo as well as Harari's a brief history of mankind to all readers of this thread.

The concept of set has been theorised very late in the human history: in the 19th century. Then the concept of number has not always been the one of the cardinal of a set. This didn't prevent people from using them efficiently, thus not everything is necessarily taken for granted.

However, the concept of cardinal of a set shows that mathematics is only about relations, a set is a particular case of relation. If there are three pebbles before me, I can only count them if I imagine a relation among them: they belong to the same set or other, and there is no reason this relation be part of Nature, since I just contrived it. Actually, we project our own mathematical images upon Nature, and think that these images are Nature itself.

This means that the tribal…

+ How true, in several senses. First, sending five sturdy men instead of all to move the stones, and keeping the rest in the village would better serve a tribe. Greek mythology offers prominent examples of raping young men while the tribe was elesewhere.

Second, a mathematical description indeed depends on the level of available mathematics. Every now and then we need to invent new mathematics. What is evident today had no need of being so a century ago. A prominent example is Newton, who invented differentiation to formulate his second law.

Physics is a important example of allowing a mathematical description of the natural phenomena it analyzes. Dijksterhuis (1961) has provided us with a brilliant description of how mechanics came about, with Newton and his three laws as the crowning achievement. Not only did Dijksterhuis analyze how mechanical notions evolved but also how the intense interaction with experiment brought them into final shape. Hence I end(ed) up with my key question as to whether Nature in general allows a mathematical description and, if so, how?

Claude:

Actually, we project our own mathematical images upon Nature, and think that these images are Nature itself.

+ Maybe you do but I don't. That is (too;) quickly said but I think your statement contains too many ill-defined notions, at least for me. Sorry but I honestly admit that I do not want to clarify them right now.

Leo.

The problem of disposing of rocks in the path is a human issue only, it has nothing to do with physics, and still less with mathematicallity of the physical reality, or what is called Nature here. It is accounting, and mathematics (including geometry) has been originally invented (or discovered) for accounting purposes. Only a human being can know which is a coin and which is not, because its brain is build to recognise objects, and not to deal with quantitative relations. It is nearly impossible for it to measure the diameter of a coin or its darkness. Now the idea of coin is manifestly projected upon Nature, there is nothing such as price in physics, that is only a human convention.

I could elaborate on how perceptions work, and why our perceptions are not at all an exact replica of the external world, but I don't feel everyone here is open to consider new things and/or from a different angle.

Leo, I see your lab has hijacked this question. Anyway, I have never seen an answer dripping of contempt like your last one. That's comic, because reading you, I have the curious impression you have read only one book. If you don't want to answer, please don't answer at all, but don't imply that the other contributors are even not worth answering them, they surely are.

Seams like the answer to my question might be in Wittgenstein. Does that mean no physics can be talked about? OH NO!!

Sure we can talk among people of good will. The issue is not how to observe "mathematicallity," this is a confusion based on ill-defined notions. It is a metaphysical issue, therefore it isn't addressed by experiment, but by reason. Metaphysical doesn't mean without any interest, because mathematics are themselves metaphysical, they can't be observed. The classical metaphysicists, Descartes and Leibniz for instance, were also mathematicians who invented new mathematics, sometimes directly inspired by their metaphysics. They were not cranks, but the greatest minds of all time.

The contempt about metaphysics among the scientists is very recent, only a half century. At the time of the development of modern physics, the physicists were often also philosophers, and their achievements are without doubt much greater than the ones of today's physicists, who survive only through controlling a territory with their pals. For example, special relativity is the realisation of some metaphysical principle of Leibniz, as Einstein admitted himself.

Now about Wittgenstein, if his ideas must be adopted, then mathematics themselves don't say anything about the world, they are only logical inferences from axioms assumed to be true. It isn't even clear whether observation can falsify the axioms, since different axiom systems can be used to describe different aspect of reality. And here "reality" must be taken in its broader meaning, including for example the idea of money that is of human relevance only. Therefore to me it seems that Nature can be seen as intrinsically mathematical only if it is studied in the engineering science perspective, that is, from the point of view of human relevance only.

Now anticipating an objection, let us examine the observability of a number. I have three pebbles before me, do I observe the arithmetic number three? Sure no, since this number is not in these pebbles, but in any set whose cardinal is three. Mathematically, three is the equivalence class of all the set among which there exist a bijection, and that have one element more than all the sets with cardinal two, and so on up to the empty set. An equivalence class can't be observed, it is a fictive relation. The rational numbers can be defined similarly as equivalence classes of pairs of included sets with cardinal N and D. The real numbers are still more abstract since we must take the limit of bigger and bigger sets, limit which is never realised. Still worst, there is not enough objects in the Universe to realise all the cardinals of finite sets.

So what? There is no contradiction in that. Mathematics has been refined is not limited to what one can observe. But the scope of mathematics was not the question but only about properties of the nature. Do I miss sth or misunderstand you?

