Informative tautologies (updated)

Technically, for a logician, or a semanticist of the Snow-is-White school, tautologies convey no information;  but to linguists and pragmaticians, in context they often do.  In fact, we may state that they usually do, since otherwise why utter them?  “Business is business” is flint-hearted;  “Boys will be boys”, tenderly exculpatory.

The opposite of an utterance that pretends to contain no information (and thus, in particular, to be inexpugnable) but actually does (and often of a trenchant sort), is a definition that, ex cathedra, is all about informing, but which melts to the touch.  Cf. our essay here:

http://worldofdrjustice.blogspot.com/2015/01/pugnaciously-vacuous-definitions_24.html

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Covertly vacuous uses of technical-sounding terms  have been scouted in histories of science.  As, “Things fall because of gravity, and rise because of levity.” Or Molière’s virtus dormitiva.  But more may be packed into such terms than may be initially apparent.  As, one philosopher pointed out that “The ball rebounded to the height that it did  because of its resiliency.”  But this is informative:  the height is owing to internal characteristics of that ball, rather than from the ball’s having been dropped from a greater height, or having been dropped on a more resilient surface.

Additionally, the initial tautological character of a sentence can  so to speak  “age off”, in accordance with semantic evolution of its terms. Thus, “Atoms are indivisible” was initially as circular as “Bachelors are unmarried”, since they were defined from the outset as indivisibilia (as their name, a-tom, etymologically implies).  But on its current interpretation, the sentence would qualify as false.

A mathematician looks at Newton’s Definition 1, in the Principia:

Quantity of matter is a measure of matter that arises from its density and volume jointly.

Great acumen is hardly needed to realize that this definition is hopelessly circular, since density is normally defined as the ration of mass to volume;  but Newton’s unhelpful phrase  does have some implicit implications.  For example, we expect the mass of an object to remain unchanged if we change its shape.

-- Michael Spivak, Physics for Mathematicians: Mechanics I (2010), p. 9

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A 2015  example of the uses of tautology:

Robert Buissière on Médi1, re Presidential candidates:

Jeb Bush, frère de son frère,

et Hillary Clinton,  épouse de son époux.

As they stand, these statements are “analytic”; but we understand the import:  Jeb and Hillary got where they are today, largely owing to family association.

Cf. & contrast the common expression “He is his father’s son.”  Normally this means that he takes after his Dad, and not that he is getting any special favors from other people owing to that filiation.  To imply the latter, you might say “Daddy’s little boy” or something.  By contrast, the French phrases in the above context  do not imply that Jeb’s politics are a close match to Dubya’s, let alone that Hillary’s are a close match to Bill’s.

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A bare tautology like “Business is business”, as a free-standing statement, invites contentful interpretation via a “Gricean implicature” (specifically, the Maxim of Quantity).   The following is a syntactically more complex case, there the tautology is embedded in a subordinate clause:

Ever since self was self, nature been keepin’ folks off of red-hot stoves.

-- Zora Neale Hurston, Their Eyes Were Watching God (1937)

The meaning is:  Common sense has been extant  from time immemorial.  Operationally, the (tongue-in-cheek) interpretation is:  Go back and back in time, sampling as you go. For each sample-point, verify whether  “self = self” holds at that time; and if so, then evaluate “Common Sense is in effect? Y/N”.  The sentence, for all its folksiness, has a kind of philosophy-class spin to it; the moreso as “self = self” calls up First-Order Logic with Identity.

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Stylistic appreciation

Usually there is a summary “That’s that” finality to tautologies, whether used informatively or not;  stylistically, they are bare-bones.  But consider this:

Herod:  The moon has a strange look tonight. … She reels through the clouds like a drunken woman. … Does she not reel like a drunken woman?  She is like a madwoman, is she not?

Herodias:  No;  the moon is like the moon, that is all.

-- Oscar Wilde, Salomé (1891)

Here the barrenness of the pale white, plain round  far-floating body, is reflected in the unyielding tautological formula.

~

Enten-Eller

Philosophers have scribbled  much ink, and later worn out many a typewriter ribbon, and finally expended great bushels of pixels, discoursing upon the status of “logical truths”; such as, paradigmatically, the following:

            (I) Every man is either married or a bachelor.

(We simplify, since the matter is not really of interest; leaving out of account, for instance, the curious case of Schrödinger’s groom.)

