(Guest Post) The Emperor’s New Math

Time for another guest post from my old chum Anton, this time on the topic of mathematics. I’m not sure any mathematicians reading this piece will be too happy, but if that applies to you then blame him not me. As usual, comments are welcome through the proper channel at the bottom of the page..

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Nowadays a page of mathematics looks to a physicist or engineer like gobbledygook. This was not always so: a century ago a physicist might hope to understand everything in journals of mathematics, and even contribute to them. Fifty years ago a physicist might not be able to understand everything written there, but the mathematics would appear comprehensible in principle. A qualitative change has since taken place.

This change has coincided, roughly, with acceptance of the distinction between ‘pure’ and ‘applied’ (impure??) mathematics, and with the consequent, deliberate, emancipation of ‘pure’ mathematics. This is a new departure: for centuries mathematics evolved side by side with physics, and the mathematics that was studied was the mathematics used in tackling physical problems. Galileo had said (in his work Il Saggiatore, The Assayer) that

…the universe… cannot be understood unless one first learns to comprehend the language… in which it is written. It is written in the language of mathematics.

So the change is recent, and it is huge. I suggest that it is a change for the worse; that in divorcing themselves from physical science pure mathematicians have cut off their air supply; and that the suffocating style of modern pure mathematics is a result. Mathematics was not born in a vacuum, and it will not ultimately flourish in one.

A pure mathematician might respond that I would say that, since I am a physicist. But perhaps an outsider is needed to see the problem; insiders generally adopt the party line. The justification for my stance is this. Mathematicians acknowledge that their subject is the formal study of patterns. And mathematicians think in patterns, not formulae – which are really a highly efficient way to express their thoughts. Crucially, the patterns arising in the natural world are far richer and more diverse than the patterns that even the best pure mathematicians can pull out of their heads by introspection. Even number theory is not an exception, for the positive integers are abstractions – ideals – of the physical realisations of one, two, three etc sheep in a field, or boats on a lake.

The role of pattern explains the “unreasonable effectiveness” of mathematics in physical science (as Eugene Wigner put it), since physics is concerned with relations – correlations – between variables in space and time, and correlation is synonymous with pattern. The theoretical physicist Lev Landau vehemently believed that the best mathematics is the mathematics used in physics. An opposing point of view was taken by the pure mathematician Paul Halmos, in an essay titled Applied Mathematics is Bad Mathematics. Not all mathematicians share Halmos’ view, however. The mathematician Morris Kline was the author of many books about mathematics and its embedding in the cultures which nurtured it. In his book Mathematics: The Loss of Certainty, Kline demonstrated that the history of mathematics in the 20th century has not been the smooth progression that it appears to the outsider; and that arguments about the foundations have led not to resolution, but to schism into differing schools – based on different foundations – that do not talk to each other. Mathematics is not in fact a one-way road running from self-evident axioms to consequences, but is open at both top and bottom.

Already in the 19th century a formal style was developing in the mathematical study of logic, and such distinguished noses as Henri Poincaré (in Science et Methodes, part II) protested as early as 1909 that this tended to hide misleading or negligible content. To no avail: the dominance of the formalistic logical viewpoint led to the adoption of its house style across the whole of mathematics. Below university level, mathematics is still taught today as it used to be, with the emphasis on the understanding of ideas rather than their formal presentation. Freshmen are often shocked when they first meet the new way of doing things, in university lectures given by professional mathematicians. I doubt that the form of modern mathematical writing is governed by its content, for whenever my research has demanded I read some contemporary mathematics, and I have had to translate a piece of modern mathematical writing into something comprehensible to scientists, I have found it difficult to distinguish substantial points from trivia. When, for instance, four axioms are needed to establish a result, they will typically be presented as having equal weight, even if one is the crucial axiom that allows most of the proof to be constructed, and another is used only in closing loopholes. Acknowledging the quality of axioms, as well as the quantity, does not compromise rigour.

When I think of the work of Andrew Wiles and Grigori Perelman, I realise that magnificent work is done today by mathematicians far beyond my own competence. But might mathematicians question whether what they regard as the only way to write mathematics is actually a convention, and not necessarily a good one? If they wrote mathematics as they did fifty years ago, others might be able to see for themselves. More fundamentally, might they also realise what their predecessors understood, that by its abstraction mathematics is given an autonomy of its own, and that to look to the physical world for inspiration is not to make mathematics a slave of physics? The present divorce between mathematics and physics impoverishes everyone.

