On “The Nature of Mathematical Knowledge” (enlarged)

That question is about as interesting as the nature of our knowledge of elephants.  We are interested in the zoology of elephants, not in the specificities of classroom biology lessons, or the economics of zoos.  We are uninterested in each blind man’s subjective and partial report upon the individual organs of these splendid creatures.

Mathematical knowledge, like pachydermal knowledge, is imperfect knowledge of something real that exists independently of us. By contrast, just which images we manage to form of these objects  are very much dependent upon ourselves – and to that extent, of interest only to unemployed social workers.

As our former math teacher put it:

Mathematics has a real content which transcends the inadequacies of our efforts to formalize it.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. v.

Or, from a philosopher:

For most mathematicians most of the time, having a feel for what is evident is important, but it is also enough: There is no further need for a theory of what that feel is.

-- Shaughan Lavine, Understanding the Infinite (1994)

*

Actually, it should not be deduced from the above, that I am somehow dissing our big friends the elephants.  A Theory of Elephants is considerably more promising than that idle grail of the physicists, a Theory of Everything.

The outstanding problem in the Theory of Elephants is the ontological status of BABAR – THE KING !!!!!

His Majesty ... The King !!!

An agnostic or non-Realist version, wilfully po-faced:

Our deductivist proposes clean answers to philosophical questions.  What is mathematics about?  Nothing … What is mathematical knowledge?  It is knowledge of what follows from what.  Mathematical knowledge is logical knowledge.

-- Stewart Shapiro, Thinking about Mathematics (2000), p. 150

*

*

There is a systematic ambiguity (roughly that of actio versus actum) to the term mathematics:

(I) The praxis of mathematizing.  This is a human pastime, comparable to needlework or basketball.

(II) The truths of mathematics.  Or, more or less synonymously: The real (though invisible) world.  These, in themselves, bear no dependency upon human practice or to any species whatever;  they existed before we were born.

The above may count as a polemical reformulation of roughly the dichotomy in the title of Hao Wang’s fine essay, “The Theory and Practice of Mathematics”.

*

In 1950, Raymond Wilder gave an address, “The Cultural Basis of Mathematics”, reprinted in  various places, and later wrote a whole book on the subject, Mathematics as a Cultural System (1981).   Bien-pensant commentors treat these with grave respect;  but the notion is practically nonsense.  For, if we take mathematics in sense (II) – the only sense of interest to us here – that is like saying “The cultural basis of elephants”:  there is none.  There is a cultural basis of circus stunts, of mouse- and peanut-myths, of Dumbo, but not of elephants themselves.  Their basis is their own four feet.

Why should the culture of mathematics  (necessarily, sensu I) – that is, the foibles of mathematicians – retain our attention?  The purely “human side” of mathematicians  is generally less interesting than that of country music stars.  A lot of mathematicians are pretty Asperger’s, frankly.
(For a poignant illustration of this, read The Genius in My Basement.)

There is, we grant, a certain interest in the sociology of mathematics, or in biographies of the great mathematicians. Intellectually, it is on a level with gossip about the off-court antics of basketball stars.  Fun, but of no mathematical (or basketball) interest.   It’s just a way for the mind to chew gum while it’s too exhausted to do anything substantial.  To get real, do math (or play basketball).

Not to come down too hard on the small geeky community that does follow the doings of math and physics whizzes; I number myself among them.   It would even be neat  if, instead of collecting baseball cards, people collected mathematician cards (“Trajea two Steven Smales for a John Milnor!”) .  -- By “people”, I here mean “eight-year olds”.

*

More interesting is the purported “reduction of mathematics to logic”.  It is not initially clear, however,  to what extent this program, if successful on its own terms, would enlighten us as to mathematics-sensu-(II), as opposed to the sense-(I) territory of our own mathematical formulations and formalizations (these being, after all, largely for mere convenience).  It might be more along the lines of the demonstration of the equivalence of the Heisenberg-style matrix-mechanics formulation with that of the Schroedinger-style wave formulation, of quantum mechanics. That feat didn't tell us all that much about the actual phenomena of physics,  apart from the fact that the world is a many-splendored thing, and can be described -- blind-man-fondling-elephant-fashion -- in a variety of ways.  It’s more like deciding whether today’s symposium shall be conducted in English or in French.

