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!

*

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.

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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.)

*

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  ]

*

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

*

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

On Excessively General Questions (further generalized)

In our essay, “On What There Is”,  we confessed ourselves unequal to the task of addressing the question of Being, bare.   Compare further:

You asked me the use of criticism.  You might just as well have asked me  the use of thought.

-- Oscar Wilde, “The Critic as Artist” (1891)

Bizarrely, though the question was meant satirically, that very phrase occurs as a chapter-title in a book written by the philosopher and historian Ernest Gellner  in all seriousness:  “The Uses of Thought”.

The succeeding chapter-titles are equally grandiose:  “The Uses of Doubt”, and “The Stuff of Change”.  This occur in his volume from 1964, the even grander Thought and Change.  Yet this is odd, since Gellner’s general thrust  is deflationary.

Cf. also the classic Heideggerian title, Was heisst denken?, to which we have often had occasion to allude.

A Dive to the Depths (expanded)

A phrase you will often meet in higher mathematics, and almost nowhere else, is:

“a deep result”

O loveliest of monostichs, thou !

The very notion of what ‘deep’ means, in such a context, is itself deep;  indeed, too deep for me, at present.   This, owing to a crippling condition of mathematical oligophrenia.  --  which, however, I pray that time and diligence might partly palliate.  (For a glimpse into the terrible sufferings of mathematical oligophreniacs, click here, if you dare.)  Yet I am putting up this skeletal promissory-note of a post, so that there will be a space to scribble insights as they wake me in the night.

First off -- the term "deep" does not mean simply ‘difficult’; indeed, though such results lie in the depths, and are not to be had for the asking, once you have somehow managed to fish one up, it may seem clarity itself.  Nor does merely being difficult make anything deep.  Any humongous brute-force calculation falls into that category;  for a more-substantive example, consider the Four-Color Hypothesis, which people suspected should be deep, but the proof that changed "Hypothesis" to "Theorem"  is a combination of clever tricks and elbow-grease.  The response of the mathematical community was disappointment:  "So, it turns out it wasn't an interesting conjecture after all."  (Of course, it may yet prove to be "interesting" in our cognitive human sense; that awaits a proof of an entirely different kind.)

We may go further, and put forward that an overarching purpose of mathematical research is to reveal something previously difficult  as now simple, when seen in the right way.  Again and again this has happened in history, beginning with the replacement of finger-counting by symbols, and of clunky symbols like Roman numerals by decimals.  For a more recent example:

Although Beurling’s own proof [characterizing invariant subspaces of an operator on Hilbert space] was quite involved, it is by now simple to prove;  it depends on hardly anything more than the geometry of Hilbert space.  The profitable point of view  is not sequential but functional.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 17

Nor is there any simple similarity or opposition between being “deep”  and using what are now called “elementary” methods in number theory (as opposed to analytic methods), e.g. Alte Selberg’s proof of the Prime Number Theorem by elementary methods, compared with earlier analytic proofs by Hadamard and others.  Typically, proofs that restrict themselves to “elementary” methods are harder than those that permit themselves a more capacious toolkit;  but whether they ever, or generally, gain depth via this austere discipline, I have no idea.

In the meantime, some related posts outside of a mathematical context  are these:

            On Depth and Breadth

            On Scope and Difficulty

As appetizers, try the following hors-d’œuvres platter -- to follow which, however, we have as yet prepared no meal (as with our early essays on the Realist vernacular, this is more by way of linguistic warm-up):

The connection between linear transformations  and bilinear functionals  goes quite a bit deeper …

-- Paul Halmos, Introduction to Hilbert Space (1951, 2^nd edn. 1957), p. 38

The Riesz representation theorem depends on some of the deeper parts of the theory of measure and integration.

--George Simmons, Introduction to Topology and Modern Analysis (1963), p.

The analogy between cyclotomic fields  and fields formed from the points of finite order on elliptic curves   is very deep.

-- Neil Koblitz, 1993

The notion of Kan extensions is the deeper form of the basic constructions of adjoints.  We end with the observation that all concepts of category theory are Kan extensions.

