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.)
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:
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.
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:
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 … ]