*

That said --

We argued here that the axiomatic method is cognitively post-hoc, and that  only in cases where (as with the Euclidean axioms) their positing is transparently motivated by our experience of the sensible world, is a top-down, axiomatic presentation  pedagogically sound.   Thus similarly in physics:

In lecture after lecture, and essay after essay,  Einstein began, not with an introduction to the subject at hand, but with an overview of how he’d arrived at that subject, or of how scientists in general  arrive at subjects in general. … For Einstein himself, the results of science had become incomprehensible without an understanding of the processes that led to them.

Richard Panek, The Invisible Century (2004), p. 153-4

C'est exact;  and the farther physics wanders from our human experience, and the father math develops beyond anything the world has seen before, the more necessary such a psycho-cognitive ladder does become.

*

Something like the dichotomy outlined above  must have been behind André Weil’s tart remark, in “History of Mathematics” (reprinted in Collected Works v. III as (1978b)):

Some universities have established chairs for “the history and philosophy of mathematics”;  it is hard for me to imagine  what those two have in common.

For:  the one is situated and contingent, the other timeless and beyond place.

*

Footnote:   These remarks about mathematics  apply  mutatis mutandis  to the Deity.  Deliberately confusing the distinction between truth and praxis, Karen Armstrong wrote a book -- a minor best-seller -- with the impudent title A History of God.  (At least she put A, not The; probably saved herself an extra millennium in Purgatory right there.)

*

Lakatos’ classic dialectical-dialogue Proofs and Refutations (you see the Hegel-style paradox already in the title), though focussing on the (as he persuasively argues, in the course of a detailed case-study spanning many decades) micro-level mess of actual mathematical progress, is yet Realist at its core:  the subtitle is “The Logic of Mathematical Discovery”, not “The Sociology of  ‘Mathematical’ Invention”.   We quoted him in another context  thus:

As far as naïve classification is concerned, nominalists are close to the truth when claiming that the only thing that polyhedra have in common  is their name.  But after a few centuries of proofs and refutations, as the theory of polyhedra develops, and theoretical classification replaces naïve classification,  the balance changes in favour of the realist.

-- Imre Lakatos, Proofs and Refutations (1976), p. 92

In an appendix to the main work, he offers a Hegelian formulation, one which (by the time the reader has progressed this far) has a certain paradoxical piquancy:

Mathematics, this product of human activity, ‘alienates itself’ [in the sense of Hegel and Marx] from the human activity, which has been producing it.  It becomes a living, growing organism, that acquires a certain autonomy [emphasis in original] from the activity that produced  it.  … The activity of human mathematicians, as it appears in history, is only a fumbling realisation of the wonderful dialectic of mathematical ideas.
-- Imre Lakatos, Proofs and Refutations (1976), p. 146

Plato, in his Paradise, smiles.

The Ladder of Abstraction (with added rungs)

The following  logically belongs in the “Abstraction” section of our essay Consilience in Mathematics.  But as that effort is growing overlong,  we begin to cultivate here a particular idea  building upon that of abstraction simpliciter :  namely, the tendency, in modern mathematics -- and indeed this may serve virtually as the defining characteristic of modern (even: modernist) mathematics -- to abstract from any given abstraction, layer upon layer, rise upon rise, to a virtual (topless/cloud-topped) Babel, reaching to the Beyond.

(Oh, and here again we have a term from the arts, Modernism, which, as it includes “abstract art”, metaphorically applies to mathematics.  Compare our earlier essay on Minimalism in Mathematics.)

In normal practice, mathematicians mostly talk to one another -- and indeed, mostly just to those within their own hyperspecialized neck of the woods.  But occasionally, one writes an undergraduate textbook, and thus must descend to earth, if only for the nonce, and address the laity.  Thus:

This “intrinsic” formulation of Calculus, due to its greater “abstraction”, and in particular  to the fact that, again and again, one has to leave the initial spaces, and to climb  higher and higher  to new “function spaces” (especially when dealing with the theory of higher derivatives), certainly requires some mental effort, contrasting with the comfortable routine of the classical formulas.  But we believe that the result is well worth the labor, as it will prepare the student to the still more general idea of Calculus on a differentiable manifold.

