Consilience and Cognitive Science

In a recent essay, the polymath philosopher Ian Hacking, as the first and most fundamental of his list of “applications” of mathematics, puts “App 0:  Math Applied to Math”.

Why should there be so much ultimate connectedness …?  If we follow the cognitive scientists who think that there are distinct mental modules for arithmetical reasoning and for spacial reasoning, Descartes’ Geometry of 1637 is all the more astonishing.  This question needs a lot of philosophical work, right now.

-- Ian Hacking, “Why is there Philosophy of Mathematics at All?”, repr. in Mircea Pitici, ed., The Best Writing on Mathematics 2012, p. 248

Well, here’s a tidbit towards that.

Hacking is right, that the Idols of the Neuroscientists make the success of Descartes’ breakthrough the more astonishing, as indeed they do any of the insights of mathematicians.  But that simply redounds to the discredit of the Neuroscientists.

Hacking happens to have chosen an instance of Consilience that neatly matches two pre-posited “modules” of the CogSci crowd.  (Such “modules” can be had for the asking, according to fancy;  there is no net, in that tennis-game.  -- Table-tennis, rather.  -- Right now, my “Stuff and Nonsense!” module is blinking red.)   But they would be rather more hard-pressed to come up with a separate “point-set topology module”, “abelian group theory module”, “number-theory module”, “operator theory modules”, “ring theory module”,  ET cetera ET cetera, to attempt to account all the sundry other intersubdisciplinary interilluminations which keep arising within mathematical practice -- deep connections among concepts that never fell within human ken, back when our geometry module, in collaboration with the arithmetical module, was calculating the angle and distance to the rampaging mastodon, yet which now yield such fruits as the beans Jack received in barter for a cow.

Perhaps the explanation lies well apart from the contingent biochemicalisms of the grey-matter blancmange in our brainpan, modular or otherwise.   Mayhap there is an actual unity -- a unity in variety, which we do but discover --

in bits and glimpses,

in pieces and snatches,

by dint of much study,

and by grace of Grace.

~

We are making a New Year’s Resolution, not to be so darned snarky.  (Having successfully shed nearly twenty pounds this past calendar year, we are ready to take on even greater challenges.)

Accordingly, so as to redeem our good standing among our Cognitive Scientist brethren (with whom we used to rub elbow-patches daily, back at UC Berkeley, in the permanent “temporary building” T-2), we herewith offer an extra-virgin olive branch:   preliminary results from a massive structural-neuro-longitudinal study undertaking by the Pataphysical Department of the World of Dr Justice (headquarters:  Geneva), abundantly funded by the grateful taxpayer.  With this we obtain direct insights into the child’s developing mind.   As our investigations prove, this comes pre-compartmented into the following empirically buttressed cubby-holes:

(1)  The “Everything is What it is and Not  Another Thing” foodstuff module.

The vegetables are bad enough, but when they touch the mashed potatoes -- Eeewwww.

(2)  The “Mine and Thine” module.

Though discernable in outline already in the early embryo, this module does not become activated until much later in development;  and in some specimens (B. Madoff, D. Trump) never.

Prior to activation -- much like the notocord that serves the embryo until the spinal column takes its place -- behavior is governed by the following, non-modular (“opportunist”) pragmatic maxim:

     “My bear.  My toy.  Mine.”

(3)  The “Girls Have Cooties” module.

For reasons not yet well understood, this module is activated only in immature males;  some genetic link to the Y chromosome is suspeced.  In any event, it is normally de-activated at puberty.

Gradus ad Parnassum

We are familiar with the genre of spiritual (auto)biography.

First, our young enlightenee-to-be  experiences little but Unordnung und frühes Leid;  then, groping and grappling with shadowly intuited truths; then at some point there supervenes something supernatural -- most starkly, in the form of angelic intervention. 

Thus, Muhammad of Mecca, “enwrapped” (a detail telling in its biographical specificity -- this is not just all made-up -- it’s like the detail of Jesus doodling with his finger in the sand), alternately sweating and shivering in his Cave of Retreat, at last is confronted with an Archangel, who (after some preliminaries which it would delicious to retell, but which space does not permit), says:  Iqra’! (“Read!” -- or rather, “Recite!” or “Repeat after me!”)  and reveals the Qur’an.  