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26 Responses to “(Guest Post) The Emperor’s New Math”

  1. David Brown Says:

    “… in divorcing themselves from physical science, pure mathematicians have cut off their air supply …” Should M-theorists make a maximum effort to explain dark matter, dark energy, and the GZK paradox. Google “nks forum” and look at the “Applied” postings. IF IT IS GOOD IDEA TO POSTULATE A MINIMUM NONZERO PHYSICAL WAVELENGTH, WHY IS IT NOT A GOOD IDEA TO POSTULATE A MAXIMUM PHYSICAL WAVELENGTH THAT SUPPORTS THE FREDKIN-WOLFRAM DIGITAL PHYSICS MODEL?
    In “Fredkin-Wolfram digital physics: 3 decisive tests” I give 3 empirical tests THAT CAN BE CARRIED OUT WITH CONTEMPORARY TECHNOLOGY and if any one of the tests fails, then my theory is a crackpot theory.

  2. Anton Garrett Says:

    David,
    Fredkin’s everything-is-discrete hypothesis will have testable consequences, but is this not where we were 110 years ago before quantum theory? I am sceptical of its extension to space and time themselves, although if somebody wishes to propose feasible experimental tests which clearly distinguish between that and the present paradigm then I wish them well. [To Peter: GZK is definitely your department.] Wolfram is one of increasingly many scientists who in some of their writings confuse information with physics. I don’t buy that.
    Anton

  3. David Brown Says:

    Anton,
    I think that the Fredkin-Wolfram information process is the unique, physically valid computational method for M-theory. If my theory is wrong, it should be ignored. I think that if my theory is correct, every M-theorist in the world should be considering it. Unfortunately, I need the M-theorists to calculate the corrections to the Bekenstein-Hawking radiation law. I have to convince the M-theorists that:
    EVERY PHYSICAL CHANGE IN ENERGY IS AN INTEGRAL MULTIPLE OF THE MINIMUM NONZERO ENERGY. (In other words, I need to convince the M-theorists that the Fredkin-Wolfram constant is the fourth constant that they need. I also need to convince them of some extremely bizarre assumptions — or convince myself that my theory is a crackpot theory. It does seem to have some of the attributes of a crackpot theory.) Google “feynman fredkin wolfram” for the background of my theory.
    “Nowadays a page of mathematics looks to a physicist or engineer like gobbledygook.” I think that almost all the mathematics needed by physicists and engineers was already developed by the year 1935 CE. Another problem is that applied mathematicians are in demand outside of academe but pure mathematicians are not. Therefore, academe gets overrun by pure mathematicians. If M-theory can make testable predictions the situation might improve slightly — but unfortunately not much.

  4. Anton Garrett Says:

    David,

    While I take your point I would not say that maths up to 1935 will do for all of physics indefinitely. (For example, period-doubling in chaos has been observed experimentally and needs the chaotic maths developed by Feigenbaum in the 1970s to describe it.)

    Frankly I think that much of superstring theory and M-theory is science fiction, but I’ve already been controversial enough in the original post without opening a second front.

    I wish you well in the testing of your theory. If you grumble that it is hard to get funding – you now share that problem with scientists who are regarded as part of “the establishment”.

    Anton

  5. Note that many mathematicians, such as G. H. “I have never done anything useful” Hardy, worked on what was at the time “pure” mathematics, but their discoveries later had extremely practical applications.

  6. Dipak Munshi Says:

    >Crucially, the patterns arising in the natural world are far richer and more diverse than the >patterns that even the best pure mathematicians can pull out of their heads by introspection.

    I don’t think this correct. Most of the mathematics that were later used in describing
    real world existed before they were found useful, e.g. non-Euclidean geometry was
    not developed to be useful for describing space-time physics, but as a natural
    extension of Euclidean geometry if one removes or replaces 5th postulate.

    Mathematics is an art. As in case of abstract art, there can always be many logically correct mathematical systems which don’t describe our nature. However every physical
    phenomena can be modeled with some consistent mathematical system to be called
    science.

  7. Anton Garrett Says:

    Phillip: Absolutely. Hardy was fond of toasting “To pure mathematics, may it never be any use.” But his own work has since been applied to derive partition functions, among other things. Hardy is credited with importing continental rigour into the British mathematical tradition, but he never fell for the modern impenetrable style of mathematical writing that supposedly goes with rigour. I have his book on inequalities and it is lucid. My comment that mathematics is the formalisation of pattern also goes back to Hardy (at least).

    For brevity I cut from my polemic a discussion of the relation between the terms “pure mathematics,” “applied mathematics,” “mathematical physics,” “theoretical physics,” “practical physics,” and “applied physics.” As regards mathematics I prefer the categories “mathematics not currently applied in physics,” and “mathematics currently applied in physics.” The traditional categories depend partly on the writer’s motivation.