-- Jean Dieudonné, Foundations of Modern Analysis (1960), p. 141

We dub this the “ladder of abstraction”, taking the phrase from our teacher of yore. Referring likewise to ascent into functions-of-functions, and function spaces, and functions from one function space to another, and to the duals of all that:

Detached from any context, this construction is a pointless formality.  But as we move up the ladder of abstraction, we find that constructions such as this  become commonplace …

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 43

The metaphor of ascent is well attested.  Gödel speaks of

the infinite series of ever stronger axioms of infinity, each of which expresses a new idea or insight.

-- quoted in Hao Wang, From Mathematics to Philosophy  (1974), p. 325

A mathematician writes of

the inferential staircase  leading from the laws of physics  to the world that lies about us …

-- David Berlinski, “The End of Materialist Science”, collected in:  The Deniable Darwin (2009), p. 160

~

Saunders MacLane,  in his book Mathematics:  Form and Function (1986), p. 36ff, has a section called “Mathematical Activities”, structured somewhat like our own in the Consilience essay.  Some of the topics are the same (analogy, abstraction, generalization), while others, not relating to consilience especially, differ (conundrums, axiomatization, proof).  One, intrinsic structure, seems to relate to consilience, but is only briefly developed; and the last, completion, we have treated under the more Quinean label of rounding out.

Now, abstraction and generalization are related notions, but neither entails the other.  MacLane acutely adduces the example of group theory.  Originally, this grew out of the concrete examples known as groups of transformations.  Later, algebraists abstracted into abstract groups.  Whether a generalization has thereby been achieved, is (as Chomsky likes to put it) “an empirical question”;  and in this case, it turns out, it has not. “No new groups turn up in this process, in view of the famous theorem of Cayley, which asserts that every (abstract) group is isomorphic to a group of transformations.”  Thus, in this case, the ladder of abstraction has only one rung.  (By contrast, abstract rings do turn out to generalize upon their original model, rings of integers.)

~

Seeking analogues of the Ladder of Abstraction  outside of mathematics proper, I happened upon this:

Quine suggests that levels of abstractness, modeled on Russell’s Theory of Types, might be established.  “In the beginning  there are only concrete objects.”  These constitute type zero and are the values of bound individual variables.  “To be is to be a value of a variable.”  Next comes first-order classes and relations:  they constitute entities of type 1 and are the values of bound predicate variables.  Classes of classes, and relations, constitute entities of type 2;  and so on.

-- Harold Lee, “Discourse and Event”, in: Hahn & Schilpp, eds., The Philosophy of W. V. Quine (1986), p. 297

The resemblance to Russell’s Theory of Types had not escaped me, but I rejected mention of it, since, rather than leading -- as the Ladder does -- to ever greater depth (the metaphor is here in distress -- maybe think of it as a ladder down a mineshaft), it seems to lead mostly to More of the Same.  In other words, forming those strata, as described above, is less like the dizzying and ethereal Abstract Ascent of mathematics, than simply forming new sets via the Power Set operation (a new and larger set consisting of all the subsets of the original set).  Now this, if we start with a finite set, leads absolutely nowhere.  It’s just like counting.  If you start with the whole of the Natural Numbers, now the Power Set operation does become more powerful, leading to new and incomparable levels of infinity.    Whether this leads to true new depth, or is rather a mere formal exercise, I do not know, since I lack all intuition of any infinities beyond the countable, let alone the Power of the Continuum or Measurable Cardinals.  Perhaps it does;  espresso-sodden Berkeley conversations about Quality emerging out of Quantity, return to mind.

Still, I am inclined to doubt it.  The very fact that the fellow can say “and so on”  virtually proves as much.  For there is no “and so on” to true mathematical abstraction.   There is nothing mechanical about such ascent -- it is more like a miracle.  You can proceed only one step -- nay rather, one leap at a time;  and the interval between leaps may take decades or even centuries.   Above the calculus lies Function Theory;  above that, Topology.  Above that, Algebraic Geometry. Far, far above us, hovers Category Theory, unreachably aloft.  And far, far above and beyond that, soars Topos Theory.  What comes next  is known only to angels.

To vary Nestroy’s celebrated epigram -- “Bis die Topologie gehts noch, aber von da bis sheaf theory  zieht sich der Weg.”