-- Saunders MacLane, Categories for the Working Mathematician (1971; 2^nd ed. 1998), p. vii

The continued-fraction representation of real numbers is deeper than the decimal expansion.

-- Roger Penrose, 2004

There are deep ties between enumerative geometry and Ramanujan’s tau function.

Contrast:

The useful fact about products of projections  lies near the surface.

-- Paul Halmos, Introduction to Hilbert Space (1951, 2^nd edn. 1957), p.  47

In the same work, while acknowledging the possibility of a lattice-theoretic formulation of the subspaces of Hilbert space, he dismisses this possibility as “trite” (p. 22).

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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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Much commoner than “trite” is trivial, which is virtually a terminus technicus of mathematical practice.  Let one quote stand for all:

One of the useful conclusions we can draw from Theorem 2 [to the effect that the norm of a Hermitian operator equals the supremum of its eigenvalues] is that the spectrum of a Hermitian operator  is not empty.  This is not a trivial conclusion.  We shall obtain the corresponding fact for normal operators  only after the application of a lot more relatively deep analysis.

-- Paul Halmos, Introduction to Hilbert Space (1951, 2^nd edn. 1957), p. 55

Psychosociological note:   Attempt to imagine the effect on our pumped-up math-major  freshman or sophomore brains, hearing dismissed as “trivial” (with a wave of the hand) propositions which, but a year before, we ourselves would not have begun to understand, and which indeed most of our countrymen will go peacefully to their graves without understanding.  (For more on the hubris involved, click here.)

Also note:  What counts as “trivial” is relative to where you stand.  Thus, in the very next sentence, Halmos adds:  “We hereby report that the spectrum of an arbitrary operator is also not empty;  since we shall have no occasion to make use of this fact, we shall not enter into its proof.”  The proof, one gathers, is more difficult still.  But when once you have mounted, and stand upon that summit, the fact that Hermitian operators in particular have eigenvalues, is trivial indeed.

Leave it to mathematics to recruit even the notion of triviality into some highly non-trivial constructions.   E.g.

A topological space over X is called a locally trivial fibration if every x in X has a neighborhood over which Y is trivial.

-- Klaus Jänich,  Topology (1980; Eng. trans. 1984), p. 129

And:

Sard’s Theorem … is … a highly non-trivial  theorem  which is elementary in the sense that it uses only the notion of a differentiable map.

-- Shlomo Sternberg, Lectures on Differential Geometry (1964)

The terms deep and elementary (here in the everday sense, and not the special number-theoretic meaning mentioned above) are not antonyms, but they do contrast:

The spectral theorem implies that every normal operator has a large supply of invariant subspaces;  this is classical  and can be considered elementary by now.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 17

This dependence of “depth-perception” upon experience accumulated with the passage of time, does not refute the notion of relative mathematical depth as ill-defined.   What is intuitive though difficult to put into words  is a notion of “deeper than” rather than of absolute depth.  To the giant, neither the pond nor the puddle appears deep;  but the pond is deeper  for all that.

~

Re the classification of simple algebras (“simple”, to be sure, in a certain technical sense, meaning roughly: incredibly complex and difficult):

The tools employed  are not deep; they are just, so to speak, linear algebra  raised to the nth power.

-- Irving Kaplansky, “Lie Algebras”; in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 122

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Further attestations.

A pioneer of Hilbert space theory, expounding the then-contemporary state-of-the-art for a nonspecialist mathematical audience, particularly as regards dilations and extensions of operators:

There do not seem to be any conspicuous and challenging yes-or-no questions that serve to indicate the direction in which the search for new results might begin,  but I have faith.  There is depth in the subject;  the trouble is that the surface has not been explored enough  to show where the deepest parts lie.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 17

Writes a premier algebraic-topologist:

There are geometric problems which require the use of the multiplicative structure of the topological invariants.  Such problems are deeper than those which can be solved by considering the additive structure alone.

