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

Arnoldian Epigram

            Proofs are to mathematics  what spelling is to poetry.

            -- V.I. Arnold

Abstraction and Generality

In the following post (q.v.)

http://worldofdrjustice.blogspot.com/2011/10/consilence-in-mathematics.html

we examined some related ideas -- analogy, generalization, abstraction -- that characterize the practice of doing math.  Herewith some further terminology along similar lines:

Geometry (especially differential geometry)  clarifies, codifies, and then generalizes  ideas arising from our intuitions about certain aspects of the world.

The theory of differentiable manifolds  is a natural result of extending and clarifying   notions already familiar from multivariable calculus.

-- Jeffrey Lee, Manifolds and Differential Geometry (2009), p. xi - 1

I find it unsatisfactory to “classify” partial differential equations:  this is possible in two variables, but  creates the false impression that there is some kind of general and useful classification scheme  available in general.

-- Lawrence Evans, Partial Differential Equations (1998, 2^nd. edn. 2010), p. xix

In contrast to ordinary differential equtions, there is no unified theory of partial differential equations.  Some equations have their own theories, while others have no theory at all.  The reason for this complexity  is a more complicated geometry.   In the case of an ordinary differential equation, a locally integrable vector field (that is, one having integral curves) is defined on a manifold.  For a partial differential equation, a subspace of the tangent space  of dimension greater than 1  is defined at each point of the manifold.  As is known, even a field of two-dimensional planes in three-dimensional space  is in general  not integrable.

-- Vladimir I. Arnold, Lectures on Partial Differential Equations (Russian edition 1997; English translation 2004), p.1

His Berkeley colleague concurs:

There is no general theory known  concerning the solvability of all partial differential equations.  Such a theory is extremely unlikely to exist, given the rich variety of physical, geometric, and probabilistic phenomena  which can be modeled by PDE.

-- Lawrence Evans, Partial Differential Equations (1998, 2^nd. edn. 2010), p. 3

This shows a becoming modesty, as against the media-physicists “quest” for a “Theory of Everything” -- there may be no ToE even for PDE’s.  Yet the reason he cites for their presumable non-existence seems weak, compared with that given by Arnold:  the unexpectedly rich variety of applications of this or that area of mathematics  is precisely what gave rise to the marveling at “the unreasonable effectiveness of mathematics”.

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~ Commercial break ~

Relief for beleaguered Nook lovers!

http://murphybros.blogspot.com/p/read-murphy-on-your-nook.html

We now return you to your regularly scheduled essay.

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In this post

http://worldofdrjustice.blogspot.com/2011/06/on-vulgar-numbers.html

we described the undergraduate ‘crush’ upon abstraction, almost for its own sake.   Herewith a caution, from a very old hand in the game.

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

Indeed, the point is not one merely of “simple” versus “difficult”:  rather, the “simplest situation” he is referring to is typically the motivating example of the theory.   Thus, the notion of a Boolean semi-ring was inspired by the facts about the Natural Numbers.

He goes on:

Although it is usually simpler to prove a general fact  than to prove numerous special cases of it,  for a student  the content of a mathematical theory is never larger than the set of examples that are thoroughly understood.  That is why it is examples and ideas, rather than general theorems and axioms, that form the basis of this book.

Bringing it all back home:

We could perhaps refer to the fact that both these statements have already been proved in Chaper III … but we prefer to prove them here  without getting involved ... with other more general problems.

-- A. D. Aleksandrov, “Non-Euclidean Geometry”, in: Aleksandrov et al, eds, Mathematics: Its Content, Methods, and Meaning (publication in the original Russian: 1956;  Eng. tr. publ. 1963), III.125.

There is wisdom in this -- not as regards the essence of mathematics in its Platonic sphere, but as regards mathematical truth proportionate to our understanding.

As:   You, the father, could say to your two-year-old, who has just snatched a cookie from the trembling fingers of his little sister:   “No!  Bad!  No steal cookie!”   -- Or you could say:  “Ahh, my young fellow!  A perfect illustration of the application of the Categorical Imperative, presented to the world by Emmanuel Kant.  Thus, let us take as given, that ….”

~

Again, the dialectic, or at least the give-and-take:

Too large a generalisation  leads to mere barrenness.  It is the large generalisation, limited by a happy particularity, which is the fruitful conception.

--  Alfred North Whitehead, quoted in James R. Newman, ed. World of Mathematics (1956), p. 411

A sharper form of generality is duality.   In its full precision, this is a concept by itself, and deserving a separate essay.  But in the following informal treatment, the term is introduced as a sort of way-station:

Before leaving 1-forms,  we digress to point out  that there exists a form of duality between the analysis and the geometrical notions …

Curves: γ is closed iff ∂ γ = 0

1-forms: ω is closed iff dω  = 0

-- Creighton Buck, Advanced Calculus (1956, 3^rd edn. 1978), p. 506

And likewise for ‘bounding’ vs. ‘exact’.

Now -- this might strike you as striking  for the wrong reason:  ‘closed’ means something different in either case, as do curly-d and d;  these terms and symbols were chosen with insight aforethought, and in themselves indicate nothing.   The real meat comes in the theorems, e.g. every closed 1-form is exact iff every closed curve is bounding.