Additionally, Quine introduced the term semantic ascent.  There is some similarity to Gleason’s ladder of abstraction, but the ascent doesn’t go very high, and Quine himself -- perhaps surprisingly for a logician -- is wary of the upper reaches, preferring basic-level entities  behaviourally grounded.

Here the Russian author A. D. Aleksandrov, instead of envisaging a ladder,  uses the metaphor of layers  or (appropriately enough) of nesting, like Russian dolls, in the procession to affine or projective geometry and on to topology:

The properties of space are stratified … with respect to their depth and stability.  The ordinary Euclidean geometry was created by disregarding all properties of real bodies other than the geometrical;  here we perform yet another abstraction within geometry.

-- Aleksandrov et al, eds, Mathematics: Its Content, Methods, and Meaning (publication in the original Russian: 1956;  Eng. tr. publ. 1963), vol. III, p. 133

~

The more I think about it, the more this Ladder of Abstraction idea seems possibly fruitful.  Not so much as in the Theory of Types, but as in the scala naturae, which encompasses angelology.  (Compare also graded algebras.)

By contrast, mere ungraded “abstractness” in itself is of little interest. Thus, to take MacLane’s Group Theory example:  the so-called “abstract” groups (MacLane himself uses the sneer-quotes here) mean to lift aloft from Groups of Transformations, in that they retain the laws (associativity, inverses, and all that) while becoming agnostic as to the nature of the elements of the group.  But, first of all, groups of transformations are, compared with, say, pickles, already quite Abstract;  so the word adds, really, nothing.  Indeed, as soon as you say that two apples plus two apples are four apples, and that in the same sense  two penguins plus two penguins make four penguins (well, and a few more, after a while, if the sex mix is right), you are already indulging in such abstraction.

~

I tried looking up “abstraction” in the index of the various math textbooks and philosophy treatises on my shelves, and basically came up with  bupkes.  Thus, in Dummett’s omnibus volume, Truth and Other Enigmas (1978), we find no reference to abstraction per se, let alone to the Ladder of Abstraction, but only to “abstract objects” -- i.e., pickles versus the Meaning of ‘Pickle”,  the Idea of a Pickle, the set-containing-a-pickle, the… sandwich containing a pickle, the -- but enough.  Mathematics is so far beyond this, no comment is required.

~

The ethic -- even, the aesthetic -- of abstraction for its own sake, sociologically chronicaled here (“On Vulgar Numbers”), eventually evoked a backlash.

The Bourbaki group sought to present the entire abstract structure of all mathematical concepts in one set of volumes, the Eléments de Mathématique. In that treatise, the real numbers, which most of us regard as a starting point, only appeared midway into the series, as a special “locally compact topological group”.

An opposing idea, promoted especially in the Russian school, is that a few well-chosen examples can illuminate an entire field.

-- David Mumford, Forward to Mircea Pitici, ed., The Best Writing on Mathematics 2012, p. xv

~

For the latest in fine reading, check this out:

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

For more about abstraction, here:
         http://worldofdrjustice.blogspot.com/search/label/abstraction

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

  

On Reading in Someone Else’s Traces

As an impecunious graduate student, and later as a threadbare lexicographer, I bought used copies of books  whenever possible.  Sometimes, these had been ill-used.  This happened especially in the case of volumes currently being used as textbooks.   Absurdly overpriced when new, these might come within the price-range of the elbow-patched pauper  after having passed through the hands of some subsidized undergraduates  who never really should have taken the course in the first place, and who now sold their texts, out of sight out of mind,  allowing us (as Leviticus prescribes) to survive on their leavings and gleanings.   These individuals would often underline in ink, or highlight in yellow (or, horresco referens, pink), sometimes every other sentence or paragraph:  and the color would bleed through the page, spuriously hemi-highlighting many a quite random passage.   To read through the work, I had to wade through the swamp of their mediocrity of mind.   Why couldn’t they at least highlight in pencil?

At present, I am reading a used paperback copy of Gordon Craig’s classic 1978 history of Germany.   And as the chapters go by, it becomes increasingly apparent that the previous owner had been a scholar, or scrupulous autodidact.   The passages marked are few, and always in faint pencil.  Moreover, these do not constitute “highlights” in any obvious sense;  rather, they illustrate some theme which that reader was pursuing in his mind, no longer apparent to this one.   His interests, whatever they might have been, do not match mine:  the phrase “civil service” merits, in his recension, a rare double-underline.  Moreover, there are occasional pithy marginal notes, but penciled-in so small that, even with a magnifier, I cannot decipher them -- in part because he uses personal abbreviations, in part because the thought is not predictable.  But one sigil thus used  I do understand, and it marks him as my Sinnesgenosse:  that little pyramid of three dots, which signifies “therefore” to a logician.

