Why lionize mathematics in science/engineering?

Hi Biswajit,

0. It's a pleasure receiving thoughtful comments--regardless of whether one agrees completely with them or not.

 1. About my own post. On second thoughts, I think I have put in too many points in a single post. This has made the writing meander through several sub-threads. But then, it's just a post on a blog--not even a Web article... I believe that unlike a tightly written article or a highly structured debate, blog replies should try to retain that quality of fluidity of exchanges which can so naturally occur in real-life conversations.

2. The points of contention you pick aren't the ones I would pick as my core concerns. I suppose this much is clear from my post.

For example, I write about the primacy of induction and its applicability in mathematics, something which none of your summary points includes or even hints at.

 3. The main issue in this thread is, to repeat, the practice of lionizing deduction and mathematics at the expense of induction and physics. The closely related issues are: To leave mathematics unexplained i.e. unconnected to reality (in teaching). To elevate symbolic manipulations over reality (in evaluation of theories/approaches). Etc.

Now, all these are deep cultural issues; each has an immediate philosophical import. These issues are not so superficial as, e.g., the change of some priorities in science management. The issues here are those related to the *culture* of science, of ideas, and on a historic scale. As other examples of the philosophic points closely related to what I make here, please see: http://www.capmag.com/article.asp?ID=4823 and http://www.capmag.com/article.asp?ID=4818

Let me emphasize: The idea is not to take away the convenience that individuals comprising a research community might feel in their communications to each other. The issue is deeper. It is: why no such individual ever bothers to spell out the physical connections that the abstractions he uses has? Or is it that it's all talk in the thin air? Isn't it so much more easy to actually spell out the requested connection than to argue why the request is unreasonable?

4. There would be n number of ways of lionizing deductions at the expense of inductions.

Increasing the number of papers is just one minor way to do it--which, please note, I do not mention very prominently in the above post. From your today's post and activities at iMechanica, I guess, you were simply catching up, and in the process, imported it from another post of mine.

In a way, yes, I do suppose increasing the sheer # of papers could be one way of lionizing deductions.

But far more importantly, a culture that had a misplaced respect for deductions won't be able to make out when someone was simply spinning ideas--and not doing any serious or original research. This is a very serious point. I gather this is what David Harriman has been emphasizing too, in relation to the Bogdanov brothers controversy. See: http://www.capmag.com/article.asp?ID=3147

5. There are many other ways to lionize deductions that might strike one. Let me jot down some examples....

Simply appearing abstruse enough to generate a favorable impression about oneself. Here, elevating deductions above inductive generalizations can be both the means (or the mental method involved) as well as the end (or the purpose) to be achieved in reality. So, this could be one way to "lionize...".

Not being willing to consider physical arguments so long as these have not been transformed into the form of mathematical equations is another way.

Not recognizing an actually new physical theory, by incorrectly branding it as just a new interpretation of the same old mathematics, is still another. Let me give an example. According to this method of lionizing mathematics, the paradox of the wave-particle duality had already been resolved in the 1920s and 30, and so, the quantum riddles that you and I were perplexed over since high-school days were all non-issues. If you don't believe this, check out the Nobel prize's official Web site--the simplified stories written for the layman. So, that's another way of "lionizing...".

The willingness to accept, absorb or internalize any symbolic representation even if its physical meaning is unclear, just for the deductive pleasure of it, without giving any thought to the purpose of such mental activities, is yet another way.

Giving better recommendation letters for admission to a better-ranked US university to that student who can better spin out deductions is still another.

Providing tenure to a "spin-doctor" is still another...

The list can get far longer... You are welcome to add to this list too... One doesn't have to be creative here--just being a little observant is enough.

6. About 1D/2D/3D.

In general terms, the issue here is similar to invoking the assumptions of linearity, conservation/path-independence, infinity of extent, etc.... These are all assumptions which can only be justified by an appeal to the point you make, viz., that they let us have a qualitative understanding.

But then, one wonders why the support for a qualitative understanding is not so strong for those theories that have not been put in the form of symbolic mathematics or equations.

The main reason I picked out the example of micromechanical modeling (I could have picked out anything in analytical procedures for stress analysis) is that in stress theory, not only does the qualitative nature of the solution change in a major way between 1D, 2D and 3D, but also that the solution in 2D is *less* conservative--i.e. is potentially more dangerous--from the design point of view. Further, notice, crack propagation is often a catastrophic event.

So, the argument you present is, at best, valid only half-way through. In particular: The 2D models are useful in suggesting new ways of materials design, but they are dangerous when it comes to mechanical design.