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Adventures in Algebraic Geometry

The closest I ever came -- and that  unwittingly -- to a brush with algebraic geometry,   was in freshman calculus.  (This was back in the ‘sixties -- before your time.)  The instructor was Robin Hartshorne, a young and winsome elf of a man.  He was an engaging lecturer, teaching from  or at least in parallel to  a beguiling text (Spivak’s Calculus, so utterly different in spirit from the dry Thomas treatise that had repelled me in high school to the point of dropping the class);  moreover he was -- now that I think back on it -- the first deeply intelligent teacher I’d ever had (though there were to be others -- notably Gleason):  up through high school, there had been no hint that such creatures even existed.

Still, he wore his learning lightly, like his tweeds.  The class was fun.  Particularly endearing -- though also startling, at the time -- was one day in the second semester, when he was presenting the topic of definite integrals -- painful but necessary, rather like a rectal exam.   He was chalking away, when all at once he seized up, staring in bemusement at the blackboard;  then turned to us with a sheepish grin.

“It’s been a long time since I’ve done one of these,” he said.

~

Now -- if the anecdote stopped there, it would be just one more ultimately stupid instance of Genius Porn :   the populace cooing contentedly when told that Einstein flunked grade-school math,  or that Gauss was late to learn to speak, or that Erdös tried to cut a grapefruit with a butter-knife  (which indeed he did, though it was a craftily calculated move).   These falsely flatter our vanity.  They are the opposite of a much better genre of joke, which you need a bit of math to actually understand, such as the one about von Neumann and the summation of infinite series.

For the incident, trivial until viewed in the light of later developments, did  there and then  plant the seed of doubt and wonder, as I sat theretofore clueless in the second row.   His being momentarily at a loss  struck me  at the time  as quite surprising, almost inexplicable:  as though a test-pilot, stepping from his aircraft into his roadster, were to stare at the ignition and say, “Remind me how to start one of these.”

Had I continued with a chemistry major as originally planned (well, originally-originally an English major, until I realized, with chill horror, the error of my ways), and thus retreated or perhaps advanced  depending on how you look at it, into ever-more-technical intricacies  and mechanical practicalities, the significance of that incident would never have become apparent.  But as it was, Hartshorne’s class (and Spivak’s sparkle) were instrumental in turning me towards a concentration in pure mathematics -- which was the only kind of mathematics they really taught at Harvard (golden memories of that climate of abstraction here), and eventually towards having a go at Berkeley towards a Ph.D.   Accordingly I was to be introduced to as-yet-unsuspected levels of intellectual depth, such that each, compared with the one before (I speak loosely;  they are basically incomparable), is as the definite integral to 2 + 2:  rising like the serried ranks of angels.  And though I myself never progressed beyond the level of the cherubim, it was sufficient to glimpse that empyrean wherein, indeed, you might forget the particular monkey-tricks used to solve thorny individual definite integrals (basically you just memorize these, storing them for reference like tools in a toolkit, unless you’re Euler or von Neumann, in which case you re-derive them instantly from scratch, or simply perform a brute-force numerical calculation in your head).

As each glowing level is added, the one below  becomes obsolete ...

Moreover, the tired old cart-horse of the calculus was, it turns out, very far from the centers of Hartshorne’s research interests, which are almost unimaginably abstract.   In that pokey little classroom in Massachusetts, he was really only on loan to us from Sagittarius, having once studied with Grothendieck, a confirmed extraterrestrial.  It was bruited about that he had something to do with something called projective geometry;  but only much later was I to learn that he is one of the pioneers of …

sheaf theory

… a topic so ferociously abstract, that even to define what sheaves are  is utterly beyond me.  (Wikipedia doesn’t even try, observing that “their correct definition is rather technical”.)  Nay, wert thou to gaze upon this theory naked -- not even to speak, not even to breathe of its further generalizations in topos theory -- ‘twould make thine eyes, like stars, to start from their spheres, and thine each particular hair -- nay more, thou wouldst  in sooth  explode in flames,  as did dame Semele,  when  all unheeding she beheld,   unveiled,  great Zeus in all his lightning !
 

~

~  Posthumous Endorsement ~

"If I were alive today, and in the mood for a mystery,

this is what I'd be reading: "

I Don't Do Divorce Cases

Murphy on the Mount.

(Je m'appelle Evariste Galois, and I approved this message.)

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As so often when some movement of math has been seen streaking off westwards  out into the void, presumably never to been seen again by mortal man, it reappears shining in the east, reborn in some applicable form -- thus suggesting, you will notice, that the global topology of the noösphere is toroidal.  In the present case, algebraic topology has come to be crucial in such applications as: string theory (via its prior discovery of Calabi-Yau manifolds), coding theory,  cryptography and steganography -- which means that the juicy bits are probably highly classified.

And this raises the interesting paradox, which Epimenides would have relished, whether someone like Hartshorne is Cleared for the contents of his own head.

Robin Hartshorne as seen in a recent spectral image.  Since the old days, he seems to have sprouted quite a bundle of fibres over his base-space.

