On “The Nature of Mathematical Knowledge” (enlarged)

That question is about as interesting as the nature of our knowledge of elephants.  We are interested in the zoology of elephants, not in the specificities of classroom biology lessons, or the economics of zoos.  We are uninterested in each blind man’s subjective and partial report upon the individual organs of these splendid creatures.

Mathematical knowledge, like pachydermal knowledge, is imperfect knowledge of something real that exists independently of us. By contrast, just which images we manage to form of these objects  are very much dependent upon ourselves – and to that extent, of interest only to unemployed social workers.

As our former math teacher put it:

Mathematics has a real content which transcends the inadequacies of our efforts to formalize it.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. v.

Or, from a philosopher:

For most mathematicians most of the time, having a feel for what is evident is important, but it is also enough: There is no further need for a theory of what that feel is.

-- Shaughan Lavine, Understanding the Infinite (1994)

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Actually, it should not be deduced from the above, that I am somehow dissing our big friends the elephants.  A Theory of Elephants is considerably more promising than that idle grail of the physicists, a Theory of Everything.

The outstanding problem in the Theory of Elephants is the ontological status of BABAR – THE KING !!!!!

His Majesty ... The King !!!

An agnostic or non-Realist version, wilfully po-faced:

Our deductivist proposes clean answers to philosophical questions.  What is mathematics about?  Nothing … What is mathematical knowledge?  It is knowledge of what follows from what.  Mathematical knowledge is logical knowledge.

-- Stewart Shapiro, Thinking about Mathematics (2000), p. 150

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There is a systematic ambiguity (roughly that of actio versus actum) to the term mathematics:

(I) The praxis of mathematizing.  This is a human pastime, comparable to needlework or basketball.

(II) The truths of mathematics.  Or, more or less synonymously: The real (though invisible) world.  These, in themselves, bear no dependency upon human practice or to any species whatever;  they existed before we were born.

The above may count as a polemical reformulation of roughly the dichotomy in the title of Hao Wang’s fine essay, “The Theory and Practice of Mathematics”.

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In 1950, Raymond Wilder gave an address, “The Cultural Basis of Mathematics”, reprinted in  various places, and later wrote a whole book on the subject, Mathematics as a Cultural System (1981).   Bien-pensant commentors treat these with grave respect;  but the notion is practically nonsense.  For, if we take mathematics in sense (II) – the only sense of interest to us here – that is like saying “The cultural basis of elephants”:  there is none.  There is a cultural basis of circus stunts, of mouse- and peanut-myths, of Dumbo, but not of elephants themselves.  Their basis is their own four feet.

Why should the culture of mathematics  (necessarily, sensu I) – that is, the foibles of mathematicians – retain our attention?  The purely “human side” of mathematicians  is generally less interesting than that of country music stars.  A lot of mathematicians are pretty Asperger’s, frankly.
(For a poignant illustration of this, read The Genius in My Basement.)

There is, we grant, a certain interest in the sociology of mathematics, or in biographies of the great mathematicians. Intellectually, it is on a level with gossip about the off-court antics of basketball stars.  Fun, but of no mathematical (or basketball) interest.   It’s just a way for the mind to chew gum while it’s too exhausted to do anything substantial.  To get real, do math (or play basketball).

Not to come down too hard on the small geeky community that does follow the doings of math and physics whizzes; I number myself among them.   It would even be neat  if, instead of collecting baseball cards, people collected mathematician cards (“Trajea two Steven Smales for a John Milnor!”) .  -- By “people”, I here mean “eight-year olds”.

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More interesting is the purported “reduction of mathematics to logic”.  It is not initially clear, however,  to what extent this program, if successful on its own terms, would enlighten us as to mathematics-sensu-(II), as opposed to the sense-(I) territory of our own mathematical formulations and formalizations (these being, after all, largely for mere convenience).  It might be more along the lines of the demonstration of the equivalence of the Heisenberg-style matrix-mechanics formulation with that of the Schroedinger-style wave formulation, of quantum mechanics. That feat didn't tell us all that much about the actual phenomena of physics,  apart from the fact that the world is a many-splendored thing, and can be described -- blind-man-fondling-elephant-fashion -- in a variety of ways.  It’s more like deciding whether today’s symposium shall be conducted in English or in French.

