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.
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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.
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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.
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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.
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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.
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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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