Samuel Eilenberg, “Algebraic Topology”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 105

Re Lie algebras and a closed subgroup H of the full linear group:

It is furthermore true (and this is deeper) that these one-parameter subgroups  fill a neighborhood of the identity in H, and consequently generate H if H is connected.  …The converse part of the correspondence  involves a subtlety of the type that makes the study of Lie groups a quite sophisticated topic.

-- Irving Kaplansky, “Lie Algebras”; in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 118

Re the Hodge conjecture:

It arises deep within the subject, at a high level of abstraction;  and the only way to reach it  is by way of those layers of increasing abstraction.

-- Keith Devlin, The Millennium Problems (2002), p. 9

This is a truly adult depiction of depth: as lying deep within the layers of the enigmatic onion.  It is not a case where you can just swallow some peyote and see it all in a flash.  
(For more, compare:  The Ladder of Abstraction.)

Writes a philosopher:

The axioms are not logical truths ... Their truth is established by intuitions which lie too deep for proof, since all proof depends on them.

-- Roger Scruton, Modern Philosophy (1994), p. 392

An example from outside the field of mathematics -- though it is a mathematician who is writing this:

Just how a protein manages to organize itself in space, using only the sequence of its own amino acids, remains a mystery, perhaps the deepest in computational biology.

-- David Berlinski, “What Brings a World into Being” (2001), collected in:  The Deniable Darwin (2009), p. 243

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Related vocabulary:

Related to the concept of depth (which focusses on the root of things) is that of richness (regarding the blossoms that bloom from this root).   Hadamard adopts this metaphor explicitly:

Application’s constant relation to theory  is the same as that of the leaf to the tree:  one supports the other, but the former feeds the latter.

-- Jacques Hadamard, The Psychology of Invention in the Mathematical Field (1945), p. 125

Further examples:

The algebra of composition of maps  resembles the algebra of multiplication of numbers,  but its interpretation  is much richer.

-- Lawvere & Schanuel, Conceptual Mathematics (1997), p. 11

… a recent, and surprising, theoretical advance by Winkler.  He shows that, for many … algebraically closed fields, the “free Skolemization” has a model companion.

-- Angus Macintyre, “Model Completeness”, in: Jon Barwise, ed. Handbook of Mathematical Logic (1977), p. 164

The concept of thickness [in graph theory] is the deep mathematical idea that underlies the recreational puzzle of Earth/Moon maps.
-- Ian Stewart,  How to Cut a Cake (2006), p. 126

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Having convinced ourselves, by the examples of mathematics, that there is more to this assessment-word deep than an emotional or impressionistic grunt,  we look to some cases outside of mathematics where an idea has been similarly assessed.

Some discoveries provide answers to questions.   Others are so deep  that they cast questions in a new light,  showing that previous mysteries  were misperceived.

-- Brian Greene, Fabric of the Cosmos

We are not at home in the world, and this homelessness is a deep truth about our condition.

-- Roger Scruton, Modern Philosophy (1994), p. 464

T.S. Eliot affirms that what is past and what is present, even what might have been, indicate a present purpose.  This is a metaphysical point of great depth.

-- James Schall, S.J., The Order of Things (2007), p. 69

And, more prosaically, but no less tellingly for all that:

Although running Bain Capital required a lot more brains and savvy than playing roulette does -- a lot more brains and savvy than most of us could even pretend to possess -- the job was not conceptually deep.  Romney did not develop a model of the world from the business of private equity. … “He’s not a very notional leader,” [said] Romney’s campaign spokesman …

-- Louis Menand, “Money Pol”, The New Yorker (19 III 2012)

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This is quite aside  from the path of mathematics, but -- it may be, that such depth is displayed in quite distantly allied regions:  all tracing back to Him, perhaps by some functorial construction.  In that spirit, this:

http://www.youtube.com/watch?v=s5xDEyuXaOY&feature=fvwrel

The final anguish  of the Asian bride  suggests the depth  of the Riemann Hypothesis.