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That said --

We argued here that the axiomatic method is cognitively post-hoc, and that  only in cases where (as with the Euclidean axioms) their positing is transparently motivated by our experience of the sensible world, is a top-down, axiomatic presentation  pedagogically sound.   Thus similarly in physics:

In lecture after lecture, and essay after essay,  Einstein began, not with an introduction to the subject at hand, but with an overview of how he’d arrived at that subject, or of how scientists in general  arrive at subjects in general. … For Einstein himself, the results of science had become incomprehensible without an understanding of the processes that led to them.

Richard Panek, The Invisible Century (2004), p. 153-4

C'est exact;  and the farther physics wanders from our human experience, and the father math develops beyond anything the world has seen before, the more necessary such a psycho-cognitive ladder does become.

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Something like the dichotomy outlined above  must have been behind André Weil’s tart remark, in “History of Mathematics” (reprinted in Collected Works v. III as (1978b)):

Some universities have established chairs for “the history and philosophy of mathematics”;  it is hard for me to imagine  what those two have in common.

For:  the one is situated and contingent, the other timeless and beyond place.

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Footnote:   These remarks about mathematics  apply  mutatis mutandis  to the Deity.  Deliberately confusing the distinction between truth and praxis, Karen Armstrong wrote a book -- a minor best-seller -- with the impudent title A History of God.  (At least she put A, not The; probably saved herself an extra millennium in Purgatory right there.)

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Lakatos’ classic dialectical-dialogue Proofs and Refutations (you see the Hegel-style paradox already in the title), though focussing on the (as he persuasively argues, in the course of a detailed case-study spanning many decades) micro-level mess of actual mathematical progress, is yet Realist at its core:  the subtitle is “The Logic of Mathematical Discovery”, not “The Sociology of  ‘Mathematical’ Invention”.   We quoted him in another context  thus:

As far as naïve classification is concerned, nominalists are close to the truth when claiming that the only thing that polyhedra have in common  is their name.  But after a few centuries of proofs and refutations, as the theory of polyhedra develops, and theoretical classification replaces naïve classification,  the balance changes in favour of the realist.

-- Imre Lakatos, Proofs and Refutations (1976), p. 92

In an appendix to the main work, he offers a Hegelian formulation, one which (by the time the reader has progressed this far) has a certain paradoxical piquancy:

Mathematics, this product of human activity, ‘alienates itself’ [in the sense of Hegel and Marx] from the human activity, which has been producing it.  It becomes a living, growing organism, that acquires a certain autonomy [emphasis in original] from the activity that produced  it.  … The activity of human mathematicians, as it appears in history, is only a fumbling realisation of the wonderful dialectic of mathematical ideas.
-- Imre Lakatos, Proofs and Refutations (1976), p. 146

Plato, in his Paradise, smiles.

E8: a Riposte (concluded)

Let us examine a bit more closely  Synge’s picture of physics as bricolage,  where theories have the intellectual status of just-so stories, and are really little more than pragmatic techniques, or tools -- Newtonian mechanics and relativistic mechanics each useful in its own sphere, like screwdrivers and spoons, but of little interest in their own right.   Now, this is not to knock the status of a toolkit -- my respect for competent carpenters and electricians borders on reverence -- but fundamental physics is not like that.

            Synge presents the Newtonian view as having not been replaced or refuted by relativity;  it rules as before in its own realm.  Newton’s good for some things, Einstein for others, and Wiccan no doubt for others still.   But this view assumes a confusion.  For it is not the case that Newtonism and relativity are independently valid in their own way but incompatible;  rather, Newtonism is the limiting case of relativity, in a way very familiar in mathematics;  its continued use in everyday life is simply a calculational convenience, a shortcut.   To continue the tool metaphore:  Einstein and Newton are not like screwdriver and pliers, but like a hammer, and an old shoe used as a hammer, good enough for the task at hand.