It is agreed that such a sentence tells us nothing about the world, unlike that time-honored exemplar of informativeness,

            (II)  The cat is on the mat.

which has been so oft repeated. down the years, that said cat has achieved the immobility and timelessness of an Egyptian idol.  (Presumably the mat lies in a patch of sunlight, so why ever move?)

            And yet its affordances are quite different from those of another statement of the same logical form; say:

            (II) Every number is either even or odd.

For, although the sentence (I) does not perhaps baldly state anything substantive about the world, its presuppositions speak volumes.  For one thing, it gives us to understand that there is a sharply defined institution, Marriage, into which a man may enter or not; and that his resultant state is either-or. Even so much will give our Martian anthropologists  sufficient grist  for many turns of the mill.

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The above are mostly individual linguistic parlor-tricks.  Much more generally, there is the matter of the status of the equations of mathematics.  A view put forward by the dessicated and ennervating tendency called formalist (nominalist)  maintained that, being mathematical, they are tautologies, and being tautologies, they are uninformative -- semantically vacuous.  Practical experience shows that doctrine to be false.  As a way to see how such equations manage to be informative, consider the number pi.

Pi can be defined, qualitatively, in a number of different ways, most familiarly as the ratio of a circle’s circumference to is diameter.  It  is, moreover, a Given of the invisibilia, woven into the fabric of the noöspheric pattern;  and -- crucially -- it can be reached in a startling variety of precise quantitative ways, along this strand or that of the warp and the woof -- and the woorp and the wahf, along any of the dimensions of the multidimensional mathematical textile.   Pi is itself ever one, the peak of the Golden Mountain;  but the paths that climb to it are numberless.  It can be expressed as a definite integral; as an infinite continued fraction; as an infinite product; as an infinite sum; with many variations of each.  (Behold some of them at https://en.wikipedia.org/wiki/Pi.)  That fact that any one of these equals π is informative and indeed amazing, for in effect each one constitutes hiking instructions -- a trail-map -- to the summit, each from a wildly different base-camp.  And each one of these infinitely intricate expressions  is equal to any other, in an equation which is as distant from tautological or uninformative as can be imagined.

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[Update 13 May 2020]  An aborbing article by Evan Osnos, in the 11 May 2020 issue of The New Yorker, maps out the paths by which the Greenwich Connecticut  tennis-and-boating-club crowd  came to support Trump.  The notables of that town have a patrician heritage, in principle at variance with the flashy style of the vulgarian from the Bronx, but as one blue-blood testifies, his conversion came while witnessing an early speech by the candidate:  “He had that line that he would use: ‘Folks, we either have a country or we don’t.’ And I felt the chill .. I’m, like, ‘Oh, my God, this is a really good line.’

Apart from the formally tautological character of that line, it puzzles by its vagueness:  out of context, it is unclear what at all is being hinted at.  Presumably the line is a dog-whistle, a bit of Rorschach rhetoric from which the listener will extract whatever meaning he likes.

The line does nothing for me; but it does recall such successful political antecedents as “The business of America is business.”

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[Update 14 May 2020] 

Some of you may be familiar with the British sport of trainspotting.  That may or may not be in accordance with current U.K. guidance on coronavirus lockdown.  But here’s a hobby you can practice in the safety and comfort of your own home:

Tautology-Spotting !

As:

Headline in this morning’s New York Times:

The People Behind the Counter Are People

Remember this the next time you order takeout.

That one recalls those sleep-inducing example-sentences from introductory logic class (All brave Athenians are Athenians).  But in this case, it has a punch, and the source of that punch is not logical but lexical.  For, people has a variety of senses;  from the neutrally classificatory

(a)  an instantiation of the species Homo sapiens, near-cousin of Pan troglodytes, sometimes known as “a forked radish”;

to the “pregnant sense”  (I phrase it freely)

(b)  an ensouled being created in the image of the Lord of Hosts, whom Christ died to redeem.

In the cited sentence, the first occurrence of the word people has the (a) meaning; the second occurrence, (b).

