Reading carefully the question, it raises the issues of both "analogies" and "qualitative explanations". We can have analogies in a mathematical formalism, and actually Mathematics is built around the definition of common structures, and it can be pushed at the highest level, for instance in category theories. So the use of analogies is certainly not foreign to the mathematical formalism in Physics. What is more interesting is the second part : qualitative explanation. They are based on the definition of objects with properties, which are not necessarily precise, but which can be associated with visible phenomena. For instance location, material body, solid,, force,...They are not only useful, because a theory is a construct which starts in the brain, using images put into relations with each others, they are also necessary because they are the link between the mathematical objects  and the data that we collect from experiments. In order to establish a firm foundation of QM, there has been a move towards a purely mathematical vision of physics : concepts such as location or particle are either useless, or misleading, and the only safe way would be to give credance to the mathematical objects themselves. A system would be no more than an algebra of operators. Whatever one believes about these attemps (at least they try to understand QM) the drawback is that the gap between the physical phenomena and the measure is larger than ever. This makes difficult to check the physical laws (what is an observable ?) and it reduces the building of a theory, that is a consistent set of  laws which explain what happens, to a pure mathematical endeavour.

Human needs dreams, even if they are not the exact reality. It is good to keep the dreams in check through experiments, it is bad to kill them because from dreams come intuition. A batch of data will never replace a good story.

n

Scientists that use bad analogies often haven't understood the matter at hand, even if they have worked on it for a long time.  So they think non specialists can't understand either and they explain something else more easy.  Analogies can go very far with an in depth and thorough understanding of the concept.  Often, mathematics is just a way to make things precise and quantitative, but are not necessary to know what it is about.  As an example I will give the Fermat principle, which seems very mysterious when seen through the mathematical formulation, but is a staigthforward consequence of interference.

An example of "anthropomorphizing" is the least action principle.  Nature would be lazy and spend the minimum of a currency called "action" to realize a given move.  That has been the basic of a whole philosophy, and in particular it is how the Fermat principle is mathematically formulated.  But mathematics may be misleading as different equivalent formulations can exist.  Actually, virtually any equation can be written as a variational principle, and that entails the beforhand definition of the end point, which is another "anthropomorphizing" similar to finality or fate.  Seen as an interference effect, it could have led to quantum mechanics, and beside the lack of experimental necessity, the reason why it wasn't discovered is because the least action principle is so "anthromorphically" and mathematically appealing.  That shows that not only analogies possess this deficency, and they have very often led to important discoveries.

I don't thing Kant said exactly that.  There is much mathematics in string theory, and in many other theories that have been forgotten since, but there is not a fig of truth in them.  The "reine Vernunft" is what it is: pure.  Mathematics is about truth, but not physics.

In a nutshell, Kant said that our knowledge can't surpass the limits of experiment.  That means we can't get any truth about the real world with mathematics alone.  Physics can only describe and predict reality, since experiment is necessarily finite.  Mathematics is an idealization that deals with infinites.

To understand electric circuits, the analogy of hydrodynamics can be used.  That's because both obey the same set of equation.  Then physics can't be reduced to mathematics.  The reason is the description may be incomplete, and the only way to know that is experiment.

Reading carefully the question, it raises the issues of both "analogies" and "qualitative explanations". We can have analogies in a mathematical formalism, and actually Mathematics is built around the definition of common structures, and it can be pushed at the highest level, for instance in category theories. So the use of analogies is certainly not foreign to the mathematical formalism in Physics. What is more interesting is the second part : qualitative explanation. They are based on the definition of objects with properties, which are not necessarily precise, but which can be associated with visible phenomena. For instance location, material body, solid,, force,...They are not only useful, because a theory is a construct which starts in the brain, using images put into relations with each others, they are also necessary because they are the link between the mathematical objects  and the data that we collect from experiments. In order to establish a firm foundation of QM, there has been a move towards a purely mathematical vision of physics : concepts such as location or particle are either useless, or misleading, and the only safe way would be to give credance to the mathematical objects themselves. A system would be no more than an algebra of operators. Whatever one believes about these attemps (at least they try to understand QM) the drawback is that the gap between the physical phenomena and the measure is larger than ever. This makes difficult to check the physical laws (what is an observable ?) and it reduces the building of a theory, that is a consistent set of  laws which explain what happens, to a pure mathematical endeavour.

Human needs dreams, even if they are not the exact reality. It is good to keep the dreams in check through experiments, it is bad to kill them because from dreams come intuition. A batch of data will never replace a good story.

n

Dear Mohamed, may I ask that you elaborate a bit on the concepts in your paper? Or give a good reference to a compact but good introduction to the applicable ideas? I have but a basic idea of category theory and am now very interested in what you mean by "fundamental monads". (Ironically, this request of mine is indeed a request for a qualitative explanation of a mathematical concept/entity.)