∴

~

Someday, when I am gathered  to that great library in the sky,  my own annotated holdings will flood the market (if it still exists;  perhaps paper will be obsolete, and everyone on Kindle).   And the purchasers will puzzle over my own arcane jottings.   To aid later philologers, I shall mention here, that many of the abbreviatory symbols stem from lectures by Gleason or Quine:  the rounded curly-d for ‘boundary’, an acutely downhooked upright for ‘restricted to’ (whence ‘only, just’), a perpendicularly downhooked horizontal for ‘not’, a square for ‘necessarily’ (whence ‘must’), a diamond for ‘possibly, maybe’, a thick-shafted arrow (=>) for ‘causes, gives rise to’, an upside-down A for ‘all’, a backwards E for ‘there is, there exists’, an inverted point-triad for ‘since, because’, and so forth.  Or perhaps, like my own sad ashes, they will simply all be pulped.

[Note:  The more usual symbol for restriction of a function to a subset of a domain  is simply vertical-bar.  But that has many, many other meanings;  so I follow Gleason in adding a hook, which  quite appropriately  depicts the restrictor as a grappling-iron …

Likewise, there are many traditional symbols used for ‘not’, all of them grievously ambiguous.  I follow Quine in adding the disambiguating hook.]

The “Idea” Idea (with an excursus on ideation and subvocalisation)

Much of the most important and vital work done in the last half-century  depends [not upon experiment or brute calculation, but] upon new ideas;  and new ideas are notoriously exceedingly difficult to grasp.

-- Louis J. Mordell, Reflections of a Mathematician (1959), p. 11

We previously stated that mathematics is best characterized as the science, not of number, but of structure (or of pattern -- at this level of generality, either term will do).   As MacLane phrases it:

This chapter introduces the idea of the formal  in terms of certain basic structures:  Set, transformation, group, order, and topology.  With Bourbaki, we hold that Mathematics deals with such “mother structures”.  Against the historical order, we hold that they arise directly from the basic stuff of Mathematics.

Saunders MacLane,  Mathematics:  Form and Function (1986), p. 7

That last bit, you will note, is unabashedly Platonist, counterposing contingent human praxis  to transcendent time-independent Truth.  (We discuss this contraposition here.)

Voilà  le hic

But beyond that, or rather as an animating force within it,  and distinguishing mathematics from such structure- or pattern-centered enterprises as architecture or the plastic arts, is the central role of ideas. 

MacLane puts the matter well.  Re the derivation of Hamilton’s equations from Lagrange’s:

What appears as a trick is in fact an idea -- an idea which must have been clear to Hamilton when he did it.  But we claim that in general  most of the formal tricks appearing in Mathematics  are really ideas in disguise -- ideas presented as manipulations  because the manipulations can be made explicit, while the ideas are a bit nebulous.

-- Saunders MacLane,  Mathematics:  Form and Function (1986), p. 284

In a previous series of essays, we put forward certain particular “mother ideas”.  Here we reserve a meditation-space  for musing about “Ideas -- the very idea”.

~

Hadamard comments on Rodin’s testimony that, throughout the process of sculpting, he must keep the “global idea” in mind, even while working on the smallest details;  and that “this cannot be done without a very severe strain of thought.”

I do not feel that I have understood [a mathematical argument] as long as I do not succeed in grasping it in one global idea; and, unhappily, as with Rodin, this often requires a more or less painful exertion of thought.

-- Jacques Hadamard, The Psychology of Invention in the Mathematical Field (1945), p. 65

Hadamard scoffs at the account given by Souriau in his Théorie de l’Invention:  “Does the algebraist know what becomes of his ideas when he introduces them, in the form of signs, into his formulae?  Undoubtedly not,”  but just turns the crank of mechanical calculation.  Apparently Souriau never consulted an actual mathematician, says Hadamard:  the mathematician trusts his idea, his insight, his intuition, more than he does his calculations, which after all are not infrequently in error  (Hadamard confesses that he, like Poincaré, was but an indifferent numerical calculator):  If these clash, you first redo the calculations, before tossing overboard the Idea that motivated the whole thing.

