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A PDE: Between Physics and Mathematics

  1. Jan 5, 2019 #1
    This is perhaps the single most important mathematical physics papers I have ever read; I think everyone - especially (theoretical) physicists - interested in theoretical physics should read it. In fact, read it now before reading the rest of the thread:
    Klainerman 2010, PDE as a Unified Subject
    The author, Sergiu Klainerman, essentially argues that the premature christening of 'the theory of PDE' by mathematicians in the last century was completely unnatural, based on incorrect presuppositions and expectations which has consequently lead to an almost universally pervasive misunderstanding among all physicists and mathematicians about the very nature of both mathematics and physics and therefore also a misapprehension of the goal of mathematicians and physicists.

    Klainerman's vision is a call for a basic reevaluation of how exactly mathematicians and physicists should naturally view their subject based upon what they do (and historically have always done) instead of what they think they (should) do; such a change of perspective has extremely nontrivial consequences for it directly quite literally shapes not only subjects (and so conjectures, research programmes, the boundaries of fields and careers) but literally also creates conventions of what is considered to be mathematics and what is considered to be physics.

    Moreover, taking Klainerman's vision seriously also means that many popular criticisms and complaints by physicists and scientists leveled at string theory, multiverse theories and so on by Woit, Smolin, Hossenfelder et al. are indeed somewhat unfair premature mischaracterizations of the natural progression over time of theories which lay between physics and mathematics.

    The artificial strict boundaries by which they judge theories, imposing upon the theories a completely artificial immediate necessity of either directly conforming to being 'physics' or 'mathematics', may be a wholly misguided endeavor, one which occurs due to modernly arisen incorrect idealized definitions of physics and mathematics, which reflect the modern conventional idea of what logically these subjects should be, not what they naturally are.

    Indeed, if we look at history, in particular to the universalists who were both mathematicians and physicists starting in the 17th century with Newton & Leibniz, going on to Laplace, Lagrange, Euler, Fourier, Gauss et al. all the way up to and including Poincaré, we clearly see that their mathematics is indistinguishable from their physics. Today on the other hand we have so much specialization that not only does practically no one seem to be able to tell the forest from the trees, many often do not even realize that they may be studying the exact same mathematical structure as others in other branches are doing or have already done; worse still, often when this is pointed out to them many still do not recognize that this is actually so.

    Prior to my formal introduction to physics, I was naturally a classical pure mathematician at heart; despite my abhorrence for engaging in modern formal mathematics, in many respects I still am. I think it is no accident that a very productive way of looking at almost any unsolved problem in pretty much any science is by comparing a potential solution, i.e. a mathematical model, to another already solved problem and its respective mathematical model; a return to our roots, like Klainerman is proposing here, therefore sounds like music to my ears.
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  3. Jan 5, 2019 #2


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    I do like the article, perhaps I should read it again to refresh my memory, but I think you are reading too much between the lines. I am not sure if Klainerman would agree with all of what you say that he says.
  4. Jan 5, 2019 #3
    Perhaps I am, but I am merely carrying the presented view to its logical conclusion in order to compare it with other views; I believe such a framing is very useful because it is, as Feynman liked to say, a way of 'taking the world from another point of view'; such an alternative point of view might turn out to be very valuable for me or others for whatever reasons.

    For example, some things I'm stating in the OP, e.g. that Woit, Smolin, Hossenfelder et al. may be judging the progress of some physics programmes unfairly based on non-existent ideals of what physics should be, most definitely is my own interpretation of Klainerman's view if one is to fully take Klainerman's message to heart; I am not personally acquainted with him so I can not speak about his actual views on such matters, but it doesn't seem to be inconsistent with them either.

    To be clear about this, my above interpretation of Klainerman's message isn't my actual position, but - myself being another 'believer' in ideal distinctions - actually almost the exact opposite, with my views tending to be more or less aligned with Woit, Smolin, Hossenfelder et al.; I am actively trying to challenge my own point of view in case it might actually be an incorrect view and at the same time exposing it and opening it up to criticism by posting it in this forum.
  5. Jan 5, 2019 #4
    Mathematicians are a strange bunch is all I got after reading this. Math does not predict anything, all it does is state some truths based off the logical conclusions we can make from some axioms. That's all, it just states truths. No where in math does "throwing a stick" come about. It's only when someone asks a questions, "I wonder how far this rock will go if I throw it" and actually want to predict how far it will go BEFORE I measure it, will I need math. Otherwise, I just throw it and go "Wow, that's pretty far!" Do mathematicians actually think their abstractions are any different than me describing a unicorn in English? Does anyone think Hilbert Spaces exist in nature? Mathematics is useful in physics because it is the language system needed to paint the picture.


    The author then states on page 23 of the PDF (301 in the book?) " THE SITUATION OF SINGULARITIES IN GENERAL RELATIVITY IS RADICALLY DIFFERENT TO THE TYPE OF SINGULARITIES EXPECTED HERE IS SUCH THAT NO CONTINUATION OF THE SOLUTIONS IS POSSIBLE WITHOUT ALTERING THE PHYSICAL THEORY ITSELF THE PREVALING OPINION IN THIS RESPECT IS THAT ONLY A QUANTUM FIELD THEORY OF GRAVITY COULD ACHIEVE THIS". Right, something that isn't stopped by mathematics. Only when we consider the physical aspect of the theory do we stop... but naturally, a mathematician has no reason to stop.

    To restate my first paragraph: there is a fundamental difference between the two subjects. Physics is a predictive science. Math is a logical system built up on axioms. No different than grammar structures set forth by languages. Do we also need to set the standard between linguists and physicists as well?!