For more on perspectival semantics and pregnant senses, check out this essay:

https://worldofdrjustice.blogspot.com/2019/10/words-of-day-sublunary-postlapsarian.html

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In all the instances above, tautology is used in its traditional logico-philosophical sense.  But technical terms sometimes get picked up by a wider audience, where their use may be lax.  Thus, the literary critic V.S. Pritchett, in his article on the novelist Anthony Powell, wrote:

Mr. Powell is excellent with the raffish…. I think the sententious irony succeeds. … It adds a very English flavor, either of comic tautology or deflation.

The sense of tautology is unclear here.  Perhaps it refers to mechanical repetition, a standard component of broad humor.

The Double Truth

It is clear by now that the old Averroist doctrine of the “double truth” (veritas duplex)  must be revived:  this time, not along scientific-vs-theological lines, but political.  Radio talk-shows have long recognized the incommunicability of one set of truths, to partisans of the other, by offering separate Republican vs. Democrat call-in lines.

So, let us consider that simplest possible area of human belief and inquiry:  arithmetic, the discipline that offers up such hoary verities as “2 + 2 = 4”.  

In modern mathematics, the study of such numbers (called integers)  has exfoliated quite a bit, and indeed has bifurcated into named subdisciplines.   Thus we have, on the one hand, analytic number-theory,  which deals with such things as the Riemann hypothesis and the Goldbach conjecture, using techniques from complex analysis, and on the other, algebraic number-theory, involving extensions of the integers such as algebraic number-fields.   Both subfields have developed to such an extent, that specialists in the one will be scarcely able to understand, let alone prove, theorems in the other;  thus reproducing, unintentionally, a simulacrum of the current partisan Dialogue of the Deaf.

And now we have two new entrants to the disciplinary matrix:   Democratic number-theory and Republican number-theory.

Democratic number-theory is characterized (up to isomorphism) by its claim that all numbers are equal.   The denial of that doctrine is termed ‘fascism’.

Republican number-theory, by contrast, generalizes the ancient concept of a perfect number (equal to its aliquot sum) to that of the awesome numbers, which are greater than the sum of everybody else.  The leftovers are called the  pathetic numbers (also termed ‘losers’).  This theory focuses exclusively on the former subset (although the membership in this set  can vary capriciously  from day to day).

So far, neither body of theory has produced interesting theorems.

[Update 8 Jan 2018]   No sooner had I posted that would-be satirical thought-dream, than I happened upon this,  by the philosopher Stewart Shapiro:

Hartry Field takes mathematical language at face value.  Since he holds that mathematical objects do not exist, mathematical propositions have objective but vacuous truth-values.  For example, he maintains that ‘all natural numbers are prime’ is true, since there are no natural numbers.

-- Thinking about Mathematics (2000), p. 226

So once again, reality beggars satire.  (Apparently this Field fellow is some sort of an academic, rather than an inmate or homeless-person.)

Note, a propos, that the Fieldian assertion “all natural numbers are prime”  is actually an axiom of Democratic number-theory.   (“Everybody gets a trophy.  Everyone is special.”)

[Historical footnote]    If satire is the production of humor by clever exaggeration of a real situation, then here again the satire lacks its bite:  for such ideologically infected strains of the hard sciences  are already exemplified in history -- not merely in the musings of philosophical fantasists, but under dictators with the power of life or death.   Thus Hitlerian notions of Aryan science vs. Jewish science, or the various scientific orthodoxies under Stalin.    Under those two regimes, mathematics, fortunately, was relatively spared distortions of actual content (so long as the formulaic obeissances to ideology were packed into the preface),  though  in human terms  there were profound and pernicious effects on who was allowed to be mathematical practitioners.   The neo-Stalinoid zealots of “anti-racist mathematics” go further, prescribing severe limits to content and presentation as well.

On Crunchy Numbers

In the Ike era, we grew up on Wonderbread® :  a sort of Brot ohne Eigenschaften whose edulcorated transmogrification is known as Twinkies.

Since that time, we have learned to abjure, not only such treif, but anything not calling itself wholegrain.   Or, better still, multigrain: some brands boast seven grains, a few claim twelve; disparate mixtures  full of gritty, grainy, crunchy goodness.

 

Now  our local upscale supermarket offers a variety of own-branded bread, that boasts (in large letters) no fewer than

27 GRAINS

That really surprised me.   It’s one of the main points of Jared Diamond’s Guns, Germs, and Steel  that digestible, domesticable, feasibly growable grains  are not to be had for the asking;  there just aren’t that many of them.   The explanation is that, a little lower down and in smaller font, the label reads

AND SEEDS

In other words, this brand of bread is equally at home in the bakery and in the bird-feeder.