By the way. In physics, there is a difference between entities that can communicate with one another, against those which cannot. Assuming the pebbles do communicate , by way of forces between them or because their lightcones overlap, then this would be a reason for regarding them in the same physical set, and not just in the same set because humans see them that way.

First, let me quote two remarks of Claude,

1) I could elaborate on how perceptions work, and why our perceptions are not at all an exact replica of the external world, but I don't feel everyone here is open to consider new things and/or from a different angle.

+ I have spent half of my life on answering the above question of how perception works. Accordingly, I do not feel any need to delve into it here.

2) Leo, I see your lab has hijacked this question.

+ If I can solve them, I love hijacking good questions ;-) By the way, hijacked entities keep their identity so that I don’t see anything wrong here. But to send the above remark to hell, where it can then get fried forever, see my first two, partial, replies to Steve’s original question in Biol. Cybern. 97 (2007) 1–3 and 100 (2009) 1-3, and in full detail in Biol. Cybern. 108 (2014) 701-712, which was submitted in 2013. That is, I preyed on mathematization long before my `prey´ was available.

To finish, I can only say: Sorry, folks, instead of continuing a discussion you create a cauliflower of generalizations, which I consider counterproductive. As for Wittgenstein, I prefer quoting ``What can be said, can be said clearly. And what cannot be said, keep silent about it.´´ That is what I will now do.

Take the example of the tribe. Contrary to what has been stated, it is not a question of the number of stones, but of a bijection between the set of stone and a subset of members of the tribe. An arithmetic number can't be given any absolute meaning, it is always a relation, and a bijection is a particular case of relation. Then, the number is but an invention used as an intermediary in some reasoning, just like currency is an intermediary in some transaction.

Numbers is one of the more primitive concepts, so that there is an innate notion of number. For the humans, it goes up to about hundred. Even some bird species have this notion, up to about three for the crows. It is an ability, and so resides in the brain of these animals, not in Nature.

Now, how do we do to integrate the concept of number in axiomatic and logical mathematics? That's not obvious, and is not innate at all. The three axioms of Peano are about zero and next number. With that only, it is possible to define arithmetic operations and to prove theorems about them. There is a beautiful theorem stating that the cardinal of a set is independent on the order in which its elements are counted. That's a bit more subtile than expected. Counting is building a bijection, and in this point of view, the procedure of counting has nothing evident.

Thanks for this point Claude, technically, you are right. The notion of a number can be traced back to even more, say, fundamental mathematical concepts. I won't object to this. On the other hand, I would even regard this as a support of my previous comments, or at least not contradictory to them. Mathematics has been refined and made more general throughout its history. The notion of sets and the study of maps between them as we know them today was a major achievement of 19th century and 20th century math. However, even before I think such concepts may have existed at least in an intuitive sense or subconsciously in those days mathematicians. I am not sure this is right because I cannot give evidence for my thought but as these notions can be obtained as relatively strong abstractions from the physical world, it is at minimum not fully farfetched on the plausibility level.

Mathematics is in its scope not bound to physical reality in my experience and up to my present knowledge. Still it is undoubtedly the medium and the toolkit applied mathematicians and physicists employ to model the real world whatever that may mean if I think of some conjectured theories in high-energy physics.

Steve, physics is a science in different scales. In principle it is possible that a stone emits a photon and communicates with another stone. But this is to our present knowledge not an organized but rather a disordered phenomenon. On the level of stones at rest, the concept of light cones is not false but from the condensed explanation you gave, I can unfortunately not see what the concept's benefit is for the present, as classical as possible situation we discuss, is. Perhaps you could clarify your answer?

Best regards

David

Let's see how we could define a "physical" set. If we have PV = nRT, the equation of a perfect gaz, can we say that the gaz is a set of n moles of molecules? Only if the system we chose is these n moles, and not a sub system, or a super system. We also need to have prepared the system so that is exhibits that law. For an extensive quantity like the volume, its size is the number of elementary volumes whose side has the length of a standard, measured by a comparison procedure. But the value of the whole volume is only one of the possible volumes. As for a intensive quantity, there are always two terms, and it is always measured through these two terms, so that it amount to the same. For example a manometer has a predefined surface.

Then in physics, we have to define the system under study, possibly prepare it, and assume it is isolated. Only under these conditions can we have a mathematical description. In Quantum Mechanics, we know that the notion of isolated system is dubious. Moreover, the choice of the system is of mere human point of view, it is a system we need to predict the behaviour of, either for using it, or for taking action. Only systems that have an interest for the human is prepared. Now interest, usefulness, are not part of Nature, they can't be falsified experimentally. They are abstract like mathematics. All these purely human considerations are projected upon Nature, while Nature has not the purpose of being useful, nor even understandable, by the humans.