In general, the "qualitative help" argument has far better validity in the context of *vector* fields. But the stress/strain tensor fields, by definition, carry tight coupling of differential terms across the spatial axes. This is quite unlike the usual vetor fields. Hence, in stress analysis, a 2D result for a 3D problem is comparatively far less valuable--and much more dangerous.

Now that this is the nature of the problem, would the research community deliberate in any significant way over this matter? Does one easily run into such deliberations as easily as one runs into yet another minor variation of a micro-mechanical model? In 15+ years of my watching research activities in this field, I have not. This habit is typical. As noted elsewhere--probably by Hardy--mathematicians, with their "pride," are prone to simply stop talking about the incovenient problems they cannot solve, rather than admit the limitation forthright in a sporty spirit and go ahead. Though not necessary, people invest sentiments in mathematical limitations the way they would never imagine in physics. None takes it a personal assault if light does have a speed limit. Most mathematicians would take offence if enough tact does not go into pointing out their inability to solve a class of problems. Why? Because mathematical concepts at core are concepts of method, that's why! It's funny, but despite knowing the origin, people continue to take offence! (Mathematics! LOL!!)

7. Regarding Galerkin's method.

I spoke of the *extended* version of it. (See the post.) I fail to understand how come comments such as what you have made can at all come up!

Of course, it's true that in the context of reversible linear isotropic elasticity (and I can't be sure if I didn't miss an applicable qualification or two), there *does* exist that energy interpretation. But is the existence of such an interpretation a cause for celebration? Are physical interpretations so rare in mechanics, or in its exposition, that specific books--from amongst hundreds--have still to be singled out? Doesn't then simply reinforce the point I am trying to make here?

And, then, given the physical interpretation of Galerkin's method (these days available in most any book on FEM), two questions immediately strike me--one very obvious, and the other one, not so very obvious.

The obvious question is: What *physics* does correspond to the extended version of that method?

Corollory: If we don't understand the physics of it, how can we assert the viability of the technique for its *application* to *physical* problems?

Now, this is a gross error--so gross, in fact, that one would be actually baffled if one were asked to identify the precise epistemological nature of the error.

So, let me give just an indicative example. The entire domain of business accounting does not involve any mathematics beyond what is covered in high-school calculus. The most advanced mathematical issue in the core of accounting, arguably, is: continous compounding of interest. Does it then mean that every science-stream high-school graduate qualifies to be an accountant? Forget whether the law of the land allows this or not. The real issue here is: Does every science-stream high-school graduate *know* accountancy simply because he has known the related mathematics? And if not, why should we take the extended Galerkin method to be at all applicable to, say, a nonlinear problem? To a fluid dynamical problem? To a fracture mechanical problem? Et cetera?

Let me add: I have no issue if people want to try out mathematics ahead of either identification of physics of it, or of applications.

If *some* people find it easier to "play around the equations" the way Dirac did, that's their habit or their idiosyncrasy of developing science. But note, such "play" cannot be taken as the final content of science....

So, I do have an issue if a purely mathematical method is simply pushed as worthy of solving physical problems, ahead of any careful identification of its physical nature. But this is precisely what FEM community routinely does, for generalized WR (i.e. Galerkin) methods. I have issue with *that* practice of the FEM community. And I have even a more serious, philosophical, issue if playing around equations is being elevated above physical discovery--the way often it does get portrayed in culture of science (not just popular but practising culture).

No matter how much it hurts, FEM community ought to stop and think about it. Believe me, identifying physics of the extended Galerkin (or WR or any other similar mathematics) would only have helped the FEM community by now--not hurt it.

If someone has good points to show that identifying physics hurts mathematics, let him at least say so in a reply post here!!

Now, a little about the non-obvious question: Here are two methods. One is mathematically more challenging and sophisticated--but its physics is not known, or, even if known, has never been published. Then, there is another method. It is mathematically much more simple. Its correspondence to the basic physics involved in the situation is also very straight-forward to see. Yet, none even thinks of applying it for any of the solid mechanis problems. Why?

In case you didn't guess it so far, the second method, obviously, is FDM.

Why does the research and application community have such a mental block in simply using it in solid mechanics?

Isn't it because FDM fails the "lionization of deductions and mathematics at the expense of induction and physics" test? Think about it. If you do, you will realize that Zienkiwicz's answer in favor of FEM falls short. (I picked his name simply as an example. Almost anyone else of his generation would do for the purpose. It's *not* bad that they pursued development of FEM. It *is* bad that FEM's advantages and FDM's disadvantages, are blown out of proportion.)

