Monday, January 26, 2015

Minimalism in Mathematics (further updated)

A disclaimer:   What follows is not a substantive proposal, but a suggestive meditation, turning over this minute but multifaceted notion of “minimalism” and seeing how the light glints off.  It is neither better nor worse than a metaphor.

A couple of years ago,  a book-length treatment was published  that similarly plays with the notion of (in this case) “modernism”  -- which, like “minimalism”, is originally a term of the arts -- in relation to math:  Plato’s Ghost:  The Modernist Transformation of Mathematics, by Jeremy Gray.   To the extent that such an enterprise is worthwhile, it is in casting a bit of light from innovative angles, rather than deepening one’s understanding of math itself (though it did manage to get published by Princeton University Press):  it is more like a bull-session than a milestone.    Reviewing the book for American Scientist (Sept 2009), the mathematician Solomon Feferman sums up by quoting a remark by the historian Leo Corry, to the effect that
Extending the appellation modernism to mathematics … is like “shooting an arrow and then tracing a bull’s eye around it.”

Our own effort, in seeking resonances with the prior notion of minimalism, in mathematics, physics, and linguistics, is open to the same remark;  but it is what it is.

In the stylistic spirit of minimalism (and of that pointilliste Wittgenstein), we shall begin with a Delphic  epigram:

Logicism:  a kind of reductionist minimalism.


Considering that he took on the whole universe, in his methods  Newton was surprisingly Spartan.  Not only as regards “hypotheses non fingo”, but methodologically:

Newton consistently preferred Euclidean-style proofs.  He used his own calculus only where strictly necessary, and barred algebra from his treatise  entirely.
-- Leo Corry, “The Development of the Idea of Proof”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008).

(Cf. a laborious non-analytic “elementary” proof in number theory.)
As fastidious as an Intuitionist!


Not a matter of method, let alone of taste, but sheer fact (albeit initially so counter-intuitive as to have been dubbed a "paradox"):
The Löwenheim-Skolem theorem: if a first-order theory has a model, then it has a countable model.

The attempts, lasting centuries, to do away with the Parallel Postulate by deriving it from the other Euclidean axioms, represent a remarkable early manifestation of the minimalist instinct.  Success would not have added to our fund of theorems about geometry, nor led to more perspicuous proofs.  The impulse was in part aesthetic.

A topic to explore:  the relation between abstraction in mathematics (an intellectual quality) and mathematical minimalism (which is not antecedently defined, but I have in mind the aesthetic, even spiritual side).

Contrast Finitism, Intuitionism, etc.:  Not Minimalism, but self-castration.

Zijn lange, magere  maar gespierde gestalte,  zijn scherp ascetische gelaatstrekken…

There is also a sterile sort of minimalism:  as, the replacement of the standard set of logical symbols AND, OR, NOT, by a single one --   NOR or  NAND (Sheffer’s stroke).  It led nowhere.


A variety of the Minimalist instinct  characteristic of abstract mathematics  is the notion of elegance.   Its role in mathematical practice (it has no purchase on mathematical fact) is reminiscent of, though practically distinct from, that of beauty in the practices of physics.

This, from a man with one foot firmly in either camp, math and physics:

The development of mathematics may seem to diverge from what it had been set up to achieve, namely  simply to reflect physical behavior.  Yet, in many instances, this drive for mathematical … elegance takes us to mathematical structures and concepts  which turn out to mirror the physical world in a much deeper and more broad-ranging way…

-- Roger Penrose,  The Road to Reality (2004), p. 60

(This is the "unreasonable effectiveness" motif.)


We earlier noticed what we called “the Dialectic of the Topological Enterprise” -- abstracting-away from rich familiar entities, extracting what seem the essentials, and seeing what happens.   The first step might seem Minimalist, but the consequence is an effusion and exfoliation of new spaces which meet the newly relaxed criteria, and which turn out to have an even richer riot of properties than we began with.   Per se, there is little in all this that might justify bringing in the aesthetically-tinged label of “Minimalist” (not a traditional term in mathematics; the closest you get is “abstract”):  but the aesthetic ethos is there, for all that.  Thus Shing-Tung Yau, The Shape of Inner Space (2010), p. 77:
We start with some raw topological space, which is like a bare patch of land that’s been razed for construction.  On top of that, we’d like to build some kind of geometric structure that can later be decorated in various ways.


