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.

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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!

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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.

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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.

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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.)

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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.

[Footnote 2026:  

> like a bare patch of land that’s been razed for construction

 

Terrain vague, quand tu nous tiens!

More here: 

http://worldofdrjustice.blogspot.com/2026/08/une-promenade-aux-terrains-vagues.html  ]

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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.)

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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

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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...)

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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.

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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: )

http://www.linguasacrapublishing.com/justice.html

QUIN? – ECCE!

Those of you who may have had occasion to follow, in another place, my periodic outpourings, my strange and mystical musings upon the penguins (shimmering in perfection,  upon the mind’s own ice),  may well yet ask yourselves:  What might all this lather of math  have to do with any of that?

            And the answer is simplicity itself.  We, the race of Men, having been created but a little lower than the penguins…. No, excuse me, of course I mean:  … created (as regards  complexity) but a little lower than the angels, right, the angels   – and, in obstreperousness, but little better than the devils – anyhow we, we are much too much of a mouthful/an eyeful/an earful/a soulful/    to survey all at once.  And so I …  project us onto the penguins,  these being (to use the topological term) a hyperplane in Peoplespace;  thére   the better to survey our kind  in petto.  -- Even so do the very angels, peering over their edge of clouds, survey (e’en)  us.

In Vitro/In Vivo

It often happens that, for ethical or political reasons, someone wants us all to stop doing A, and do B instead.   In addition to the straightforward though subjective arguments, we are sometimes told that B is “safer and cheaper and more effective” than A.

This is disingenuous.  Put on your logic cap -- If that were true, you wouldn’t need legislation to force people to do B.

The latest example is in an article in this morning’s Washington Post.   It’s on the front page of the print edition;  yet, with uncanny regularity, precisely the articles I find most interesting  get buried on the Web site; I don’t see it on their homepage anywhere.  Anyhow, here it is:

http://www.washingtonpost.com/national/health-science/chimpanzee-research-an-endangered-species-as-experts-debate-usefulness-ethics/2011/08/12/gIQAGt0xDJ_story.html

We there are informed that “chimp research is waning with the emergence of lower-cost, higher-tech alternatives.”  Oh well then, no controversy, right?  No, huge controversy.  Europe has banned the practice, and American facilities are under intense pressure.

I won’t wade into the substance of the debate, which is fundamentally ethical and thus it is hard to argue the other party around.   The only contribution of an armchair logician is to point out cases where conviction or emotion is dressed up as scientific fact.    As to the thorny complexus of scientific facts concerning medical experimentation, I have no idea; but then neither do the “parade of politicians, activists and famous faces, including former New Mexico governor Bill Richardson and chimpanzee champion Jane Goodall” who “mounted an uprising”.  There is, though, one overarching general point to be made.

Simulation of complex systems -- biological systems especially -- is very difficult.  Findings in the area of nonlinear dynamics have shown some sorts of simulation even of inanimate systems to be futile beyond a point.   To figure out what things can, and what cannot, be realistically simulated, and by what means, is itself a major scientific undertaking, by no means resoluble by pointing to some technique or other as being “higher-tech”.

Real-world problems, apart from their ‘local’ complexity, have indeterminate boundaries:  It’s unclear a priori what might be relevant.  Thus, take the recent snatch of Ben Laden.  As you can read in the current New Yorker,

http://www.newyorker.com/reporting/2011/08/08/110808fa_fact_schmidle

the op was meticulously simulated and practiced over and over again, using a replica of the Abbottabad compound.  Only, the walls of the actual compound were solid, whereas those of the replica were a chain-link fence; and this turned out to have major aerodynamic consequences, so that our high-tech chopper crashed in its own rotor wash.  Oops.

Biological testing is even trickier.  Take the horrendous case of TGN1412. “The data it had seen showed that TGN1412 had appeared to be properly tested on animals, leading it to give the go-ahead for human trials. The regulator had approved the use of the drug at 1/500th of the dose used in animal testing, ‘so it has a high safety margin built in’ “.  No doubt the predicted risk was calculated with an impressive flourish of high-tech decimal points.

This was the result:

Six men are in hospital intensive care – two of them in critical condition – after participating in trials of a new drug intended to treat chronic inflammatory conditions and leukaemia.

The scene was a living medical hell, say witnesses. After being injected with the anti-inflammatory drug TGN1412, patients began tearing their shirts off, screaming that their heads were going to explode. One patient's head swelled to triple its normal size, and patients were soon passing out, vomiting, or screaming in sheer terror.

Oops.

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You can get the impression that, as science marches on, discoveries become routine, predictable -- perhaps programmable.   And it is true that we have discovered much, even in biology.   But much of the progress has been by serendipity -- the polite name for a happy accident -- and by fiddling around in the lab or the field, and keeping your eyes open.  The discovery of penicillin is the best-known instance. Here is a homelier example:

In 1989, a group of Canadian researchers studying a blood pressure drug were astonished to discover that drinking a glass of grapefruit juice dangerously increased the drug's potency.

They were testing the effects of drinking alcohol on a medicine called Plendil. The scientists needed something that would hide the taste of alcohol so that subjects would know only that they were taking the drug and not know whether they were drinking alcohol with it.

"They thought it was a joke. We had trouble getting it published in a major medical journal."

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On economic modeling:

Seldom are the premises of such models examined closely.  Their appeal is in the chrome and roar of the engine, not the velocity or destination.

-- Edward O. Wilson, Consilience (1998), p. 203

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By contrast, one area in which computer modeling has been spectacularly successful  is space-flight. 

This makes sense.   The deal here is gravity, and its detailed mathematical study goes back to Newton.  Since the spacecraft itself weighs so little, we are dealing with a one-and-a-half-body, sometimes a two-and-a-half-body problem.   Both masses and speeds are too low to require relativistic corrections.  So we are dealing with just a plain old inverse-square law.  Piece of cake.

As a result, NASA has come up with aero-mathematical tours de force -- dynamical chaos, microsteering, sweet spots in the gravity-landscape…

[Update 17 Nov 2011] A highly interesting new article here:
http://www.slate.com/articles/health_and_science/the_mouse_trap/2011/11/lab_mice_are_they_limiting_our_understanding_of_human_disease_.html

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I have been careless, above, using “model” and “simulation” as quasi-interchangeable.   Here is a distinction -- whether a description of actual current practice, or a proposal for tidying our vocabulary, doesn’t matter;  the distinction is a good one:

The difference between a model and a simulation  might be that a model is for the purpose of understanding the phenomenon of interest, whereas a simulation’s purpose  is forecasting.

-- Russell Standish, Theory of Nothing (2006; 2^nd edn. 2011), p. 29

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