iMechanica - Comments for "Why lionize mathematics in science/engineering?"
https://www.imechanica.org/node/1440
Comments for "Why lionize mathematics in science/engineering?"enRe: induction vs. deduction
https://www.imechanica.org/comment/2997#comment-2997
<a id="comment-2997"></a>
<p><em>In reply to <a href="https://www.imechanica.org/comment/2978#comment-2978">Induction vs Deduction</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Hi Temesgen,
</p>
<p>
Wow!
</p>
<p>
I have enjoyed reading Professor Morris Kline's history of mathematics book too, but that was some 15 years back, so I don't really remember any specifics of it. All I remember is that I had liked his writing style (which was very simple and clear) and that the first chapter was more philosophical and so valuable from the induction-deduction viewpoints.
</p>
<p>
... The reason I said "wow" is that most people, when I mention induction vs. deduction, and in reference to *anything* to do with mathematics, go like: "Yeah! The method of induction is great! I always loved it too!! You take n = 1, n = 2, n = 3..." So, you can see why I would so appreciate the kind of comment you have made! :) ... Nevertheless, some clarifications are in order....
</p>
<p>
The issue is not superiority--the issue is primacy. Of course, deduction is useful too. But the thing is that any deductive inference can be reduced in the form of the basic Aristotlean syllogism, and, if you want to avoid infinite regress in a syllogism, as Aristotle pointed out, you have no option but to define your starting terms using a process of induction.
</p>
<p>
So, it is induction, and not deduction, which is actually the process of generalization.
</p>
<p>
But I yes, I do want to emphasize my agreement that deduction is very *useful*. The best example of deduction in physics/engg. I know of is the prediction of EM waves from Maxwell's theory.
</p>
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</ul>Wed, 30 May 2007 08:23:20 +0000Ajit R. Jadhavcomment 2997 at https://www.imechanica.orgBut, Zhigang, science is not a "story" (of fiction), is it?
https://www.imechanica.org/comment/2996#comment-2996
<a id="comment-2996"></a>
<p><em>In reply to <a href="https://www.imechanica.org/comment/2959#comment-2959">Multiple versions of a story</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear Zhigang,
</p>
<p>
The theory of inductive roots of mathematics is not at all well developed (to my knowledge). I won't be surprised if some of the points I make or cite here are later found to be original ones.
</p>
<p>
If, after 100 years of existence of a theory, good books to explain all its aspects cannot still be had or readily cited without attendent arguments--even if the theory had become the dominant mainstream theory--does the fault then lie with the the kind of genius mankind possesses for learning, or does it rather lie with the theory itself? (This was re. relativity theory.)
</p>
<p>
English, and to the extent I know about it, Chinese, are languages. But mathematics is not. Mathematics has a certain content the way languages don't--it has a definite subject matter, a definite body of *facts* that are organized as and according to certain principles. In contrast, Shakespeare's works, for example, are only *expressed* using the English language. But the English literature, even if taken in its totality, cannot still be identified as the *content* of the English language. So, when people say that mathematics is a language, it is wrong.
</p>
<p>
But giving them some benefit of doubt, it can be taken to mean that mathematics has a special *notation*. Even if the mathematical *notation* itself is considered to be a separate "language," it is a highly restrictive form of a language--it only allows quantitative relations to be expressed, no other. (The restriction actually gives the mathematical notation a kind of power that is not possible while using languages.)
</p>
<p>
But I whole-heartedly agree with the your last point, namely, that a richer variety of pedagogical devices, tools, and technologies are so much for the better.
</p>
<p>
Thanks for your kind comments. ... On a personal note, I wonder how my name would be written using Chinese alphabets.
</p>
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</ul>Wed, 30 May 2007 08:06:28 +0000Ajit R. Jadhavcomment 2996 at https://www.imechanica.orgInduction vs Deduction
https://www.imechanica.org/comment/2978#comment-2978
<a id="comment-2978"></a>
<p><em>In reply to <a href="https://www.imechanica.org/node/1440">Why lionize mathematics in science/engineering?</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Hi Ajit,
</p>
<p>
A lot of interesting issues are mentioned in your blog. Here I will simply give my opinion on induction vs deduction.
</p>
<p>
I<br />
think both methods of reasoning are very important and I don't think we<br />
can take one to be superior over the other. What is debatable is the<br />
order of which comes first. I share Kliein's opinion (author of 'why<br />
johnny can't add') that induction is the natural way of attacking new<br />
problems and deduction comes later. You need to used induction to get a<br />
feeling and deduction to come up with generalizations.
