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*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.Abstract said:Given that one of the goals of the conference is to address the issue of the unity of Mathematics, I feel emboldened to talk about a question which has kept bothering me all through my scientific career: Is there really a unified subject of Mathematics which one can call PDE? At first glance this seems easy: we may define PDE as the subject which is concerned with all partial differential equations.

According to this view, the goal of the subject is to find a general theory of all, or very general classes of PDE’s. This “natural” definition comes dangerously close to what M. Gromov had in mind, I believe, when he warned us, during the conference, that objects, definitions or questions which look natural at first glance may in fact “be stupid”.

Indeed, it is now recognized by many practitioners of the subject that the general point of view, as a goal in itself, is seriously flawed. That it ever had any credibility is due to the fact that it works quite well for linear PDE’s with constant coefficients, in which case the Fourier transform is extremely effective.

It has also produced significant results for some general special classes of linear equations with variable coefficients. Its weakness is most evident in connection to nonlinear equations. The only useful general result we have is the Cauchy-Kowalevsky theorem, in the quite boring class of analytic solutions.

In the more restrictive frameworks of elliptic, hyperbolic, or parabolic equations, some important local aspects of nonlinear equations can be treated with a considerable degree of generality. It is the passage from local to global properties which forces us to abandon any generality and take full advantage of the special features of the important equations.

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.