Naturlich Mohamed, aber All lust will Ewigkeit.

Thank you SOOO much Mohamed. That is quite a mouth ful, so I will come back to you as soon as I worked through the material. These concepts and especially those pertaining to my quesion are sideline endeavors, but ones that I pursue enthusiastically

Would have formulated Einstein the General Relativity, if he were not thinking about acceleration in the elevator? Could have Newton understand gravity, if he had not made the analogy between the Moon and apple? As soon as experience covers with formulas, begins to seem that the result is the reason. There is something to think about, is not it?

I think the use of analogies for explaining difficult topics and concepts to an uneducated public is a very dangerous endeavor. I was not convinced by the attempts to explain the Higgs mechanism when the Nobel prize to Higgs and Englert was announced/given.  On the other hand, I have not found or invented anything better...

But if two phenomena, regardless how different, map onto the same mathematical model -- f.i., the same differential equation(s), your knowledge and intuition about one of the phenomena may be used to understand the other, through analogy, in considerable depth. Which is why I always recommend anyone wanting to understand applied mathematics to give priority to physics (or a similar science) before pure mathematics. Because the former helps you build a database of analogue situations in your brain, the latter makes you worry about (something like) your solution being Lebesgue integrable!

I demonstrated an example of this in a recent thread on ResearchGate (see the link). Not that anyone seemed to care about, or appreciate, my postings...

https://www.researchgate.net/post/Can_anyone_help_me_to_solve_the_non-linear_ordinary_differential_equation_attached_herewith_by_using_SLM-Successive_Linearisation_Method

[4]  arXiv:1504.06401 [pdf]The Benefit of Seeking a General Rule for Learning from Analogy

Eric Kuo, Carl E. Wieman

Subjects: Physics Education (physics.ed-ph)

It is common practice to teach physical concepts with an analogy. However, how such analogical instruction should be structured is not well studied. We compare two ways of using an analogy to learn a target physical principle: (i) applying the analogy to the target physical context on a Case-by-Case basis or (ii) using the analogy to create a General Rule in the target physical context. In the discussion sections of a large, introductory physics course (N = 231), students who sought a General Rule were better able to discover and apply a correct physics principle, gaining 15% more than Case-by-Case students from pre-test to a mid-test immediately after the discussion section. Later, both groups gained the same amount from subsequent direct instruction on that principle. The explanation is that students who performed Case-by-Case analyses are more likely to focus on idiosyncratic problem-specific features rather than the deep structural features that can be used to answer such problems in general.

Tomas> to convince them, it's probably better to be vague than rigorous and boring

vague and enthusiastic!

There is the side effect that a subset of people believe they fully understand the topic (in particular if they have learned a little bit of physics), and start their own very ambitious research projects. You can find reports from many such projects on these Q&A threads.

PS. Do we see you in Trondheim soon? 

Dear Kare Olaussen. As a person who is enthusiastic about education I am a bit skeptical of your approach. For vague and enthusiastic accounts there is Discovery Channel and National Geographic and other popular science programs and magazines which, quite frankly, infuriate me. There are many analogies/comparisons with limitations; and I believe that, with teaching people these comparisons, one owes the students/audience to be informed of the limitations.

I can easily profess to know something about string theory and tell a story at a bar about how the universe is really like a song being played on a guitar... but I am quite indignant toward professors who do that to their freshmen.

Petrus> ... bit skeptical of your approach

What did you interprete to be my approach?

Einstein is considered one of the most perceptive scientists in history.

His breakthroughs, and there were many, usually began with analogies and/or qualitative ideas, i.e. conceptual analysis. Then, and only then, did he find the mathematics needed to express his idea.

Here's a direct quotation from AE: "The intuitive mind is a sacred gift and the rational mind is a faithful servant. We have created a society that honors the servant and has forgotten the gift."

Perhaps that is why theoretical has come up so empty-handed after many decades of emphasis on abstract mathematical model-building and persistent ignorance of the important role that analogies, intuition and conceptual analysis play in understanding the real physical world.

Thank you Tomas. Apologies Kåre for misinterpreting.

Hi. I know this is not the sort of thing you were expecting, or I guess, but the calculations in many approaches to quantum gravity involve an analogy: that is the space-time continuum. In other subjects in theoretical physics of course this works extremely well, however the problem seems to be that it works so well, that when approaching quantum gravity, even though a physically meaningful model of space-time is needed there, we kind of forget that the continuum (as a concept underlying calculus, analysis) was only meant to be an analogy in the first place, and that's arguably why QG has come to an impasse, or so some of the experts seem to be implying, and I find this interesting. (This is one of the reasons for non-commutative geometry becoming more and more popular amongst physicists.)