</p>
<p>
Cheers
</p>
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</ul>Tue, 29 May 2007 11:22:38 +0000Temesgen Markoscomment 2978 at https://www.imechanica.orgMultiple versions of a story
https://www.imechanica.org/comment/2959#comment-2959
<a id="comment-2959"></a>
<p><em>In reply to <a href="https://www.imechanica.org/comment/2952#comment-2952">Re: Why lionize mathematics</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>Dear Ajit and Biswajit: </p>
<p>
Thank you both for discussing this topic. I myself never really understand why mathematics can be so effective to express some thoughts, but so ineffective to express some other thoughts.
</p>
<p>
For the last few days I'm giving the theory of relativity one more try. The subject has been fascinating to me for a long time, and there might be a chance that I can use the theroy in a serious way. I have tried quite a few nonmathematical books, including Einstein's own popular writings. None really has helped me understand anything of substance.
</p>
<p>
This time around I am trying an undergraduate <a href="http://www.amazon.com/Relativity-Special-Cosmological-Wolfgang-Rindler/dp/0198567324/ref=pd_bbs_sr_1/104-8492820-7134325?ie=UTF8&s=books&qid=1180308419&sr=1-1">textbook by Wolfgang Rindler</a>. The subject suddenly becomes reasonable to me. Even in Rindler's book, however, I find many passages of nonmathematical explanations superfluous, confusing and distracting.
</p>
<p>
After learning one thing or another for most of my life, and teaching for nearly 20 years, I have gained some empirical knowledge of learning, but still don't understand much about the theory of learning. Many people would like to find an efficient way of learning, have one version of truth, and remove redundancies. In writing a paper or a book, the author tries to teach other people something. But people are different, and even the same person is different from one moment to another. It is often impossible for one version of a story to be appreciated by all people at all times. Thus, it is a good idea to tell the same story in multiple versions, some in English, some in Chinese, some in mathematics, some in cartoons, and yet some in videos.
</p>
<p>
Now that the Internet has made storing and distributing all these versions nearly free, we might as well encourage all modes of expression. Let an individual learner and her mood of the moment decide which version to learn from.
</p>
<p>
Perhaps tomorrow I'll find Rindler's nonmathematical passages lucid. Perhaps not. But that is beside the point. He clearly enjoys telling the story, and I have so many sources to learn from.
</p>
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</ul>Sun, 27 May 2007 23:34:24 +0000Zhigang Suocomment 2959 at https://www.imechanica.orgReply: Lionization of mathematics and Biswajit's points
https://www.imechanica.org/comment/2954#comment-2954
<a id="comment-2954"></a>
<p><em>In reply to <a href="https://www.imechanica.org/node/1440">Why lionize mathematics in science/engineering?</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Hi Biswajit,
</p>
<p>
0. It's a pleasure receiving thoughtful comments--regardless of whether one agrees completely with them or not.
</p>
<p>
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.
</p>
<p>
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.
</p>
<p>
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.
</p>
<p>
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.
</p>
<p>
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: <a href="http://www.capmag.com/article.asp?ID=4823">http://www.capmag.com/article.asp?ID=4823</a> and <a href="http://www.capmag.com/article.asp?ID=4818">http://www.capmag.com/article.asp?ID=4818</a>
</p>
<p>
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?
</p>
<p>
4. There would be n number of ways of lionizing deductions at the expense of inductions.
</p>
<p>
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.
</p>
<p>
In a way, yes, I do suppose increasing the sheer # of papers could be one way of lionizing deductions.
</p>
<p>
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: <a href="http://www.capmag.com/article.asp?ID=3147">http://www.capmag.com/article.asp?ID=3147</a>
</p>
<p>
5. There are many other ways to lionize deductions that might strike one. Let me jot down some examples....
</p>
<p>
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...".
</p>
<p>
Not being willing to consider physical arguments so long as these have not been transformed into the form of mathematical equations is another way.
</p>
<p>
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...".
</p>
<p>
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.
</p>
<p>
Giving better recommendation letters for admission to a better-ranked US university to that student who can better spin out deductions is still another.
</p>
<p>
Providing tenure to a "spin-doctor" is still another...
</p>
<p>
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.
</p>
<p>
6. About 1D/2D/3D.
</p>
<p>
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.
</p>
<p>
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.