    Anton

  8. Anton Garrett Says:

    Dipak: Non-Euclidean geometry was inspired by the desire to generalise plane geometry to geometry on curved surfaces such as the surface of a beach ball – ie, it was inspired by experience of the physical world. Only later was it deployed in the physics of spacetime – Anton

  9. Dipak Munshi Says:

    Anton,

    There are many geometries and topological spaces which are completely arbitrary
    without any real life analogs. Not all of them have any connection with real world
    even in any abstract sense.

    Ofcourse at some level all abstract art is inspired from the real world.
    All human thoughts (even abstract ones) are rooted in real life experience.

    But, to me doing physics is trying to learn how to take pictures of already existing objects, e.g. may be with a camera. While doing maths is in some way similar to learning how to paint/sketch things which might only even exist in your own mind. The former is a very small subset of the later.

  10. Anton Garrett Says:

    Dipak: Yes, mathematics generalises into the abstract (eg, N spatial dimensions). But it generalises FROM the real (3 dimensions). I am not saying this generalisation is a worthless exercise; we are not necesssarily in disagreement – Anton

  11. Dipak Munshi Says:

    Anton,

    I only disagree with your following statement:

    >the patterns arising in the natural world are far richer and more diverse than the >patterns >that even the best pure mathematicians can pull out of their heads by introspection.

    Because mathematicians can dream of many many abstract worlds ( may be all inspired
    from our own real world). However we should agree that we only have -one- real world out of those many possibilities!

    Dipak.

  12. Anton Garrett Says:

    Dipak: We do differ here. I question whether the patterns that pure mathematicians dream up have the richness of the patterns enountered in the physical world that are described by mathematics, or of generalisations inspired thereby. Putting it metaphorically or literally as you prefer, the mind of God is richer than the mind of man – Anton

  13. Dipak Munshi Says:

    I did not think we will start discussing God here.

    But, If one day we find that we live in a multiverse where individual
    Universes (of which ours is one) have all possible set of topologies,
    geometries and field theories etc., then I will probably start believing
    that “God’s” is/are equally smart as mathematicians. Although that only
    proves doing Maths is better way to understand “god’s” mind.
    Till than I will try to compute instead of praying.

    A more trivial question dont you think ND geometries have more
    richness than real world 3D/4D geometries?

    Thanks anyway for all your comments. Bye for now.

  14. Anton Garrett Says:

    Dipak: feel free to regard my use of the word ‘god’ as metaphor, as I said – I’m not evangelising here, and I’m sure you understand what I mean even if you disagree with it.

    The multiverse requires careful definition before useful discussion can take place. There is *some* valid logical argumentation by people who invoke the concept, but many people would say that there is only one universe by definition.

    My answer to your question: Yes, certainly. But generalisation to N dimensions is inspired by what goes on in 2D and in 3D, ie by sense-based experience. The attitude of deliberate emancipation from any thought about the world, which is found in many pure mathematicians today, is bad for their own subject. What know they of mathematics, who only mathematics know?

    Anton

  15. Dipak Munshi Says:

    Just to add that both pure math and “God” are product of human imagination.
    However one clarifies thing where as the other simply is to confuse people.

  16. Anton Garrett Says:

    Dipak: Regarding whether man invented god or vice-versa, I have made no dogmatic claim on this blog (unlike you). I accept that you don’t find my use of the word ‘god’ helpful viewed as metaphor. So try this way of putting it (credit goes to Montaigne). Nature can generate a worm, but man is almost indefinitely far from being able to do that – only in the last 30 years have we even worked out its genome, and we are a very long way from understanding all the details of how that genome acts to generate the organism. And that’s just a worm, not even a mammal, let alone a human being. *That* is how much richer and more complex the patterns found in nature are than the patterns we can dream up…
    Anton

  17. Dipak Munshi Says:

    How many years has it taken nature to create an worm?
    And why it still can do that only in one particular planet called earth?
    I just think we should compare that time scale against
    the number of years since Industrial Revolution.

  18. Anton Garrett Says:

    Good questions Dipak, but a bit too off-topic for me now. Maths has been around since the ancient Greeks – Anton

  19. Dipak Munshi Says:

    To my knowledge math was practiced even before Greeks in many other countries. But,
    I guess in those days math was used to please “gods” , e.g. to find out
    a specific ratio for the sides of Pyramids, temples, etc and not to understand
    or predict nature.