Likewise the future Saint Augustine.  He led a misspent youth, at one point sinking to the actual infamy of stealing pears (!);  until one day, a unseen voice cried out: 

Tolle, lege!

(‘Pick up [the Bible] and read!’)

After these interventions, it is pretty much smooth sailing for our Chosen Ones (one of the epithets of Muhammad,  Mustafa, means precisely ‘chosen’), who never look back.

~

And now we come to the intellectual autobiography -- specifically, the mathematical memoir -- of Edward Frenkel (Френкель, Эдвард, de son vrai nom):  Love and Math.

He too grew up in somewhat unpromising circumstances, well outside of Moscow, which for a Soviet of the time  was as cruel a fate as living far from Paris, for the French.  Jewish to boot, which meant that, so far from being called (here in a secular sense summoned, rather than that of ‘having a calling’), he was actually turned away at the gate, and later (not taking nyet for an answer) had literally to scale a fence and sneak past armed guards to reach the seminar rooms of that sanctum sanctorum, Moscow State University.
(There is some takeaway here for idealistic educators:  You can paint the classrooms with colors as bright as you like, but ultimately it comes down to student capability, and motivation.)

Now, all that high adventure is fine preparation for an actor, or a novelist, or a revolutionary, but is not especially helpful to gain a grounding in the principles and arcana of contemporary mathematics:  a path that has risen at an ever-increasing pitch, since antiquity, and branched into perplexing byways, before the blessed consilience  of synthesis, such as Cartesian geometry, the Erlangen Program, and latterly the Langlands Program, forged new anastomoses, reknitting the whole thing.  Yet at the age of sixteen, when most of us are just learning to shave (or looking forward to needing to -- meanwhile, these pesky pimples), he somehow comes to the attention of a world-class mathematician, who refers him to the special care of one of the archangels of the field, Prof. Dmitry Fuchs.  Fuchs hands him an article from the forefront of breaking research, and says:  Tolle, lege.  (Or, one supposes, принять! читать!)   And the next thing you know, our shaveling is attending the legendary evening seminars of that god among men, I.M. Gelfand -- the Wiener Kreis of Soviet mathematics --  understanding everything, and swiftly publishing a research breakthrough of his own.  By the time he is twenty-one (barely old enough to vote, when I was that age), he has been summoned to Harvard.

(For anecdotal evidence about how hard this stuff is, even for people who have been doing math their whole life, try this:  Oligophrenia mathematica.)

~

Now, if you have never yourself grappled with research-forefront mathematics, you will have no idea how extraordinary, almost preposterous, that account is.  The epiphany-stories of Muhammad and of Augustine, which theophobes will dismiss out of hand as «miraculous» (as though the presence of a miracle itself suffices to spoil the tale, like a fly in the soup), are humdrum by comparison.

For, both those chosen were presented with texts in a language they already knew (Arabic and Latin respectively), and which  moreover  had been composed specifically to be received by the masses (with imperfect understanding, it may be, but getting the gist and the uplift).  The Qur'an, indeed, helpfully mentions that it has been revealed «in plain Arabic».   Whereas the Fuchsian manuscript presupposes millennia of progressively more successful wrestling, with abstruse insights, by the finest minds on earth.
So:  Either Professor Frenkel is embellishing just a bit, or rather compressing, in retrospect, or else this scene indeed was:  a miracle.  For, for anyone else, that manuscript would have been a book  of seven times seven seals.

~

Frenkel's heart is in the right place.   He has joined with such popularizers as Stanislas Dehaene, in suggesting that more or less everyone has la bosse des maths,  the little darlings need merely be placed into the right pot and watered, and they will bud and blossom.   The invariant come-on is a pointing to results «beautiful and elegant».   On the very first page of his book, attempting to explain the public indifference or actual aversion to what they imagine to constitute math, Frenkel writes:

What if at school  you had to take an 'art class' in which you were only taught how to paint a fence? ... While the paintings of the great masters are readily available, the math of the great masters  is locked away.

True, and nicely observed. But such beauty and such elegance are perceptible only to the mind prepared -- otherwise it is like playing Bach to a baby.