The enigma of a woman’s heart,

finally espied  by a Private Eye,

for less than the price  of a Valentine …

This Rose

[Kindle]  [Nook]

~     ~     ~

Somewhat less far off the path …  Deep is indeed the term of art  in mathematics, antonymic to trivial.   Now compare, from another discipline, the word profound, in reference to Newton’s perplexing, little-known  philosophical-speculative opus:

That it is exclusively mystical  I do not believe -- that there is a mystical element  seems certain.  I hope that  one day  some profound student -- no one less will suffice -- will study this mass of papers.

-- E. Andrade, quoted in James Newman, ed. World of Mathematics (1956), p. 273

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Above, we saw the distinction deep vs. difficult.  Here now even the latter concept is bilayered:

Although Beurling’s own proof [characterizing invariant subspaces of an operator on Hilbert space] was quite involved, it is by now simple to prove;  it depends on hardly anything more than the geometry of Hilbert space.  The profitable point of view  is not sequential but functional.

-- Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 17

In infinite dimensions, it is tedious but not difficult  to construct spaces  strictly  but not uniformly  convex.

-- Prof. Lewis, University of Alberta, course in Functional Analysis, 1982

The proof that Reidemesiter moves  and planar isotopy  suffice to get us from any one projection of a knot  to any other projection of that knot  is not particularly difficult;  however, it is technically involved.

-- Colin Adams, The Knot Book (1994), p. 15

And here the original distinction is made even sharper:  though they are not interchangeable, depth tracks with generality, which in turn tracks (on the plane of praxis -- of proof) with simplicity:

Re the Denjoy-Young-Saks Theorem on the derived numbers of functions:

As we would expect  in view of the great generality of the final statement of the theorem,  the proof due to Saks is of extreme simplicity.

-- F. Riez & B. Sz.-Nagy, Leçons d’analyse fonctionelle [references to the English translation, Functional Analysis, 1955], p. 17

~

Here a leading mathematician laments the shallowness of his understanding of something he himself proved (regarding representations of a Kac-Moody algebra, as it happens):

My proof of this result was technically quite involved.  I was able to explain how the Langlands dual group appeared, but even now, more than twenty years later, I still find mysterious why it appears.  I solved the problem, but it was ultimately unsatisfying to feel that something just appeared out of thin air.

-- Edward Frenkel, Love & Math (2013), p. 181

This is setting oneself high standards indeed.   Shakespeare probably did not lie awake o’ nights fretting how the devil he ever came to write Hamlet;  Mozart did not find the bread of pleasure at having written the Sonata in A  turning to ashes at the thought that it might have been dictated to his unconscious  by an angel.

~

A near-synonym of the math-word deep, but shorn of all irrelevant aesthetic echo, is:  “highly nontrivial”.   The term is decidedly commendatory, though to a layman it might sound like faint praise, as were one to dub one’s lady-love “seriously unugly”.  The expression may be extensionally impeccable, but ‘twould never pass in a sonnet.

 Further:

In set theory, a forcing extension in Cohen’s sense  is reminiscent of algebraic extensions of a field, but

… the forcing method is far more complex, both conceptually and technically, involving set-theoretic, combinatorial, topological, logical, and metamathematical aspects.

-- Joan Bagaria “Set Theory”,  in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 625

Although he does not use the term "deep" here, that expanded characterization  "complex, both conceptually and technically", especially the "conceptually" part, points in that direction.

~

One motive for Frege’s choice  was again generality:

Does not the ground of arithmetic lie deeper than that of all empirical knowledge, deeper even than that of geometry?

-- I. Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 184

… [Cantor’s] remarks on functions of several variables (where the provability of theorems  was deepening the level of rigour in analysis)

-- I. Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 223

 

.

"Die Vermessung der Welt"

 Gauss saw things in terms of sheets embedded in three-space, the natural abstraction from his experience as a land-surveyor.  Riemann shifted the view to that interior to the space.

-- John Derbyshire, Prime Obsession (2004)

In an earlier essay,  we exulted (as sober man of science) and lamented (as blogging satirist) that there exists very little indeed by way of “math porn” (in the non-sexual journalistic sense), as against “physics porn”.  However, we did come across a (rather pale and marginal) example of that seldom-met genre, and now share it with you here.