Petrus Johannes, you can go as far as you want with analogies. But it has to be tested afterwards. Gravity can be written in the same form as electromagnetism, where the motion of a mass creates a magnetic gravity field in addition to the Newtonian gravity. This is called gravitomagnetism. The Gravity Probe B experiment, a satellite containing gyroscopes, has confirmed the correctness of this analogy by measuring the magnetic gravity effect of the spinning Earth upon the torque of the gyroscopes.

To Thierry,

I am interested by your remark. When one treats the gravitational field as any other field in a Yang-Mills like model, using the fiber bundle and connections formalism (which is similar to the well known tetrad method) there is no need to derive the gravitational field from the metric (the metric is a byproduct of the tetrad) and to limit ourselves to the Lévi-Civita connection. The computations are both more general and much simpler than usual. In this the key variable is the Riemann tensor (analog to the strength of the field), and it shows that the gravitational field has a transversal and a rotational component, in a decomposition similar to the polarization of the EM field (magnetic and electric field). This is observer dependant, in that the "time vector" plays a crucial role in the decomposition, but the time vectors of all observers have the same constraint. So, in some way,the transversal part of the gravitational field would be the component which has a cosmological meaning (the motor which powers the expanding universe) and the rotational part is actually the gravity as we feel it (in the space slice), and this would explain the discrepancy of strength between the usual gravity and the other force fields.

So, are there any reference to the gravitomagnetism ?

To Jean-Claude: Thanks for commenting. When we get back to Petrus' question, my point initially was that analogies can be found valid if they hold after checking. But Petrus' question even went further. Like he probably suggested when I read between the lines: if the qualitative explanation of a theory, possibly by some analogy, make us raise eyebrows, probably the theory itself doesn't make any sense. Several adopted theories in the domains of gravity, elementary particles and cosmology are poorly credible, abnormally difficult to explain either with or without analogies, and they are questionable, if not already falsified by some unnoticed researcher.

The success of the simple analogy between electromagnetism and gravitomagnetism (suggested by O. Heaviside in 1893 and further developed by O. Jefimenko) shows that the universe is not as tricky as theoretical physicist like to present it. It also gives us a great guidance for further developments.

When gravitomagnetism is applied to the shape of SN1987A, to non-exploding fast spinning stars, to Saturn's rings, or to asteroids' distribution statistics, it immediately explains the phenomenas.

This shows that it is sometimes a useless waste of time to invent new theories that, with great detours, still come out on the analogy when it comes to hard facts.

Thanks Thierry,

I fully concour with you. Physics should not be the result of contrived theories. I checked a bit on Wiki about gravitomagnetism (that I did not know under this name),and it is a property of the field created by rotating bodies. What I pointed about is that, in a non exotic formulation (with fiber bundles it is quite easy) one finds that the Riemann tensor has more physical meaning that it is usually perceived, and, because this is a 2 form valued in a fixed vector space, it admits the same decomposition in an "electric" part (which involves the time components) and a "magnetic" part which involves the other components, that is the ones that we use in the usual space. And we have actually the same decomposition inthe scalar curvature (this is not obvious with the usual formalism, but it is there nevertheless). So I wonder what physical meaning this decomposition may have.

Robert, this quotation is fake:

http://www.quora.com/Did-Einstein-ever-say-%E2%80%9CThe-intuitive-mind-is-a-sacred-gift-and-the-rational-mind-is-a-faithful-servant-We-have-created-a-society-that-honors-the-servant-and-has-forgotten-the-gift-%E2%80%9D-and-if-so-what-did-he-mean

Based on the opinion of the medieval scholar given at that link, one might conclude that the first sentence sounds like Einstein, but it has not been verified (or shown to be a fake). The second sentence does seem to be generated by Sample (1976) rather than Einstein.

So here is another quotation that is widely attributed to Einstein and appears to be from a book he co-wrote with Infeld.

"Books on physics are full of complicated mathematical formulas, but thought and ideas ate the beginning of every physical theory."

Cheers!

To Jean-Claude: what you desribe unfortunately depicts the problem of actual schools regarding general relativity and particle physics. They play with lots of maths, and afterwards they try to give of physical meaning to the results, hoping so to predict something.

But I say: observe the physical facts and find the most easy and evident explanation to it. Then only, put it into maths. Or try out an analogy. There is still time to try predicting something more afterwards.