 

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I recently happened upon an essay by the usually abstruse Samuel Eilenberg (of Eilenberg-and-Steenrod notoriety), written for a collection sponsored by the Office of Naval Research.  In deference, perhaps, to the needs of our seamen, Professor Eilenberg permits himself some observations and analogies   that lie within the reach of the common folk.  Thus:

The analogy between sheaves and covering spaces  is very close.

-- Samuel Eilenberg, “Algebraic Topology”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 110

Bingo !  Covering-spaces I get.  Of course, I don’t actually see the analogy, but at least it’s a comfort, knowing that there is one.

[Update, Thanksgiving 2014]   Edward Frenkel, noticing my perplexity, kindly sent in this explanation:

Coverings and sheaves are related. And it's not just an analogy. A covering space is an example of a sheaf (the simplest example): It is a sheaf of finite sets (provided that the covering is finite).

Namely, given a covering p: C --> X of a manifold X, and given an open subset U of X, the set of sections of of the corresponding sheaf over U is just p^{-1}(U) [the preimage of U in C under p]. Note that the stalk of this sheaf over a point x of X is just the fiber over x [the set of points in C, which project down onto x under p; p^{-1}(x)].

The covering gives us a way to "glue" these fibers together (indeed, set-theoretically, C is the union of these fibers -- but it's more than that, because C is a manifold, just like X; so C is not a "disjoint" union of these fibers, they are really "glued" together in a particular way).

The simplest covering space is the trivial one: a union of N copies of X, each mapping identically to X under p.

Here is a non-trivial example: let X be a circle. Now take the Moebius strip in which this circle is the circle "in the middle." Take the "edge" of the strip -- this will be your C. Notice that for each point in your original circle X, there are two points in C. But C is NOT the union of two circles (which would be the trivial double covering). In fact, C is just ONE circle, covering another circle (our X) in a non-trivial fashion.

A general sheaf is very similar. The difference is that the fibers could be infinite, or they could be vector spaces, etc. But the idea is the same.

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We earlier discussed  the memoir Souvenirs d’Apprentissage, by a pioneer of algebraic geometry, André Weil.   Avid for more, we got hold of a copy of the memoir Random Curves (2008), by a contemporary algebraic geometer,  Neal Koblitz, well known to anyone with an interest in Elliptic Curve Cryptography. (It came in via InterLibrary Loan -- interestingly, the lending institution turned out to be the U.S. Naval Academy in Annapolis.    Compare the publishing venue of the Eilenberg article referenced immediately above.  We salute the broad interests of our midshipmen!)

Both authors have a wide range of interests and experience outside of mathematics;  both engaged in extensive foreign travel;  and it is of this that they principally write:  the reader will enjoy these accounts for their own sake, but we set down the volumes with a twinge of disappointment, that we are no closer to insight about algebraic geometry than we were before.  However, one biographical detail did strikingly stand out.  Both authors took a principled stand against unjust wars:  and this, not simply by penning valiant Letters to the Editor from the safety of their studies, but by a brave and almost reckless defiance while actually serving in the armies of their respective countries:  actions that could easily have led to their injury or even death, and which did actually lead to their imprisonment (and, in Koblitz’s case, to a severe beating).  But what is truly remarkable is that, in both cases, they used their time in the slammer far more profitably, mathematically, than most of us  use ours  even in the best of circumstances.   It was there that Weil did his seminal work, and there that Koblitz returned in concentrated form to the practice of algebra which he had largely abandoned during two years of political turmoil.

Intrigued, I wrote to the latter author, inquiring whether, from the standpoint of Kolmogorov-style measure-theoretical probability theory, we may validly generalize from this sample of two (2);  and he was kind enough to reply:

I love generalizations based on small samples.  I'm sure a lot of algebraic geometry was nursed at the Indiantown Gap army stockade!

Thus encouraged, I here make bold to speculate about the martial philosophy of Robin Hartshorne.  I know nothing of his politics, but truly cannot imagine the man wielding an M-16.   Or a flyswatter, for that matter.   Were a mayfly to venture into his office, Hartshorne would no doubt observe the pattern of its flight (musing all the while on the brevity of this earthly life) and calculate whether that trajectory describes an elliptic curve. -- Which, come to think of it, it just well might.   Many mathematical treasures remain to be unearthed in the field of biology!  (For a few of these, see the fine book by Ian Stewart, Life’s Other Secret.)

~

A rather huffy response to modern algebraic geometry, which it is a pleasure to reproduce  mainly because I do not understand the subject, and which suggests (like the characterization of Category Theory back when I was in college, as “the higher macramé”) that (as with these new-fangled things called “computers”) I am perhaps not missing much:

Attempts to extend the geometry of second-order surfaces  and the algebra of quadratic forms  to objects of higher degrees  quickly leads to  the detritus of algebraic geometry, with its discouraging hierarchy of complicated degeneracies, and answers that can be computed only theoretically.

-- Vladimir I. Arnold, Lectures on Partial Differential Equations (Russian edition 1997; English translation 2004), Preface.

Harumph!  Hear hear!