The suggestion that one can chug one's way to the top of this particular ethereal Parnassus, simply with hard work and the right attitude (I think I can, I think I can), fits in well with the myth of the Little Engine that Could, that I and my playmates were brought up on, pluckily chugging uphill.  Whereas in practice, the brave little engine makes it only as far as the first false-peak, never ascending the Ladder of Abstraction that lies beyond; while one of your company  suddenly sprouts wings, and is halfway up the slope.

The position that we are all Gausses in nuce, if only we were given half a  chance, likewise fits in well with the anti-innatist, doggedly/dogmatically environmentalist political-correctitudes of our own day.   Yet I am here not really plunking for either side of that false dichotomy.   Yes, both are necessary, sweat-equity and the right genes;  but beyond that, something mysterious ... Call it Grace.

Well;  bless him.  May his infinite series never fail to converge, may his commutators ever commute.  For the rest of us, we must be content with a Pisgah-glimpse.  And to reconcile ourselves to the following refractory, diamond-hard truth:

Not that many are even called,

and precious few are chosen.

~     ~     ~

[A note to my readers, puzzled  perhaps  by a sudden change in punctuation-style.   My word-processor, for reasons best known to itself, between the hour at which I posted the beginning of this essay, and a moment ago when I posted the rest, has suddenly and inexplicably switched from American-style quotation-marks  to the angled version favored in France (or, in reverse order, in Germany).  Apparently the software has been favored with some sort of epiphany, to which I myself am not privy.
Perhaps, as the day wears on, the keyboard will begin printing in Cyrillic.  And yea, I shall be baffled thereby, and sore afraid.
Then strange symbols, and equations, will begin creeping in:  and I shall shake, in fear and trembling.
But then a voice from on high rings out --

TOLLE -- LEGE !

And all will be well.]

~

Further reading:

Beauty as a criterion of truth in math and science.

Mathematical autobiography: 

            André Weil

            Neal Koblitz

Psychologia mathematica:  Invention and Insight 

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 … ]

The Psychology of Mathematical Invention

In the introduction to his classic study The Psychology of Invention in the Mathematical Field (1945), Jacques Hadamard writes, with great acuity:

Our title is “Psychology of Invention in the Mathematical Field” and not “Psychology of Mathematical Invention”. … Mathematical invention is but a case of invention in general…

We do not aim at any such generality, and thus choose the more circumspect title.

The other aspect of Hadamard’s title we are tempted to change, is “invention”, vice discovery; for we have argued throughout these essays that, just as in physics, we discover mathematical truths that exist antecedently and independently of the fumblings of any particular forked-radishes -- though our characterizations and purported “proofs” of these truths, are  to be sure  fallible and contingent.

It turns out, though, that Hadamard himself  is a proper Platonist, and we disagree not at all:  “We speak of invention;  it would be more correct to speak of discovery.”  He goes on to make several acute remarks about the distinction in general between discovery and invention, which we hope, in good time, to treat in another place.   (Of course, by the time I get around to it, I might myself be in Another Place, in which case the practicalities of publication might become problematic.)

[Note, by the way, that this distinction between discovery and invention is, for a philosopher, more than a niggling nicety;  for indeed it turns the significance of his opening statement, “Mathematical invention is but a case of invention in general”, quite upon its head.  As stated, and as it would normally be understood, it assimilates the -- let us call it, neurtrally,  attainment of mathematical truths, to the invention of, say, the cuckoo-clock, or the beer bong.  Whereas, substitute discovery, and now we have the Platonist position in full:  you discover Riemann surfaces they way you discover the origins of the Nile.]

The final thing I wish I could change about Hadamard’s title, is “psychology”.   For, at this level, the mental foibles of hominids or turtles  is of very little interest in itself.   But -- we are incarnated.  (I almost wrote, “Alas!”, save that Our Lord, Himself, did not disdain to don a human frame.)   And at least we are treating of cognitive psychology, and indeed that of creative mathematicians, rather than the psychology of nihilists or Donald Trump.