~

Ideation and subvocalisation

Hadamard then makes an excursus  rather off the path our our principle inquiry;  yet we shall follow him a little ways.  He confronts the question of whether language be the key to thought;   and waxes indignant at those who, like Max Müller, dogmatically assert that, without language, thought itself must needs collapse:

I had a first hint of this when I read in Le Temps (1911):  “The idea cannot be conceived otherwise than through the word, and only exists by the word.”  My feeling was that the ideas of the man who wrote that  were of a poor quality.

-- -- Jacques Hadamard, The Psychology of Invention in the Mathematical Field (1945), p. 66

Likewise, the behaviorist J.B. Watson says somewhere that “thinking is nothing but our talking to ourselves”.

The devotees of this position  point to the dual meaning of the early Greek word logos -- ‘word, language’ and ‘reason, thought’;  and would by implication deny that our diminutive and prickly friend, the humble hedgehog, could really know One Big Thing or even a little weentsy one.

Hadamard, by contrast, is virtually a militant in the opposite camp:  “I fully agree with Schopenhauer when he writes, ‘Thoughts die  the moment they are embodied in words.”  This even applies to algebraic symbolism:  too cumbersome to actually think with;  you mostly only use them when checking your work.

The Dutch Intuitionist mathematician L.E.J. Brouwer is of similar mind:

De woorden van uw wiskundig  betoog zijn slechts de begeleiding van een woordloos wiskundig bouwen …

 

(Caption quotation from Dennis Hesseling, Gnomes in the Fog:  The Reception of Brower’s Intuitionism in the 1920s (2003), p. 38.)

The Neothomist philosopher Etienne Gilson  seconds the opinion of his countryman:

Si un linguiste me dit que c’est notre langue qui modèle d’abord  le monde que nous pensons,  je sais qu’il ne me parle pas en linguiste, mais en philosophe, qui se dispenserait d’ailleurs de me donner aucune justification philosophique de son opinion.  Non seulement je ne sais pas si elle est vraie, mais je ne sais même pas pourquoi elle lui semble vraie.

-- Etienne Gilson, Linguistique et philosophie (1969), p. 51

A noted Freudian psychiatrist agrees:

Every single thought, before formulation, has gone through a prior wordless state.

-- Otto Fenichel,  The Psychoanalytic Theory of Neurosis (1945), p. 46

A contemporary philosopher goes even further:  some ideas may be not only pre-linguistic, but even pre-conscious:

We may not be aware of our ideas.  An idea  in this sense  is a tendency to accept routes of thought .. that we may not recognize in ourselves, or even be able to articulate.

-- Simon Blackburn, Being Good (2001), p. 3.

The epigram "We may not be aware of our ideas" is deliberately paradoxical.  Blackburn means "idea", not in the sense of the completely conscious  "I have an idea, let's...", but of something like the often tacit metaphysical underpinnings of mentation and investigation, which we treated of earlier.  -- Blackburn extends this notion (in a way reminiscent of, but antedating, Freud):  "A permanent strand in Christian thought  is that we have no insight, or even lie to ourselves, about our heart's desires." (id., p. 30)

We close this excursus with an epigram of William Hamilton  which Hadamard quotes:

Speech is thus not the mother,

but the godmother of knowledge.

~

The reason such musings lie off our main track, is that we are largely uninterested in psychology, or thought-processes, or any of the hunches & hiccups that fallen Man is heir to  as he struggles to comprehend all that His hand hath made.  With Hadamard, we conceive that there are cognitive activities for which vocalization is neither required nor especially helpful:  say, playing Go, or basketball.  

There is an epigram, variously ascribed, that has always fascinated me:

“How can I know what I think

 until I see what I say ?”

On the face of it, this would appear to be anecdotal evidence for the thought-needs-language thesis.  But upon nearer inspection, it might argue rather the opposite:  That thought rose from some wordless region of the self, and only became an object to critical consciousness after having been concretized by transformation into words.