I recently attended a screening of the 2012 German film “Die Vermessung der Welt” (literally geo-metry, in the etymological Greek sense) presented to a select circle of the deutschgesinnt.  What moved me to tear myself away momentarily from my usual daytime occupation of hand-to-hand rooftop combat with our adversaries, was the fact that the film was billed as a sort of dual biography of two scientific figures very much worth biographizing:  Alexander von Humboldt, the naturalist brother of the philologist Wilhelm, and Carl Gauss, probably the greatest mathematician who ever lived.   We watched it off a DVD;  had I known that the original theatrical version was 3-D, I would have lowered my expectations accordingly.  (Actually, 3-D could be put to very good use in the exploration of the Gaussian geometry of manifolds, but that was not its use here.)

The movie begins, as all Gauss sagas must, with the tale of how the young schoolboy, given a pensum  along with his fellows  of reckoning up the sum of the integers from one to a hundred, shot back an answer instanter, by finding a clever shortcut, rather than, as John von Neumann would have done, simply adding the series instantly in his head.  (That’s a joke.)   Our eighth-grade algebra class was regaled with this (and wisely so;  it’s one of the few things I remember), and our son, in the Princeton Friends School at a tender age, was instructed in the same as well.   But in the film, the anecdote was given what I take to be a possibly Germanic twist, for the scene opens with the explicitly filmed rhythmic  

    thwack,

                     thwack,

                                      thwack

of a supple and vicious-looking cane upon the bared buttocks of a lad of around nine;  graphic enough as it was, but probably even more disquieting in a theatre, with Surroundsound and 3-D.   Somehow, this sequence alone marked the movie out as not of American provenience -- here, you might be sent to prison for even watching it.  (Later, after Gauss has solved the arithmetic problem, the scene is repeated with Gauss as victim, for any viewers who didn’t manage to come to climax during the first sequence.)

So:  a rather pornographic presentation of what was in reality an utterly asexual and indeed incorporeal milestone in the annals of mathematical awakening.   But as long as we’re here, let us dwell -- as the film alas did not -- for just a moment  on the math part.

That sequence 1 + 2 + 3 + … + 100   equals, as it happens, 5050.  That fact is of no mathematical interest whatsoever, but belongs rather to the Museum of Particular Results.  (We presented a jolly fable of this notion here.   Be sure to click on that essay, it’s full of woodchucks.)   Of marginally more interest is the shortcut found by young Gauss:  pair the outermost integers in turn and you get 50 × 101.   That is clever enough;  but at the lowest level, it might be simply one of an unrelated jumble of dodges used by a Calculating Idiot-Savant, and thus belong to the Museum of Particular Tricks, just one step up from brute-force addition.  A significant step up from this recognizes that the trick is (with some tiny extra cleverness) generalizable to any sequence 1 + 2 + … + n.   Now it has risen to the level of a general trick, and thus belongs to the Museum of Particular Algorithms.   But then this finding generalizes to the idea of summation-formulas überhaupt:  for instance the sum of the squares of the first n integers, or the cubes, or any power.   The resulting infinite collection of formulas belongs to the Museum of Particular Strokes of Genius.   Striving to generalize these, you eventually wind up with Analytic Number Theory, and its sought-after crown jewel, the Riemann Hypothesis;  which is where things stand today.

None of this is even hinted at in the movie;  but really, such a development is the only reason to treasure that Gaussian anecdote:  otherwise the whole thing can seem a mere transient bit of precious cleverosity -- as it did (in the film’s telling) to Gauss’s schoolfellows, who give him a beating for his trick, and no doubt to the bulk of the audience.   And this is the “math porn” aspect of the presentation:  Even in the absence of any mathematical understanding whatsoever, we spectators are nonetheless supposed to be tremendously impressed with young Gauss, who is presented as a romantic and tragic figure, his attraction being thus, not Gaussian, but Byronic.

A superb mathematician -- and you can take that to the bank!