If the general relativity cannot explain the behavior of supernovas, spinning stars, disc galaxies or asteroids, in contrast to gravitomagnetism, then what is its use but a stroll in Einstein's land of fantasy?

To Thierry,

I do not disagree with you. But the usual GR formalism is actually closer to what you describe.  Why do we need to give a privileged role to the metric ? Why choose the Lévi-Civita connection among many possible others ? Why choose the scalar curvature as key variable ? The formalism through fiber bundles and connection is more natural, it is similar to the one used for the other fields (such that EM) and so closer to a physical explanation, it is more general (we do not have to choose one connection), it is fully compatible with the usual resutls (the choice of the Lévi-Civita and the scalar curvature looks then arbitrary, but is possible).  And it shows some troubling analogies (which are after all, the topic of this thread). And the traditional formalism is not fully checked (dark matter isuue).

To Jean-Claude: you write: "But the usual GR formalism is actually closer to what you describe."  Well, I don't understand it like that. Very practical issues are precisely explained by the gravitomagnetic analogy, such as the dark matter issue in disc galaxies (!), why prograde spirally wound disk galaxies are older than their windings would presume, why fast spinning stars don't explode, why Saturn's rings are so thin, how can a fast spinning star appear as if it were 10000 times more massive, and so on.

Isn't it a gigantic progress for science that we can explain such practical issues by the action of gravity?

Thierry,

I certainly agree with you, and I do not hold anything against GR and even its usual formalism. What I say is that with a more modern, that using mathematical tools which did not exist a century ago, we can do better, not less. So far, in my research, in do not see anything that contradicts the usual results of GR, but one can see more. And clearly the explanation with dark matter does not satisfy me. It looks like the Higgs boson. When there is trouble in the theory, invent a new object,it is good for the media, but we can do better.

Like many important tools invented by humanity, be they material or purely mental, analogy can be useful or destructive, depending on circumstances. One of the most beautiful examples of how analogy could help to make fundamental discoveries, is the theoretical work by Sadi Carnot (exposed in his book "Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance (Paris, 1824), in which he ibntroduced the Second Law of Thermodynamics. His understanding of thermal phenomena was based on the analogy with electricity as it was known in his time. There were two "fluids", hot and cold, like the positive and negative electric  charges; the notions of capacity, conductivity and resistance followed the same mathematical schemes and formulae. There was one special feature in the heat transfer, namely its irreversibility, which led Carnot to the Second Law and the notion of entropy.It is amazing how far he could get with the analogy which turned out to be completely erroneous (there is no such thing as "hot fluid", or "cold fluid", and the heat turned out  to be of statistical nature). As another deep and useful analogy one can cite Einstein's approach to gravity, inspired by Maxwell's Electrodynamics (similarity between the relation between the field and its sources). On the other hand, sometimes analogies may lead to blind alleys, too. Not withstanding how attractive the analogy between electricity and magnetism may be, it seems that magnetic monopoles do not exist, and the non-negligible effort for their mathematical description and experimental search could have been used for more promising purpose. Nevertheless - and this is perhaps the greatest miracle of science - even this seemingly fruitless research gave rise to very interesting developments in mathematics and in experimental physics; in this sense, nothing is lost.

     As for the second part of the question, what is the use of analogy in popularizing scientific results among people who are interested and would like to understand what is going on in science, but do not have adequate scientific background, I would be rather cautious. But I don't see any alternative - without analogies using everyday experience,  there is little hope to transmit any valuable information. Suppose you have no musical instrument at hand, but you would like nevertheless to explain to someone who has never heard of classical music, why we admire the music of Bach, Mozart, Beethoven  and other great composers. In such a situation, I would do my best to whistle the most beautiful melodies I remember - and it would be better than nothing. This is the analogy with what we try to do explaining science to laymen as well as we can!

Richard> But I don't see any alternative

Sometimes the alternative is simply to refuse to popularize!

I have accepted the challenge to talk about the possibility of extra dimensions  to a broad audience (some of which apparently believed in some related predictions from the Maya calendar), because there you at least can makes everyday speculations and analogies, with ample references to Edwin Abbots beautiful novel about Flatland. But I have refused to give a 15 minute explanation of supersymmetry --- it is almost impossible to explain fermions and bosons in 15 minutes.

Hi Kåre, thank you for the comment, which made me think over again what I wrote. Perhaps trying to convince someone of Beethoven's or Bach's greatness just by whistling some of their melodies could be finally more misleading than useful (but I am more optimistic with Mozart, whose melodies can keep the charm even when performed by ordinary people). I totally agree with you concerning the supersymmetry, fermions,  bosons and quantum statistics in general - these topics need a solid mathematical preparation, and trying to explain them via "hand-waving" could be completely misleading!