            Furthermore, it is a good thing, not a bad thing, when initially separate paths converge.  If you only know one way to climb a thing,  perhaps it is only a Potemkin mountain -- a paper-maché façade, hollow behind the north slope.   It is quite a relief -- and an ontological ratification -- to meet another mountaineering party that has scaled up the other side.

            The reader may be familiar with the story of how Schrödinger and Heisenberg separately found Rome by different roads.  Let George Gamow tell it, in Thirty Years that Shook Physics (1966), p. 3:

The simultaneous appearance of Schrödinger’s and Heisenberg’s papers  in two different German magazines … astonished the world of theoretical physics.  These two papers looked as different as they could be, but led to exactly the same results concerning atomic structure and spectra.

We are, in hindsight, not overly surprised by this, since by now we most of us accept that there is something there at the quantum level, something real, something other than subjective, to be described.   It is describable by two quite different mathematical approaches, much as our peak may be scaled by walking up the north face  or rappelling up the southern cliffs.    Nor is such ‘duplication of effort’ a waste of time, for  in this instance, not only the factual success, but even the approaches themselves retained their usefulness -- for determining energy levels, Schrödinger’s wave mechanics was calculationally more convenient; and Heisenberg’s matrix methods had the edge when it came to calculated the intensities of the radiated frequencies.   Or, alternately, P.A.M. Dirac, The Principles of Quantum Mechanics (4^th edn. 1958), p. viii:

Quantum mechanics … is known under one or other of the two names ‘Wave Mechanics’ and ‘Matrix Mechanics’, according to which physical things receive the emphasis in the treatment, the states of a system or its dynamical variables.

And (p. 115):

The Schrödinger form is the more useful one for practical problems, as it provides the simpler equations. … Heisenberg’s form for the equations of motion  is of value in providing an immediate analogy with classical mechanics.

Or again (R. F. Streater & A. S. Wightman, PCT, Spin & Statistics, and All That (1964), p. 4):

Throughout this book, states will be described in the Heisenberg picture of quantum mechanics.  The Schrödinger picture is much less convenient for the description of a relativistic theory, because it treats the time coordinate on a very different footing from the space coordinates.

And:

P.A.M. Dirac, The Principles of Quantum Mechanics (4^th edn. 1958), p. 311:

The Schrödinger picture is unsuited for dealing with quantum electrodynamics, because the vacuum fluctuations play such a dominant role in it. … They get bypassed when one uses the Heisenberg picture, and one is then able to concentrate on qualities that are of physical importance.

Dr. Matrix

Dr. Wave

Approaching an abstract but genuine reality from two different theoretical complexes  has its counterpart in different experiments, or different means of calculation, strengthen each other when they arrive at the same result.   Thus Einstein, in his annus mirabilis of 1905, when not inventing Relativity, found it worth his while  to “develop theoretically  three independent methods for finding Avogadro’s number.” (Abraham Pais, Subtle is the Lord (1982), p. 55.)   It was worth his while because, independently of our endeavors, this number is indeed there.

Summarizing:  For epistemology, the fact that two or more radically different approaches each manages to describe the phenomenon of interest, reassures us that we really do have our arms around this thing.   The lesson goes over, I would submit, in cases where what is being described is nothing so tangible as an atom (which Rutherford reportedly saw in front of his face as plainly as a spoon), but rather a four-manifold, or a simple Lie group.

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Afterword.

I recently happened across the following curious passage:

The algebras G_2, […] E_8  are called exceptional.  In 1945, Chevalley remarked  that the existence of these algebras  is a brutal act of Providence  which we must accept blindly.  Perhaps this should be revised today  to assert that the source of these algebras  is the wisdom of the Deity  in allowing the Cayley numbers to exist.

-- Irving Kaplansky, “Lie Algebras”; in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 126

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There’s one further type of brane in M-theory  that is really surprising.  This brane is the edge of spacetime. … The photons at the edge of spacetime participate in supersymmetric E₈ gauge theory.

-- Steven Gubser, The Little Book of String Theory (2010), p. 95