For us, the key question is to what extent an Idea -- one worthy of the majuscule -- can even be adequately expressed in our language.   Certainly the higher mathematics cannot be expressed in ordinary human language.  It has invented for itself a more or less arcane system of signs, obeying no human syntax;  you may, if you like, par abus de langage, call that too a “language”, but it is no natural human language, but rather an aide-mémoire cobbled together to express ideas that observe their own semantics, call that language or not.   Hadamard himself attests that human language does not serve him especially well, when he must express mathematical ideas.  Whenever he must hold forth on a mathematical topic, even one of his own devising and thus, to him, abstractly clear as a bell, he must write out the text of his lecture beforehand, lest he be left gasping and groping for words.

There is another old adage, current among linguistic philosophers:

“Whatever can be meant

can be expressed.”

At this point we hear the shade of that crusty critic of Le Temps, growling:  All that you mean, maybe. 

~

Let us put the point even more starkly.  Ask Not  (we channel Kennedy here) whether our (necessarily human versions of) ideas  could be adequately communicated to some other rational species.  Ask whether the Idea, as pre-existent in Platonic paradise, has been adequately incarnated in us.

(There now swims within my vision  the image of a category-theoretic Universal Object, with arrows slanting downwards  this way and that, as in Blake’s great painting.)

~

This is becoming interesting.  Hoping that your appetite has been whetted as well, we link to a couple of math-related installments of the “Any Ideas?” series:

Induction

The free invention of structures

Actual Infinity

~

We have tried to outline a capitalized or pregnant sense of the everyday word idea, which in most contexts certainly does not bear such freight.  (“I’ve got an idea, let’s go get pizza.”)  There is, however, another sense, which is still scientific/intellectual, yet which bears no Platonic or foundational flavor:  what is sometimes called a “bright idea”.   A bright idea is what causes a light-bulb to appear over the cartoon character’s head.  And it does represent some genuine cleverness, though its success is by no means guaranteed (and in the case of Donald Duck, will almost certainly come to grief.)

This more powerful form of inductive construction  can be deduced rather simply from the older form.  The trick is to construct, not the sequence of values, but the sequence of partial functions…

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 145

A “trick” is to an idea  as tactics is to strategy. 

Similarly:

We could prove the inequality by a limit argument from the known inequality for finite sums, but the following reasoning involves a very interesting technical device.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 195

~

We have noted before  that, once you set out to focus on Ideas per se, you keep winding up back in mathematics -- if only because there are so many of them there.  Yet more:  In our own lifetime, math itself has spawned a subfield  whose task, it would seem, is precisely the study and development of Ideas -- for their own sake, almost, and beyond such practicalities as computing the area of the field of Farmer Brown (or rather, Farmer Enkidu, since this concern goes back to Babylonia and beyond) or even its offspring, geometry, or the handmaiden of that, the calculus, or …  This field is called Category Theory, which (as faithful readers of this tragic blog will already know)  I do not personally understand:  but do note, that a recent introduction to same (subtitled “A first introduction to categories” -- the style of the title is that of children’s books;  and God willing, someday toddlers will study this stuff), by Lawvere & Schanuel, is titled:

Conceptual Mathematics

C’est un titre astutieux.  For again (this is a phenomenon which we have treated, in these essays, under the label “faux-naïf”), on the surface this might seem to be one of those liberal-feelgood substitutions for the actual hard work of thought, meant to bolster the self-esteem of slow-learners;  whereas in actual fact, it points at concepts -- what underlies such relatively superficial activities as real analysis, point-set topology, algebraic geometry (you with me, kids?), and all the rest.

[Excelsior]   There is a vast philosophical literature (and a smaller, but still substantial, linguistic literature) concerning the relations between language and thought.   To rehearse this would be pointless;  to attempt to enrich it, quixotic.   Still we may feel our way forwards, and conceivably (eventually) contribute some minim of value, by taking as our paradigm area of Thought -- mathematics, rather than cats being on mats, and that sort of thing.   And Language as comprising, not only natural human languages, but any attempt at symbolic and communicable representation of Thought. 

(For this quest, I request:  God’s guidance and Grace.  Since, sine qua, non.)

An initial linguistic bridge is provided by our remarks above about the notion idea in the sense of ‘bright idea’.   A bright idea is no mere clothing of a perception;  it is closer to an invention.   And the key term it brings us up next to is:  insight.

[TBC?  Solâ gratiâ … ]