Our son, scoffing at this terminological legerdemain, inquired why the market chose “27” of all things.   Unhesitating I replied, “Because it is three to the third power -- the trinitarian pinnacle of the Perfect Cubes”.  -- Said heir and offspring, himself a nascent mathematician, appreciated the point, but doubted that your average shopper was aware of such things.

And indeed, there is a larger point.   For number-theorists like Ramanujan, each integer has its own flavor, its own biography and backstory -- cf. the famous incident of the taxicab numbers (in which G.H. Hardy plays the straight-man or fall-guy).  But for ordinary folks, like rocket scientists (who deal with contingent analogue quantities, rather than integral transcendent entities) or English professors (surrounded by a midge-cloud of  pre- or sub-arithmetical post-modernists), all but a very few -- small -- integers  will be featureless.   Some will be visually familiar:  3 and 4 (the triangle, the square), 5 (the quincunx on dice), 6 (boxcars, ditto),  7 (a “lucky” number for the superstitious) or 23 (ditto, for the more cerebral), 10 (count your fingers), 20 (with its portrait of Jackson).   Some will be familiar for incidental, non-mathematical reasons, like 100 and 1000 (which owe their prominence to the accident of base-10 notation -- a case of decimal fetishism).   --  Those associations are widely shared;  but there may be others  more individual;  as, (mostly for girls), “sweet sixteen” (or in Latin America, la quinceañera).   Indeed,  any integer small enough that you have lived that number of years (particularly those for which you still kept track of your birthdays).    As, the poem(-collection) “Now We Are Six” (by A.A. Milne), an anniversary in memory still green (and which I teach to the neighborhood children when they reach that delightful threshold).  Or…. 27;  which, even before I had attained that age, always seemed numinous (probably, indeed, for its prime-power nature), and which was ratified as such by my marrying at exactly that age, quite close to my birthday.    Whereas, for most folks, a number like 81 (pourtant a perfect square, as well as  3^2^2, to boot) tells no tale, sings no melody.

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The preceding remarks are of psychological or anthropological interest, but of no moment for mathematics itself:  They present an external, human-centered view of the integers.  But further considerations suggest a subtle distinction between two ways a given integer may be “interesting” (a more bloodless equivalent of our anthropomorphic term crunchy).   We might dub these internal and external:  both times internal to mathematics as a whole, but in one case only, internal to number theory in its most elementary sense.

Thus, consider 5.  Number-theoretically, it’s a prime and that is pretty much that; but in the geometry of three-space, it is the number of Platonic solids.  Or 17:  Number-theoretically it is both a prime and a Fermat number; but in the geometry of two-space, it is the number of crystallographic planar symmetries.  

https://en.wikipedia.org/wiki/Wallpaper_group

Or: 230, the number of space groups.

Here, the integer in question is not a creature or crystal considered distinct and in itself, but a stopping-point one arrives at by calculating and counting.  There might have turned out to be, say, 18 plane symmetry groups, without upending the world; but the internal structures of 17 and 18 are unrelated.

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This dichotomy of internal versus external interest  chez the integers, represents ideal poles, which are not exhaustive, but bookend a spectrum.  As, probably intermediate: the “crunchiness” of natural numbers as viewed by students of finite groups.    (Here the masticatory metaphor  returns unbidden, for I always imagine finite groups along the lines of a wrinkly-surfaced walnut.  Finite simple groups -- those for which no homomorphism can split out any subchunk as its “kernel” -- the hardest nuts, the ones you can’t crack.)  Simon Norton seems to have had such an intimate gustatory appreciation of individual finite groups, before he Threw It All Away and went off to ride the bus.

~

In light of our training, we know at least one thing to do, should we ever be given an unfamiliar integer and locked in a room, with no other toys to play with.  Namely, we can probe for primality.  (An activity that becomes, indeed, crucial, for cryptographers.)

But now imagine that instead we had been handed some fraction like

      (2 +  √137) /4081

Untrained, we react with dismay.

Yet later, learning of the Golden Section, (1 + √5)/2, and its many remarkable properties -- not the least of these being that it can be represented as the infinite continued fraction consisting of nothing but ones -- we conclude that the critter is crunchy indeed;  and that there are more things in heaven and earth, than are dreamt of here below, and that we must anticipate the afterlife, before we could begin to embrace them.

