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.

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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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Realism: What

[We choose a classic Lockean title-style for this chapter, and begin with a middle-of-the-road definition.]

Realism Defined

First, a plain-man's characterization of what we might call minimalist/core/plain-vanilla Realism:

Outside our heads  there is freestanding reality.  Only madmen and a scattering of constructivist philosophers doubt its existence.

-- Edward O. Wilson, Consilience (1998)

This view may also be referred to as Platonism (though cf. a special restriction of this term  to mathematics, outlined below):

One issue that has traditionally divided philosophers  is whether ther are abstract objects.  Nominalists have held that there are not;  realists (in a special sense of the word) or Platonists (as they have been called  to avoid the troubles of ‘realist’), have held that there are.

-- W.V.O. Quine, Word and Object (1960), p. 233

Now for the more careful distinctions of professional philosophers:

A. E. Taylor, Elements of Metaphysics (1903; page references to the University Paperback reprint), p. 67:

By Realism is meant the doctrine that the fundamental character of that which really is, as distinguished from that which is only imagined to be, is to be found in its independence of all relation to the experience of a subject.  What exists at all, the realist holds, exists equally  whether it is experienced or not.

Within this, he distinguishes (p. 68) two subtypes:

Agnostic Realism, while asserting the ultimate dependence of our experience upon a reality which exists independently of experience, denies that we have any knowledge of the nature of this independent reality.

Sic:  not “full knowledge”: any knowledge.  Which is absurd.

Contrast (p. 69):

Dogmatic Realism, of which Leibnitz and … Herbart are the most important representatives … while maintaining that real being is independent of experience, at the same time  holds that it is possible to have positive knowledge  not only of its existence, but of its nature.

Sic:  postive knowledge, but not necessesarily full knowledge.  And in this form I heartily subscribe, despite the invidious label conferred by its opponent Mr. Taylor.   As William James put it, in The Principles of Psychology (1890), vol. II, p. 634:

Reality exists as a plenum.  … But we can neither experience nor think this plenum.  What we experience, what comes before us, is a chaos of fragmentary impressions  interrupting each other;  what we think is an abstract system of hypothetical data and laws.

Full knowledge -- plenary knowledge of Reality in all its fullness -- not only in some rarefied ding-an-sich sense, but in the ordinary sense of the sciences -- is doubtless impossible in the case of anything so wildly complex as an acorn or a rock;  but in the case of something simple, like Hilbert Space (infinite-dimensional, it is true, but tamed by a norm that is based upon an inner product), we can perhaps come close to exhausting most of what can be known of it.

Taylor fancies he has refuted Realism in all its varieties, summarizing his triumph thus:

Produce any instance you please, we said to the realist, … and we will undertake to show that it derives its reality for you  from the very fact that it is not ultimately separable from the experience of a subject.

Note the crucial bait-and-switch!   He whisks Reality -- an ontological category, and a big deal -- under the thimble, and takes out -- “Reality for you”, a miserable psychological gewgaw of no general interest whatever.   This latter concept is, we readily concede, drenched in irreducible subjectivism;  no need to bother to argue the point.  And bearing, it may be, little relation to Reality, whether theoretically or empirically.

For:  Reality, in reality-for-X, is an incomplete symbol.

Thus for instance:   It is not possible to survey the roster of extant men, and finally lay your finger upon one specimen, the average man.  The Average Man is not a real, but an ideal, to which various actual men may approximate to one degree or another.  And Real-for-Joe-Blow is not a real, but a figment, which may reflect more or less of an actual reality, with greater or lesser distortion.

~

~  Posthumous Endorsement ~

"Were I alive today, and in the mood for a mystery,

this is what I would be reading: "

I Don't Do Divorce Cases

Murphy on the Mount.

(I am John Locke, and I approved this message.)