In the arts, Minimalism is a preference:  which, once adopted, is striven for.  In mathematics, you might like to keep things as simple as can possibly be:  but the mathematical facts seem to have a will of their own, at times.   Roger Penrose gives several instances of this, in The Road to Reality (2004).  For instance, with real functions, you can do pretty well as you like; but complex functions have a built-in naturalness.  You can try to define one on a given domain, but they have a mind of their own, and expand to their natural maximal domain by analytic continuation.   Thus, the larger set of numbers, the complex, spanned by the reals and the imaginaries, turn out to be in some sense more ‘real’ -- more round, more natural -- than the “reals” themselves.
Or again:   Suppose, once-bitten by the set-theoretic antinomies, you become twice-shy, and (p. 373)
adopt a rigidly conservative ‘constructivist’ approach, according to which a set is permitted only if there is a direct construction for enabling us to tell when an element belongs to the set.

(I picture this hypothetical constructivist as being played by Graham Chapman doing his officer’s shtick.)   But alas!  Penrose runs through the Turing/Cantor diagonal arguments and concludes (p. 376):
What this ultimately tells us is that, despite the hopes that one might have had for a position of ‘extreme conservatism’, in which the only acceptable sets would be the ones -- the recursive ones -- whose membership is determined by clear-cut computational rules, this viewpoint immediately drives us into having to consider sets that are non-recursive. … We are always driven to consider classes that do not belong to our previously allowed family of sets.

This is either a baffling, even a provoking mystery, or a simple consequence of what the Cantorian Realist indeed believes:  that these things are Out There, independent of ourselves (this might remind you of a certain Deity), and you can’t just methodologically sweep them away.   U B the judge.

(For a similar example applied to physics, click here.)


Pedagogical observation from a wise observer, who has been around the block:

Instead of the principle of maximal generality that is usual in mathematical books, the author has attempted to adhere to the principle of minimal generality,  according to which  every idea should first be clearly understood in the simplest situation;  only then can the method developed  be extended to more complicated cases.
-- Vladimir I. Arnold, Lectures on Partial Differential Equations (Russian edition 1997; English translation 2004), Preface to the second Russian edition


The nec plus ultra  of mathematical minimalism  is probably Category Theory -- which, however, I cannot elucidate, since I do not understand it.  It contains such things as the Forgetful Functor (this pops up in several introductory treatments, so it’s not as though I’m grasping at straws), which, given an algebraic group, “forgets” the group structure, leaving you with just a set  (excuse me: an element of the Category of Sets.)   Great -- die Gruppe ohne Eigenschaften.   The only way this even begins to seem to have a point  is if you then consider the adjoint functor, from sets to… free groups (these being a desolate Last Year at Marienbad landscape, again groups with the flavor removed).   Category theory looks at the bare bones common to many a different area of mathematics -- rather as though one were to study portraiture by looking at stick-figures.
(Actually, there is an analogy with the motif-index in folklore.  So, not knocking it here...)

On Ramanujan’s notebooks:

There were thousands of theorems, corollaries, and examples.  For page after page, they stretched on, rarely watered down by proof or explanation, almost aphoristic in their compression, all their mathematical truths  boiled down to a line or two.
-- Robert Kanigel, The Man who Knew Infinity, p. 204

The reasons for this were twofold.  Ramanujan himself was not particularly aphoristic.   But he had never absorbed the modern notion of proof, which would take up so much more space;  and as a poor man in India, he suffered from a shortage of paper.


From a logician:

The power-set operation has been interpreted  in the constructible hierarchy  as thinly as possible … We might be tempted to think of [the minimal model] as realizing a sort of contrary of the principle of plenitude -- a principle of paucity, if you will.     The principle of ontological parsimony … encourages some authors to eliminate individuals and un-well-founded classes.
-- Michael Potter, Set Theory and its Philosophy (2004) , p. 254

(All so difficult.  Why not relax with a mystery story instead?  Cool ones here: )

No comments:

Post a Comment