~

Our treatment focused on what the innumerate are missing, much like what the Daltonist, unbeknownst, lacks of the hues.   But there is such a thing as unearned crunchiness --  a bogus significance assigned to certain numinous numbers, like 19 among the Baha’i’s; the “23 enigma”; 666; 1000 AD as the Millennium; the mumbo-jumbo numbers in “Lost” and "Touch"; for all which, cf. Wikipedia on apophenia.   In the face of such things (which have snared some otherwise rational people -- a close friend of mine, critically brilliant but unmathematical, fell for the “23” business), we are inclined to say, along with Nulla  extra ecclesiam  salus,  that  Nulla  extra mathematicam  ratio.

~

For a rather recondite example of crunchiness, consider the ‘amicable numbers’  (  الآعداد المتحابة) reported by the medieval historian Ibn-Khaldun in his Muqaddima .   Apparently only two of these were known to the Arabic medievals (or: they are the only two numbers characterized by a theorem of Thabit ibn Qurra), and they are not much to look at:  220 and 284;  but they meant something to contemporary practitionars of the talismanic art.

The modern view is summarized here:

https://en.wikipedia.org/wiki/Amicable_numbers

Minimalism in Mathematics (further updated)

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

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

Extending the appellation modernism to mathematics … is like “shooting an arrow and then tracing a bull’s eye around it.”

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

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

Logicism:  a kind of reductionist minimalism.

*

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

Newton consistently preferred Euclidean-style proofs.  He used his own calculus only where strictly necessary, and barred algebra from his treatise  entirely.

-- Leo Corry, “The Development of the Idea of Proof”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008).

(Cf. a laborious non-analytic “elementary” proof in number theory.)

As fastidious as an Intuitionist!

*

Not a matter of method, let alone of taste, but sheer fact (albeit initially so counter-intuitive as to have been dubbed a "paradox"):

The Löwenheim-Skolem theorem: if a first-order theory has a model, then it has a countable model.

*

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

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

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

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

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

*

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

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

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

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

(This is the "unreasonable effectiveness" motif.)

*

We earlier noticed what we called “the Dialectic of the Topological Enterprise” -- abstracting-away from rich familiar entities, extracting what seem the essentials, and seeing what happens.   The first step might seem Minimalist, but the consequence is an effusion and exfoliation of new spaces which meet the newly relaxed criteria, and which turn out to have an even richer riot of properties than we began with.   Per se, there is little in all this that might justify bringing in the aesthetically-tinged label of “Minimalist” (not a traditional term in mathematics; the closest you get is “abstract”):  but the aesthetic ethos is there, for all that.  Thus Shing-Tung Yau, The Shape of Inner Space (2010), p. 77:

 

We start with some raw topological space, which is like a bare patch of land that’s been razed for construction.  On top of that, we’d like to build some kind of geometric structure that can later be decorated in various ways.

[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

*

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

~

On Ramanujan’s notebooks:

There were thousands of theorems, corollaries, and examples.  For page after page, they stretched on, rarely watered down by proof or explanation, almost aphoristic in their compression, all their mathematical truths  boiled down to a line or two.

-- Robert Kanigel, The Man who Knew Infinity, p. 204

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

~

From a logician:

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

-- Michael Potter, Set Theory and its Philosophy (2004) , p. 254

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

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

Theorems, Propositions, Dumb Questions, Unspoken Assumptions

In an earlier essay (Andrew Gleason:  in Memoriam) we fondly recalled our favorite teacher from Harvard.   The incident below was not included;  but now, owing to recent events, it can be declassified.  We take you back to the year 1969 …

~

Since I lacked any spark of mathematical creativity (this sad fact only became apparent to me later), though otherwise technically proficient, I seldom participated in the classroom in any active way, even to ask a question.  I sat towards the back, took copious notes, and tried to follow the arguments as best I could.  Yet one day, in Gleason’s undergraduate Introduction to Group Theory class, something puzzled me  and I did speak up.   The group operation is required, by fiat, to satisfy an Associative Law -- but how, in the actual case before us now, did we know that the operation in question really did associate, in every case?