*

Für psychologisch tiefgreifende Krimis,

in pikanter amerikanischer Mundart,

und christlich gesinnt,

klicken Sie bitte hier:

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

*

~

Thus far, the perspective is that of male narcissism.   That stance applies as well to the portrayal of von Humboldt, as he goes stalking about the Amazon in his seven-league boots, freeing slaves as he goes, and making immortal discoveries.   Both portraits involve a certain taste of algolagnia (in the naturalist’s case, it involves his naked back and an electric eel -- all for science, you understand).   There is nothing explicitly homoerotic, though perhaps a touch of a repressed version of that, by implication, when von Humboldt goes apeshit upon discovering his handsome French traveling companion  dallying with a local squaw.

Subsequently, the movie tosses a bouquet in the direction of unearned autogynophilia as well, in the incident of young-man Gauss, still all sturm-und-drangy, brought wisdom from the Tree of Knowledge by a chance remark of a comely though uneducated Fräulein  posing Evelike with an apple.

Mathematically, the scene will almost certainly have soared over most of the audience’s heads.   Gauss chats about measuring the Earth (Vermessung der Welt) by adding up triangles;  the lass objects that the Earth is not flat …. (not a Euclidean surface, as we say in the trade) … portentous pause … Gauss, reflecting, says, Well, you’d need lots of leeetle weeentsy triangles (infinitessimal, mathematicae linguâ).  She sensuously/attentively pares the apple;  and the penny drops, the scales fall from his eyes, and he rushes off to scribble calculations.

What just happened -- and the viewer may well be excused for having missed it -- is that Gauss has (apple-prompted, like Newton and gravity) just discovered Gaussian curvature, differential geometry, and much of modern mathematics.  This episode will be utterly opaque to anyone coming fresh to the movie;  apparently, we are expected to have read the book, as with the Harry Potter movies (the latter of which were incoherent, and would have baffled anyone who hadn’t already read the series).  Which, indeed, the director had cause to suppose, since the movie is based upon a novel that was a humongous German bestseller.

*     *     *

~ Commercial break ~

For a mini-movie of our own, try this:

The Penitent and the Impenitent Thief

We now return you to your regularly scheduled essay.

*     *     *

The depiction of the wisdom-from-the-mouth-of-babes Mädchen is likewise Byronic -- namely it recalls Lord Byron’s daughter, Ada the countess of Lovelace, an associate of Babbage.  She has been exalted by those who go hunting in history for neglected heroines;  a summary can be found here:

http://en.wikipedia.org/wiki/Ada_Byron#Controversy_over_extent_of_contributions

In fact, though, Ada is a reasonably admirable and realistic role-model for girls, since she did hang around a really smart guy and did work hard and did achieve some understanding if not any actual original results, which is all that most of us can ever hope to do.  Gauss, by contrast, is no role-model at all, for anyone, since none of us have been born with his genius, which is almost unexampled in history.  Indeed, for any actual stellar mathematician, his example is yet worse, since he was notorious for hoarding results.  Hopeful young mathematicians would make the pilgrimmage to Göttingen to present their results (much as young Gauss himself is shown as doing, in a singularly infructuous interview with Immanuel Kant), only to be told that he himself had discovered those results long ago, and had them in his drawer, but had never bothered to publish them. (His dismissal of Bolyai in this regard  is notorious.)

Lagniappe:  Mathematically inclined lasses seeking ipsigeneric role-models would do better to follow Noether, Kovalevskaya, Julia Robinson, or Ingrid Daubechies.  Though, once you reach that level, you have come to realize that pure mathematics is entirely genderless, and even (so we have argued here and there in this series of essays) extraspecific.

Note:  Eventually, after an hour or so, weary of its pieties, and disinclined to take in  yet another sex scene (Memo to directors:  That is not why moviegoers flock to a film about mathematicians and scientists), I walked out.  So maybe I missed some dazzling final mathematical exposition.  But I doubt it.

~

Lesen Sie die Geschichte  spesenfrei !