~

In addition to Hadamard’s well-known booklet, the combinatoric mathematician George Pólya wrote a whole series of volumes  on mathematical heuristics.  One of them, How to Solve It, is quite widely known;  I myself  for some reason  could never really get into it, but many consider it a classic.

~

Andrew Gleason offers some stray comments,  which might illuminate the creative process, which I have gleaned from his one layman-accessible book.

Mathematical research  is largely a process of winnowing theorems from a melange of hunches, vague analogies, and geometrical images.

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

Gleason is here  speaking for himself and for some others, but not all.  Hadamard mentions Hermite’s actual hatred for geometrical images.

Mathematicians continue to rely on what is ultimately a subjective process for evaluating proofs.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 7

Since mathematics exceeds all other activities whatsoever in intellectual rigor, this statement should not be taken as some kind of fatal confession.   Indeed, since the defeat (at the hands of Gödel and others) of Hilbert’s well-intentioned but doomed Formalist program, mathematics itself has abounded in deep subtleties, illuminating this “subjectivity” in ways that show it to be far beyond any simple falling-short of objectivity, any de gustibus.   (Cf. startling results such that something may be uncountable within a model, but countable ‘from outside’.)

It may happen that a conditional and its contrapositive have a distinctly different intuitive flavor…

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 10

(Language as mediator  between math and the mere brain.)

Today’s mathematics is based on set theory. … Mathematics is thus reduced, in a sense, to glorified combinatorial problems.  While this approach is decried by some  for making mathematics nonintuitive, it does, in fact, lead to a new kind of intuition  which is indispensable in modern algebra …

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 55

(Psycho)logical note:   The purported foundation of mathematics upon set theory  does not subtract one whit from all that went before, but simply adds an understory.  This rhetoric of “reduced … glorified…” is therefore an example of mathematical humility, a topic whereof I hope to treat  in another place, at another time.

~

There follow some glimpses of the mathematician at his workbench -- not presenting his finished results to the world, but puzzling over how to make progress.

*     *     *

~ Commercial break ~

Relief for beleaguered Nook lovers!

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

We now return you to your regularly scheduled essay.

*     *     *

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

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

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

He goes on:

From a certain point of view, the main problem of group theory is to decide when two groups are isomorphic, and the main problem of topology is to decide when two spaces are homeomorophic.

--  Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 18

In practice,  algebraists and topologists  usually focus on a different problematics;  but this analogy does suggest a research program for Hilbert space -- which, however turns out to be difficult to pursue in practice, since such questions are “usually too broad (and too vague) ever to arrive at a satisfactory solution."  We can picture the researcher sitting puzzled at his desk:

Special cases of the problem of unitary equivalence  can sometimes turn out to be rewarding;  it is usually hard to tell in advance  whether they will or not.

-- id., p. 19

Heuristic, rule-of-thumb, seat-of-the-pants  research programs:

The hope is that, if we know all about all invariant subspaces of many operators, we might get an insight into either the construction of an operator without any, or the proof that all operators have one.

--  Paul Halmos, “A Glimpse into Hilbert Space”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 12

Topology in its full glory being intractable,

We associate with topological spaces and with continuous maps  certain algebraic objects, called topological invariants, under the hopeful assumption that algebra is easier than topology.  These invariants, in order to be useful, must be computable, which often is also just a hopeful assumption.

--Samuel Eilenberg, “Algebraic Topology”, in: T. L. Saaty, ed.  Lectures on Modern Mathematics, vol. I (1963), p. 101.

~

James Newman (The World of Mathematics, p. 2039) calls Hadamard's essay "entertaining ... but not very enlightening."


Weiteres zum Thema:
Above, we treated of the psychology of successful mathematical endeavor.   Another psychological problem is why, for humans, including professional mathematicians, mathematics is so darn hard.
For a fairly funny essay on the subject, click here:
Oligophrenia mathematica

 

From outside mathematics:

Aus den Mitteilungen einiger höchstproduktiven Menschen, wie Goethe und Helmholtz, erfahren wir doch eher, da das Wesentliche und Neue ihrer Schöpfungen  ihnen einfallsartig gegeben wurde, und fast fertig  zu ihrer Wahrnehmung kam.

-- Sigmund Freud, Die Traumdeutung (1900)

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Uniform Spaces

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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