    Now my view is expressed by a part of the preface of "Supermanifolds" by Bryce DeWitt- "A previous generation of theoretical physicists could function adequately with a knowledge of the theory of ordinary manifolds and ordinary Lie groups. With the discovery of Bose-Fermi supersymmetry all this changed. Nowadays the theorist must know about supermanifolds and super Lie groups. The purpose of the present volume is to provide him with an easily accessible account of these mathematical structures. Mathematicians will find much of this book incomplete and expressed in a language that they have nowadays passed beyond, but it is probably pitched about right for the average physicist."

    The quote shows my feelings towards the issue because it shows what physicists do: we learn new math if it helps describe reality. The way we learn the math is molded to how it helps us describe reality. Mathematicians won't like how we present the math. (To keep up the comparison with people use "text-speak" as English to communicate with each other, a notion I'm sure a lot of linguists wouldn't like.)
  6. Jan 8, 2019 #5
    Actually this simply isn't true; the utility of a mathematical model is often present regardless of the intent of why the model was constructed in the first place: the direct utility is that it offers a coherent description of some known concept, therefore potentially increasing our understanding of said concept.

    The practice of classical pure mathematics (which confusingly is practiced today by applied mathematicians) busies itself with building models for their own sake or for the sake of exploring what kinds of models can be demonstrated to exist mathematically; very often these 'abstract' models turn out to have a reflection in the real world, whether that be in physics as a natural description of some physical phenomenon or as a model of some phenomenon in any other human endeavor.
    As a physicist, I will fully agree with the sentiment - especially w.r.t. physics - that engaging in the construction not of mathematical models of phenomena or of concepts but instead axiomatic or logical models of the theory of mathematics itself is far less useful, and that such abstractions should possibly be seen - just like works of art and literature - as merely abstract works of fiction.

    It is easy to spot such mathematicians (or physicists engaged in the same activity of fruitless fantasizing): they more often than not cannot easily reduce their ramblings to any (semi)concrete examples which might pass for a description of something in the real world, but more importantly they do not seem to be interested in learning more from the real world in order to more accurately construct their models.

    Regardless of the fact that mathematical constructs and linguistic constructs are in many cases only that - constructs - it is patently ridiculous to state or to think that all mathematical abstractions are equally useful or that they are all only as useful or as real as e.g. literature or music.
    I think there would be many who would picture it the other way around, i.e. that it is nature which exists in or even be an abstract mathematical realm, e.g. Tegmark's mathematical universe. It seems that most people, including professional mathematicians, simply have never actually taken the time to think about such questions in depth.
    Most practitioners of mathematics at all levels would probably agree or have agreed at least once during their lifetimes that many things in reality are reflections of mathematical things, i.e. quantities are reflections of numbers and shapes are reflections of geometry; this point of view is quite typical and is called Platonism.
    Don't laugh too hard, this statement is obviously true; we physicists often do not look beyond very surface level descriptions, sometimes even fully eschewing sophisticated mathematics in favor of solvability (e.g. small angle approximation, perturbative methods, etc).

    The fact of the matter is that some mathematicians are able to take a deeper look at nature than physicists, because of their knowledge - not only of the models but also of the deeper mathematical properties underlying the models - e.g. smoothness, analyticity, holomorphicity, etc.

    Studying a natural phenomenon at such a deeper mathematical level than is required for the physicists' experimental comparison often has lead to the discovery of novel mathematical objects i.e. lead to new unexpected results or fields in mathematics; many models of phenomenon once considered to be uniquely part of physics were shown to be much more generally applicable to describe many things in the world and so became instead to be regarded as applied mathematics instead.

    Moreover, any newly discovered mathematical objects gained by mathematically studying physical models may or may not be reflected in nature as well, whether that be directly in the modelled phenomenon or indirectly in any other phenomenon which has some type of similarity with the modelled phenomenon, e.g. similar descriptive equations.
    This is just a red herring. It is obvious that new math should be learned if it is already known and/or has already been demonstrated that some particular form of mathematics is useful for describing some (aspect of) physical phenomena, and that this new math need not be presented or explicated in the highly abstract formalist manner that mathematicians tend to use.

    However, what is not obvious - and what the author is alluding to - is that older but unconventional math for physics (which might even already be known to some smaller subset of physicists, but typically remains unknown among physicists generally) may already be capable of describing some poorly understood aspects of physics in a manner superior to what the conventional math already adopted by physicists is capable of doing w.r.t. the specific subject in question; the fact that the elucidation hasn't occurred yet is then a mere accident of convention.
    I could care less about what they like or do not like about our presentation of mathematics depending on what they are aiming to achieve:
    - if the mathematician-in-question's goal is aligned with that of the theoretician i.e. to directly create more mathematically sophisticated models of some phenomena or concepts, based on a less conventional and/or more sophisticated analysis of said phenomena, then we should listen
    - if the mathematician's goal is simply to create abstruse propositions, definitions and axioms in order to be able to generalize and/or delineate what can be considered to be the 'true' or necessary formal properties of such models generally, then we should ignore them
  7. Jan 11, 2019 at 8:06 PM #6
    I want to share a passage from a pretty recent monograph by a mathematical physicist; it seems to me that the confusion of the goal or intentions of the practitioners of mathematical physics seems to be rife with widespread misunderstandings.

    Both the phraseology and the intent of the author here seems to be typical, so I hope this passage is able to illustrate some of the differences between the person who identifies as a physicist first versus one who identifies as a mathematician first, independent of what his degree says:
  8. Jan 13, 2019 at 4:58 PM #7
    I'll respond to your post directly when I have more time, but I think this interview with Michael Atiyah has a few bits (there are more) that are related to this thread:

    The last one (one on magnetic monopoles) is an interesting one.
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