8. In view of the generally prevalent prejudices, it might be restated that my post (or position) does not take something away from mathematics.

Now, since you mention manifolds, let me bring in another post here. Let us conduct a little experiment: I will give you (here, "you," is used only as a matter of speech) a class of one of the most talented Indian undergraduates of engineering in India, and as much as time as is reasonable--say 5 lectures of 2 clock-hours each. The students would be completely new to Solid Mechanics. You select which half of this student population you will teach. The other half will come to me. You will teach the vector bundles over manifolds etc. approach to tensor fields and solid mechanics, perhaps also covering nonlinearities etc. I will teach the other half of the class the standard topics in the mainstream engineering curricula (for example, following Shames book, or the suggestions in my post about an UG course on SM). At the end of the course, let us see if it is your students or mine who can more easily learn, on their own, certain *untaught* topics such as stress concentration (or even, stresses in plates/shells--a topic I myself still have not had the occasion to study). Let us see if the students fare better following the first approach or the second.

I am sure that just conducting this experiment in your thoughts would be enough for you to know that the vector bundles, manifolds, etc. aren't very deep or fundamental concepts at all--despite the skillful and intelligent way in which they are made out to be so. Now, that's another example of lionization of deductive mathematics at the expense of induction. Got it?

As to dimensionality and all. It's a separate debate. Let me just note that I believe that the entire physics ought to be presented with only 3 (spatial) dimensions, full-stop. (It can be done. It avoids confusions. It makes things easy. Really. And yes, the entirety of physics can be covered that way.)

9. Before closing, just an observation, because given the primary audience here, even an observation ought to suffice, I suppose; a full-fledged argument might not be necessary.

The observation is that it is pointless setting up the strawman that I am against mathematics, and then start beating it.

10. Of course, I value precision, rigor, definite-ness, etc. of the things mathematical--apart from the potential application range of the subject matter.

But even here, I seem to have a certain viewpoint that has not found adequate expression anytime in the entire history of ideas or of mathematics.

To broach this issue, just think why is it that "the right mathematics makes systematic thought about complicated physical processes possible"? Why can such a thing at all happen? Why does it need the *right* mathematics? What are the standards by which one can judge a certain mathematical method to be the "right" one? Any idea?....

This one has already become a very long reply, and so I will just provide a hint here (may be write another post sometime later on): By my viewpoint, the reason for the "why" mathemtics works lies *not* so much in mathematics itself but primarily in the inductive processes at the definitional stage of physics (or of in any other applications realm).

Thus, at least some of the credit--and this is the historically un-acknowledged part--which is routinely ascribed to mathematics actually ought to go to sciences other than mathematics.

The famous "unreasoable effectiveness" of mathematics in physics is not even difficult to reason about let alone be unreasonable. The effectiveness comes about, not because mathematics possesses some mystical powers to impart clarity to thought no matter what be the physical realm of application, but because the very phenomena being selected for study in physics themselves get selected in such a way that an existing quantitative analysis framework (or, more accurately, epistemologically, a measurement method) would be found in correspondence to it. Both isolation of physics concepts and the basics of the attendent mathematics get formulated simultaneously. That's why mathematics is so effective.

Let me not keep this abstract description floating in thin air as is the modern trends in science. Instead, let me actually give an understandable and pertinent example. Take care to remember of what it is an example.

Fluid mechanics isolated the continuity equation but had no honest idea about turbulence for a long time. For all such times, it was the simpler linear theory (ideal fluid flow) which was being hailed as the victory of the viewpoint that mathematics rules--that mathematics is the queen of sciences. "Hail abstractions and deductions!", so to speak. The issue of both viscosity and turbulence was being slipped under the carpet all this time. (John von Neumann has famously called it the study of "dry water.")

But once it became possible to quantitatively address turbulence, papers strictly understandable only to the specialist began getting published. And again, it is being made out as if mathematics rules--that it is the queen of sciences. Blah, blah, blah... "Hail abstractions and deductions!"

If deduction were fundamental, why did it take so long for the "research community" to figure out how turbulence arises? After all, Navier-Stokes equations had been with all the "deductive" mathematicians and mechanicians for more than a century! Why did it then take that long?

11. Ugly, is it not? Such a culture? Taking all the credit that is properly due itself, and then going out and grabbing as much as possible from other fields too? (And if not ugly, what word would precisely fit here?)

12. Now, a bit personal. The best part about Biswajit's post is not what he wrote, but the simple fact *that* he wrote--despite the subject matter being what it is! Thanks, Biswajit! Do feel free to add, but also, please do try to understand the philosophic matters I address too--without which, the exchange can only grow infinitely long.