</p>
<p>
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.
</p>
<p>
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.
</p>
<p>
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.
</p>
<p>
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!!)
</p>
<p>
7. Regarding Galerkin's method.
</p>
<p>
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!
</p>
<p>
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?
</p>
<p>
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.
</p>
<p>
The obvious question is: What *physics* does correspond to the extended version of that method?
</p>
<p>
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?
</p>
<p>
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.
</p>
<p>
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?
</p>
<p>
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.
</p>
<p>
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....
</p>
<p>
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).
</p>
<p>
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.
</p>
<p>
If someone has good points to show that identifying physics hurts mathematics, let him at least say so in a reply post here!!
</p>
<p>
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?
</p>
<p>
In case you didn't guess it so far, the second method, obviously, is FDM.
</p>
<p>
Why does the research and application community have such a mental block in simply using it in solid mechanics?
</p>
<p>
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.)
</p>
<p>
8. In view of the generally prevalent prejudices, it might be restated that my post (or position) does not take something away from mathematics.
</p>
<p>
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.
</p>
<p>
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?
</p>
<p>
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.)
</p>
<p>
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.
</p>
<p>
The observation is that it is pointless setting up the strawman that I am against mathematics, and then start beating it.
</p>
<p>
10. Of course, I value precision, rigor, definite-ness, etc. of the things mathematical--apart from the potential application range of the subject matter.
</p>
<p>
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.
</p>
<p>
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?....
</p>
<p>
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).
</p>
<p>
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.
</p>
<p>
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.
</p>
<p>
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.
</p>
<p>
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.")
</p>
<p>
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!"
</p>
<p>
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?
</p>
<p>
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?)
</p>
<p>
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.
</p>
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</ul>Sun, 27 May 2007 09:10:44 +0000Ajit R. Jadhavcomment 2954 at https://www.imechanica.orgRe: Why lionize mathematics
https://www.imechanica.org/comment/2952#comment-2952
<a id="comment-2952"></a>
<p><em>In reply to <a href="https://www.imechanica.org/node/1440">Why lionize mathematics in science/engineering?</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Ajit,
</p>
<p>
Let me state how I understand the points of contention that you have tried to present in your post:
</p>
<ol><li>You think that some researchers use mathematics without any understanding of or interest in the underlying physical processes for the sake of furthering their list of publications.</li>
<li>Two-dimensional models of physical processes are inadequate yet that fact is not mentioned in the literature.</li>
<li>Galerkin's method has no physical basis or its physical basis has not been explored adequately in the literature.</li>
<li>You think this is because of the tendency (wrong in your opinion) to place mathematics on a high pedestal.</li>
</ol><p>
Regarding your first point, though there may indeed be some such people, I don't think that such a generalization applies to all people. Some people might think that calculus is an unnecessary complication in engineering. However, once you understand calculus, you realize that thinking becomes much easier in that language. The same thing is true for group theory. If you want to understand quarks or crystallography easily, an intutive understanding of certains groups is essential. That is why people use various aspects of mathematics to discuss their work. Implicit in most papers is the understanding that the community that you are talking to understands the language and can move on to the content. The novice always has to struggle to get to the point when the mathematical language seems obvious or even necessary.
</p>
<p>
Regarding your second point, one- and two-dimensional models are essential for a qualitative understanding of physical processes. For some simple situations they may even suffice to provide quantitative results. I think you underestimate the power of these simple models.
</p>
<p>
About your third point, Galerkin's method is first of all a mathematical technique. That, under some circumstances, it has a physical interpretation is a bit of luck. You will find details about the physical interpretation of the method in the context of linear elasticity in a number of texts, for instance, Owen's series from the late seventies or even Hughes book in finite elements.
</p>
<p>
I think there is some justification in giving mathematics the stature it has. In many situations, using the right mathematics makes systematic thought about complicated physical processes possible. A case in point is the idea of manifolds for nonlinear dynamical systems. You can then think of high dimensional spaces as surfaces and get an intuitive feel for things that are hard to visualize. Another idea is the projection from one function space to another in finite elements. Once you realize that such a projection is similar to finding the shadow of a pole you can visualize what happens in higher dimensional spaces.
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I agree with several of your points regarding the readability of research papers. I attribute that not to malice but to poor writing skills.
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Biswajit
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</ul>Sun, 27 May 2007 02:25:49 +0000Biswajit Banerjeecomment 2952 at https://www.imechanica.org