  20. Anton Garrett Says:

    There was some mathematics before ancient Greece, certainly; the Babylonians knew how to solve quadratic equations, for example. But the ancient Greeks were the first to come up with the concept of abstract number, ie one, two, three etc, and not one, two, three cubits or sheep or whatever. The Greeks also invented the notion of formal proof of theorems. (See “A History of Greek Mathematics” by Thomas Heath, 2 vols, Oxford 1921.) Those two advances make the Greeks the inventors of mathematics as a subject. It is inconceivable that any earlier civilisation could have proved that the square root of 2 is irrational, or that there are an infinite number of primes, for example. And Euclid’s ‘Elements’ was not superseded as a textbook for some 2300 years.
    Anton

  21. Dipak Munshi Says:

    The discussion was not related to history of maths but about your claim that,

    >the patterns arising in the natural world are far richer and more diverse than the >patterns >that even the best pure mathematicians can pull out of their heads by introspection.

    This is where we disagree.

    But I think, I will stop here because I understand your views bit more clearly now.
    Thanks for all the comments.

    ps. Just a minor point, Euclid’s Element was a collection of results known previously by many many generations and was not invented by Euclid himself.

  22. Anton Garrett Says:

    Dipak: I didn’t see how your comment of 6:03pm was a response to mine of 5:54pm, which is why I changed the subject and chatted about the history of maths, which is interesting in its own right. (That’s why my post of 6:51pm re history of maths was not addressed to you by name.) Happy to stop or continue, all the best: Anton

  23. Aaron F. Says:

    “Nowadays a page of mathematics looks to a physicist or engineer like gobbledygook.”

    As a somewhat cantankerous student of mathematical physics, I’d like to suggest that this is not entirely mathematicians’ fault. I’ve gotten the impression that a lot of introductory physics textbooks bend over backwards to avoid discussing any math that gives off even a whiff of abstraction, even when an abstract point of view is by far the clearest and most natural. In /Gravity/, for example, Hartle gets through 80% of the book without explaining what a tensor is… and the explanation he finally does give is unwieldy and coordinate-heavy. In /Introduction to Electrodynamics/, Griffiths talks at length about linear differential equations without ever mentioning linear algebra; this makes everything sound about ten times more complicated than it actually is, and makes it harder for students to apply the powerful ideas that they learned in their linear algebra courses. I’ve yet to see an introductory electrodynamics text that even mentions differential forms, let alone one that uses differential forms as a basic tool—this despite the fact that differential forms are now over a hundred years old, and have become quite important in the study of general relativity and gauge theories.

    What I’m trying to say is that if some physicists and engineers see pure math as gobbledygook, maybe it’s because they’ve never bothered to learn any pure math. You could argue, of course, that physicists and engineers don’t learn pure math because pure math is useless; and this may well be true. It’s worth noting, however, that many early quantum physicists found representation theory to be useless, despite its now-obvious importance. I have no doubt that when Cartan introduced spinors in 1913, most physicists found those to be useless as well. Aperiodic tilings certainly had no physical application when they were proven to exist in 1961; fortunately, mathematicians studied them anyway, paving the way for the discovery of quasicrystals a few decades later. The fundamental theorem of finitely generated abelian groups also had no relevance to physics when it was first proven, although it now plays a key role in an important class of quantum algorithms.

    You suggest that “in divorcing themselves from physical science pure mathematicians have cut off their air supply.” I say that by losing interest in pure math, physicists have seriously weakened their ability to create new tools, and to use the tools that have already been created for them. The old tools are rusting rapidly… and when all the toolmakers have asphyxiated, where will physics be?

  24. Anton Garrett Says:

    Aaron: Yes, it is often the case that a little more maths than is formally necessary will bring out points about the physics. In that case a physics text should not hesitate to deploy that extra maths – but no further maths on top of *that*. Decisions must also be made about how much maths to teach in the book and how much to assume that the reader already knows; and also how to bring in the maths that is to be taught (in series or in parallel with the physics). But what I am complaining about in ‘goobledygook’ is the pure-mathematical penchant for *starting* a piece of expository writing in fullest abstraction and generality. That is the convention nowadays, but it is poor pedagogy. Full generality needs to be worked up to.

    I’m not sure that differential forms are suitable for an *introductory* text. And, changing the subject, both they and spinors (and also tensors) are inferior to arbitrary-dimensional real Clifford algebras and the differential calculus thereof. See the rremarkable book, “Cllifford Algebra to Geometric Calculus” by David Hestenes and Garret Sobczyk, which unifies all of these an provides a superior language for the algebraic expression of physical processes going on in spacetime. If you ever wondered how to generalise the power and beauty of complex analysis to higher diimensions, this is it.

    Hestenes, by the way, is a professor of physics…

    Anton

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