My shy query did not, we may say, turn out to open up new pathways for research in mathematics;  the great professor did not gape and slap his forehead and cry out “Mein Gott!" (mathematicians revert to German when suitably moved), "This casts Abstract Algebra in an entirely new light!”;  but nor -- and this was more surprising -- did I receive, in this instance, a satisfactory reply.   For Gleason, interrupted at the blackboard, suspended amid his lecture  chalk in hand, found the question itself … puzzling.   He shrugged, grimaced, he really didn’t know where to begin.  “It’s … obvious,” he said at last, giving up on me, and, brushing the dust from his sleeve, resumed the lesson.

Now, this hapless anecdote -- which, for shame, I have never mentioned previously to anyone, before this very date -- bids fair on the face of it to be booked beneath the scarlet rubric of Oligophrenia mathematica, which I have treated at sorrowful length in the essay “De Stultitiâ”.   And yet some recent reading frames the matter more sharply, and recalled the anecdote to mind.

The first was an article about matrix groups  like GL_n, which did not assume that matrix multiplication is associative, yet nor did it bother to prove it in any straightforward calculational way (this can be done, but is messy, and quite unilluminating), but said that since the matrices represent linear operations on a vector space, their associativity follows from the associativity of composition of the operations that underlie them.   Now, that is a thought with some content.

The second passage, which really nails the matter, comes from Tim Gowers’ lucid and insight-packed introduction to his collection of articles surveying all of mathematics.  He observes:

The associative law [says], informally, that “brackets do not matter”.  However, while it shows that we can write x * y * z without fear of ambiguity, it does not show quite so obviously that we can write a * b * c * d * e, for example.  How do we know that, just because the positions of brackets do not matter when you have three objects, they do not matter when you have more than three?

Many mathematics students go happily through university without noticing that this is a problem.

-- Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 73

And later, in an article that is a masterpiece of step-by-step exposition, Gowers shows how quaternions can be represented as ordinary matrices, and adds:

As an immediate corollary, we have a proof of a fact mentioned earlier:  that quaternionic multiplication is associative.  Why?  Because matrix multiplication is associative.  (And that is true because the composition of functions is associative.)

-- “Quaternions, Octonions, and Normed Division Algebras”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 277

~

So, to return to the tableau in which Gleason is frozen in stupefaction behind the lectern, while the dunce of the class blushes and desires to be mapped to the empty-set instanter:  It is evident, to any practiced hand, how one might prove the proposition (by induction, one supposes, on the length of the string), but the proposition itself, as Gowers observes, does require such proof, since from the simple  “(a * b) * c = a * (b * c)” we are now asserting an analogue (which, even to state, requires some symbological ingenuity) for an infinity of cases.

Now, Gleason himself was perfectly familiar with all this¹, so how did he not recognize that what I had asked was not actually such a stupid question?   And the answer is now plain:  It was not a stupid question, but it was a boring question, in a precise sense, which Gowers addresses on the same page, in light of this very example.   In the course of a straightforward précis of the meanings of the terms theorem, proposition, lemma, and corollary, he puts forward this epigrammatic distinction:

A proposition is a bit like a theorem, but it tends to be slightly “boring”.

That is, it is a statement that one needs -- perhaps, indeed, at every turn, so that in a sense it may even be fundamental -- but whose truth is utterly unsurprising, and whose proof involves no interesting insights or techniques.   During the time he paused at the blackboard,  Gleason probably (like John von Neumann summing an infinite series in his head) instantly proved the proposition in his own unconscious.

Footnote ¹:   Gleason was in fact especially sensitive to such quasi-linguistic matters of hidden assumptions.  Thus, in his text Fundamentals of Abstract Analysis (1966), he remarks that the direct-product procedure is strictly speaking not associative, but that there exists a natural bijection among the various possibilities, so that we speak simply of “the” direct product of a roster of spaces, par abus de langage.