“Der Detektiv und das Mädlein”

~

[Footnote:   For another psychologically attuned analysis of movies, click here.]

Uniform Spaces

[The following does not rise even to the level of an essay-in-progress;  more like a thought-in-progress, or even (saving your presence) a difficult bowel-movement.   But the hordes of typist-elves in the cavernous warehouses of WDJ  have yet to present anything brought to perfection this morning, and I wished not to disappoint the milling crowds that swarm this site each weekend, bringing the whole family, Sister Sue and Fido too, gawking at the glittering thoughtfronts -- the polemics, the poems, the darling little monostichs (these we can all afford) -- while shaking their heads sadly at the Trinitarian Minimalism and Cantorian Realism (out of our price-range) -- all  save one diminutive child towards the back of the bunch, eyes riveted on the prize, instinct with penetrating understanding…]

We saw here the dialectic of mathematical invention (not trying to be too Hegelian here -- think of it as an ensouled pendulum) whereby, beginning with the everyday world we live in -- I almost wrote ‘space’, but that would be to get ahead of our tale -- we abstract from the clutter of minute-to-minute experience, and conceive of it all happening within a space.   We then formalize that space with the Euclidean axioms.   We then familiarize ourselves with this new mind-environment, solving tricky problems and whatnot for a couple of thousand years, then -- since we have long effectively been working in the World of the Unseen -- very lightly generalize to Euclidean spaces of any finite dimension  -- a bit of a stretch biologically, but where, mathematically, everything works pretty much as before.

Meanwhile independently, mathematical analysis had proceeded apace, not necessarily concerned with the geometrical substrate as such, but piling up its own increasingly intricate problematics.   Then by an ideational leap which is of the essence of mathematics, and into which simply listening to lectures and slogging through the problem-sets at the end of the chapters, gives you no insight at all (executive summary:  Mathematicians are like gods), a clutch of bold spirits, bearing in mind certain delicate problems such as infinite sequences of functions and their convergence, generalized the stage on which such pageants play out, from the Euclidean to the general topological.   (The history has here been brutally telescoped.)  Something of the sort was in any case needed to save the Euclidean picture itself, since infinite-dimensional spaces were now required (even by physics),  and the finite-dimensional structures would not generalize in any straightforward way.

General topological spaces being a wildly assorted bag, various restrictions are put on them, for one purpose or another, to allow deduction and calculation.  One of these is metrizability, which we examined in the essay on Urysohn.   That has the advantage of preserving much of our hard-won familiarity with the Euclidean metric, while allowing a vast array of new metrics for particular purposes. (For example:  the by-now-familiar Lorentz metric of Einsteinian spacetime.  Once mind-boggling, yet now -- in this vaster context -- almost cuddly.)  These in turn can be slightly re-generalized, by considering pseudometrics; or further regimented, with the concept of a norm, which in turn may be relaxed into a seminorm;  and so it goes.

~

A quite different and likewise fruitful generalization of metric spaces  is the notion of a Uniform Space, introduced by algebraic geometer André Weil, in “Sur les espaces à structure uniforme et sur la topologie générale” (reprinted in volume I of his Collected Papers as [1937]).   He broaches it with a bang:

La notion de distance  est utilisée dans de nombreux travaux de topologie, [mais] l’on s’explique mal qu’elle soit venue à jouer un pareil rôle  dans une branche des mathématiques  où elle n’est, à proprement parler, qu’une intruse…

On voit apparaître ici  cette hypothèse du dénombrable (dite aussi, on ne sait pourquoi, de séparabilité),  malfaisant parasite qui infeste tant de livres … dont il affaiblit la portée  tout en nuisant à une claire compréhension des phénomènes.  … La conscience d’un mathématicien, s’il en possède [!], doit répugner à faire intervenir une hypothèse superflue …

Strong words !   The notion of metric, he claims, is not simply too restrictive, but is the wrong sort of notion for topology -- a cuckoo’s-egg in the nest.   And indeed, minus the polemics, James Dugundji makes the same point (Topology, p. 200):

A metric … can be regarded a providing a measure of nearness that is applicable throughout the space  … This notion of uniform smallness is not a topological concept :  equivalent metrics specify different sets as being equally small.

… Notice that, even in metric spaces, a continuous map may be uniformly continuous if one pair of metrics is used, but not uniformly continuous when another pair of equivalent metrics is used;  uniform continuity is therefore  not a topological concept.

(“Equivalent” metrics in the sense that they generate the same roster of open sets, which define the topology.)

Contrast a different -- and very fruitful -- restriction on general topological spaces, that of being compact Hausdorff.  This notion is strictly topological in spirit.

Footnote:   For another instance of Gallic arithmophobia, cf. the remarks of Weil’s countryman  Jean Dieudonné, in Foundations of Modern Analysis (1960), p. 141:

The fundamental idea of Calculus [is] the “local” approximation of functions by linear functions.  In the classical teaching of Calculus, this idea is immediately obscured  by the accidental fact that, on a one-dimensional vector space, there is a one-to-one correspondence between linear forms and numbers, and therefore the derivative at a point is defined [horresco referens !] as  number instead of a linear form.

In defense of Sir Isaac Newton, it must be observed, that our worthy ancestor was  quite understandably  interested in how fast something was going, at each moment:  to answer which question, he needed to invent the differential calculus.  Dieudonné, from the vantage point of centuries of progress, is looking ahead to function-spaces and dense subsets of special functions and like that.

~

The passages immediately above  evoke, unbidden, an untoward echo  characteristic of their times (the Thirties; the Sixties):  “unAmerican” and (failure to adhere to) “Chairman Mao’s Correct Line”.   But “topological” is not an all-or-nothing concept;  and we return to sanity  with jolly John Kelly (General Topology), in the chapter titled “Uniform Spaces”:

We deduce from a topological premise (that the space is compact) a non-topological conclusion (that a function is uniformly continuous).  This chapter is devoted to a study of quasi-topological results of this sort.

Even more telling is the remark by George Simmons, author of the superbly pedagogical Introduction to Topology and Modern Analysis (1963):

Some writers deal with the theory of metric  spaces as if it were merely a fragment of the general theory of topological spaces.  This practice is no doubt logically correct, but it seems to me to violate the natural relations between these topics, in which metric spaces motivate the more general theory.

Thus, it is scarcely fair, or psychologically realistic, to denounce the notion of metric as an “intruder” in topology, as Weil does.  Similarly:  you shouldn’t start off with categories and functors  before learning about  ordinary numbers and sets, even if categories prove ultimately more foundational.

That said, there does come a point where actual everyday examples impel one to consider such things as convergence and compactness  in a setting more general than a metric space.  As: pointwise convergence, which is a perfectly familiar non-exotic sort of convergence, but which cannot be seen as convergence with respect to a metric.

~     ~     ~

We have thus seen uniform space as a gentle generalization of metric spaces.  Since the point of the latter is often concerned largely with matters of limits and convergence, all we really need to know is what it means to get “closer and closer”;  we don’t need to put a number on how close, each step of the way.   This aspect was highlighted by André Weil, when he debuted the idea of uniform spaces, as a kind of intellectual hygiene.   But in practice,  quite as important to the introducer of uniform spaces is their natural application to topological groups, which come ready-made with a structure amenable to notions of nearness.

But there is more.   John Kelley, in his General Topology (1955), who devotes an entire chapter to uniform spaces, writes:

It should be emphasized that this is by no means the only framework in which uniformity can be studied.  It is possible to study a set X  together with a distinguished family of pseudo-metrics for X, or to distinguish a collection of covers of X where are to be uniform covers (roughly in the sense of the Lebesgue covering lemma).  One may also consider “metrics” with values in a structure less restricted than that of the real numbers.  All of these notions are essentially equivalent.

Such a situation illustrates a recurring intellectual theme of this series of essays (with both philosophical and mathematical applications), tagged as “Rome by different roads”.   There is a section on this notion in our essay Consilience in mathematics (indeed, in one sense, the entire notion of consilience in general  is related to this idea).