~

A more recent example of my posing a question which left the teacher speechless, apparently as being unanswerably dumb, happened a couple of years ago.  A visiting combinatorialist, scholar-in-residence at the Cryptological Museum, gave a public talk about Stirling numbers of the second kind.   By doing this, that, and the other thing, you can find all sorts of pretty geometric patterns popping out at you in Pascal’s triangle and whatnot.   Since I am these days but infrequently in the audience of a combinatorialist (since moving from Princeton, my mathematical surroundings have become quite impoverished -- really an algebraic social-worker should stop by with some charitable Ideals on Wheels), it seemed a good occasion to pose a question that has always bothered me:  what is the point of “perfect numbers” (those that are equal to the sum of their prime dividers)?   I raised my hand and ventured:

“It soon becomes obvious why prime numbers are of prime importance:  they are used for many purposes other than in the study of their own properties, and they jump out at you even when you’re not looking for them, in physics or wherever; they are part of the woodwork of the world.   Also, on a more intuitive or metaphorical level, they are the evident “building blocks” of all the integers, the way the atomic elements are the building blocks of all the molecules.   But -- “perfect” numbers.  The definition seems so arbitrary.  Why study them?  What are they good for?”

Instead of instructing the curious groundling by giving examples of their usefulness, or their inevitability, or even saying “An explanation exists but it would be way over your head, you peasant” (which, while impolite, would actually be somewhat informative), or “You’re quite right, they are purely recreational”, he simply looked blank.  The question evidently made no sense to him;  it was as though I had asked whether the value of pi were the same on the dark side of the moon, or under all gravitational conditions. (Thus, not exactly a dumb question, more like a crank question,  of the sort which is likely to spring from the lips of the unemployed middle-aged men in raincoats who wander into lecture halls in hopes of a donut and to get in out of the cold, and who have their own private but quite definite opinions about whether a circle can indeed be squared  or whether, rather, it might not be square in fact already, only They don’t want you to realize this;  the lecturer’s only defense is to decline to be drawn into debate.)  And yet, mathematicians might be characterized as people to whom such questions make a lot of sense, and are even fundamental.

Perhaps, though, combinatorialists less than other specialties.   There does seem to be a fair amount of pointless ingenuity in what some of them do, but then I’m no judge of it.  

However!  Once again, Gowers to the rescue, to clarify the sort of issues that are at stake.  On the next page of that same Introduction to mathematical terminology, he defines (or explains) the notion of definition.

Mathematical definitions are generally what linguists call stipulative definitions, essentially just rewordings or abbreviations.  “Definitions like this,” Gowers comments, “are mere definitions of convenience”.   Yet, just as in the case of the taxonomic definitions of philology or biology (Indo-European; crustacean), where the really useful ones reflect a significant amount of research and analysis leading up to them, so in mathematics;  and indeed, Gowers reveals, in some of its branches, even moreso:

Some mathematicians will tell you that the main aim of their research is to find the right definition, after which their whole area will be illuminated.  Yes, they will have to write proofs, but if the definition is the one they are looking for, then these proofs will be fairly straightforward.

-- Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 73

The author does not instance such cases where the definitions do much or most of the real work; these would have been fascinating to hear;  but I imagine he has in mind as precedents  such things as the homotopy groups, easy to define but the devil to calculate, versus homology groups, fiendish to define and visualize  but easy to compute; or the Generalized Stokes Theorem, which by our own day possesses a brief proof once all your definitional ducks are in a row -- but oh!  what ducks!

He does, though, go on to provide an example of a definition that might appeal to the readers of “brain teasers” in the Sunday papers, but which does no work at all -- one so distasteful, it is distressing even to write it down:

A number is called palindromic if its representation in base 10 is a palindrome.

Such ontological excrescences are even more thewless than “perfect” numbers, since at least the latter are independent of their inscriptional base.   (You can think of the writing of one of God’s own integers in any base  as representing a tragic demotion from the Platonic sphere, sort of like a soul’s being incarnated in the body of a frog.) 

At that point I almost skipped on to the following page, so little do I wish to learn the least thing about such concocted objects;  but Gowers goes in an interesting direction with this.  One might ask:  How many primes are palindromes?  There are some, although, in a well-defined sense, “not many” (even if there are infinitely many), examples being 919, 929, followed only much later by 10310.   And thus the question:  Are there infinitely many?  (Once you have more than about seventeen of something, that is the first question a mathematician asks:  They don’t like sequences of integers that go on for a bit and then just stop.)   The answer would be boring either way;  but unlike the “boring” propositions alluded to earlier, it would be the very Dickens to prove or disprove (and thus not worth the candle).  For,

It can be shown quite easily that  the number of palindromic numbers less than n is in the region of  √n, which is a very small fraction indeed.   It is notoriously hard to prove results about primes in sparse sets like this.

-- id., p. 75

And in any such endeavor, the “definition” of palindromic would be of no help at all, since it is “so artificial that there seems to be no way of using it in a detailed way in a mathematical proof.” (p. 76).  And that same infirmity of the beginning definition  insures that the bare answer, whatever it might be, would be uninteresting per se (although, as Gowers points out, there might be a much more general conjecture with no original connection to “palindromes”, which would be interesting and which might turn out to settle the result for palindromes as well):  for, unlike prime numbers, palindromes, being irremediably notation-dependent, do not form part of the Furniture of the Universe.  (For that concept, consult the series of essays begun here.)  That was what I had been trying to get at by my question to the itinerant combinatorialist, and which meant nothing to him;  perhaps he is not Platonistically inclined.

~

While we’re on the subject, let us consider further the question of definition in mathematics.

Re Hilbert’s approach to the axiomatization of geometry:

Rather than defining points or lines at the outset  and then postulating axioms that are assumed to be valid for them, a point and a line were not directly defined, except as entities that satisfy the axioms postulated by the system.

-- Leo Corry , “The Development of the Idea of Proof”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 139

This is not quite so radical or ‘post-modernist’ as it might sound, since traditional grammar recognizes many analogous cases in natural language, under the rubric of syncategorematic.   It is a relative notion, with a sliding scale;  but analysis will suggest that a very large set of words and multiword expressions (as, the use of a word in an idiom, especially in an opaque idiom) partake of some degree of syncategorematicity.   However, in the particular perspective of mathematics, this idea harmonizes especially well with a logicist or formalist approach to the subject:

The use of undefined concepts  and the concomitant conception of axioms as implicit definitions  gave enormous impetus to the view of geometry as a purely logical system.

-- Leo Corry , “The Development of the Idea of Proof”, in Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 139

Again, this is much less disorienting and self-bootstrapping than it may seem, since -- outside, indeed, of formal contexts -- virtually all of natural language works exactly like that; and not only expressions like whereas, the moreso as, French ne, German doch, which wear their syncategorematicity on their (empty) sleeves, either,  but plain words like bunny.   You do not learn to use such words on the basis of a definition, formal or informal -- however much it might please linguistic philosophers to invent a terminus technicus “ostensive definition”.   For, as we have seen in our discussions and parables related to matters Quinean, these don’t really work, not logically;  they work pragmatically, to the extent that they do, because (since we are all molded from the same clay; or  if you prefer, since our bloodlines have all been subjected to the rigors of Natural Selection) we are all cut to the same cloth.  (To the extent that some individuals fall outside the innate cognitive norms, they fail to acquire the same semantics that the rest of us do:  or else, like some gifted and industrious autists, they acquire this only by dint of an artificial study, like someone learning Sumerian logographics.)   Thus, the following Onomastic Primal Scene does not actually obtain in any real nursery:

“That, Timmy” (pointing -- but at or towards what?) “is a rabbit (noun count, singular).   And by this -- attend now, and please do not misunderstand me -- I do not intend to indicate the entire scene embracing carrots and furballs and playpen and binky (who left that there?) etc., let alone the cosmos as a whole (after all, one has to point somewhere), whether by itself or considered as but one flaky layer in the whole baclava-like complexus known as the multiverse;  but only the, er, furball-related entity.   And by this, I do not mean, so much, (although I do not literally not mean it, either), a pointlike or infinitessimal space-time slice of a leporiform trajectory along the world-sheet, nor a “thickened” (perceptually available) neighborhood of the same;  nor a sort of puddle of rabbit-stuff, undifferentiated from the rest of the puddle; nor a concrete instantiation of the Platonic Form, ‘Rabbit’;  nor a subobject in the Category Leporidae;  nor an agnostically structured pointset consisting of Undetached Rabbit Parts (although I sort of mean that, since, at some point, once you have hacked the poor critter to bits and scattered its disjecta membra over the face of the earth to be eaten by vermin and recycled as independent atoms, at some point, we can no longer confidently say, “That is a rabbit”, in the sense of noun count, singular),  nor -- well, dash it all, I mean just Fluffy, okay?  And by the way it looks like Fluffy wants a cuddle or something, because she is spritzing the wood-shavings in a semantophobic panic.”

 ~

An extension of this linguistic thought-thread  can be appreciated here:

   More on Mathematical Definition