~         ~

~

A more carefully phrased importation of subjectivism into the debate  is provided by Michael Dummett, in “Truth” (1959), repr. in Truth and other enigmas (1978), p. 23f:

The claim … should be rejected by a realist, who might, and I think ought to, agree to the following weaker principle:  that a statement cannot be true unless it is in principle capable of being known to be true. […]  The anti-realist interprets ‘capable of being known’ to mean ‘capable of being known by us’, whereas the realist interprets it to mean ‘capable of being known by some hypothetical being  whose intellectual capacities and powers of observation  may exceed our own.’ … The issue between realism and anti-realism … is one of the most fundamental of all the problems of philosophy.

Thus here, clearly, we are once again confronted -- like it or not -- with the question of theism.

So:  if you are a full-bore, double-barrelled, two-seed-in-the-spirit, dyed-in-the-wool copper-bottomed no-holds-barred Cantorian Realist … (we pause for the roars of approval to subside) … even in merely so much as mathematical Platonism:  must you therefore, of logical necessity, be a theist in the sense of the Abrahamic tradition?

Perhaps not; certainly, we have not shown so.  (Nor do we so much as aim to show more that that.   At this point, Jews Muslims and Christians are all singing kumbaya together in one big tent.)   You could, in principle, construct for yourself an ontological halfway-house, with a non-omnipotent, though omniscient, Knower.   And who might this Wiser Being be?  Why -- none other than Babar, the Elephant King!

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In these essays, we are concerned mainly with the form of Realism known (in a mathematical context) as Platonism, defined as

the theory that mathematics describes a realm or system of real and independently existing objects, whose nature is known to us through proof, but which are entities  over and above the proofs by whch we discover them.

-- Roger Scruton, Modern Philosophy (1994), p. 384

(This is fine, but don’t let’s squabble about “objects”; “patterns” or “truths” would do just as well.  We’re not interested in reifying anything, but merely in claiming that we’re not just making it all up arbitrarily.)

And:

… the subject matter of mathematics as realistically (i.e. platonistically) construed.

-- Colin McGinn, “Truth and use”; in: Mark Platts, ed.  Reference, Truth and Reality (1980), p. 35

 But for the record, there are other sorts:

Do we wish to say that there is a moral reality, which underpins our moral judgements  and guarantees their truth?  Some philosophers have argued for such a view (‘moral realism’).

-- Roger Scruton, Modern Philosophy (1994), p. 98

Scruton himself is far from being a moral relativist or nihilist, but seems not to feel forced to accept such moral realism, at least in the form of what we might call a moral Correspondence Theory (as opposed to the real existence of God and his laws).    Indeed, he compares (we shall call it) Aesthetic Realism:

It is obvious that St Paul’s Cathedral is beautiful, and the new Lloyd’s building  repulsive;  but is there some ‘aesthetic reality’ that makes these judgements true?

You readily see how problematic this is:  Saint Paul’s to a Wahhabi or an Iconoclast would lack appeal; and presumably the people who designed and paid for the Lloyd’s building  felt it had a certain something.   I still think a kind of case might be made out (which I’ll not pursue), that it’s not just all a matter of personal preference, and that de gustibus certe dispundandum est.   The argument would proceed by analogy with the much tighter case in mathematics.  Many things are obvious to the expert which are not obvious at all to the general public, and which indeed could never be made obvious to them no matter how hard you tried (sheaf theory, for starters).  And more tellingly,  the Man on the Clapham Omnibus will pronounce some things obvious (particularly in the area of probability, e.g. the Monty Hall problem) that are demonstrably mistaken (though some will understand the demonstration, and some will not).    We do not conclude from this that mathematical truth is just a matter of taste (“we” here referring of course to men of good sense, and excluding the Postmodernists).   Some are more qualified than others to grasp certain truths.    Quite possibly the same is true in the field of aesthetics, as regards, say, the Goldberg Variations, on the one hand, and “Who Let the Dogs Out”, on the other.   One would have to make out the case that those who exalt the latter and contemn the former have certain other things wrong with them as well.

[Update]   I have just come across an intriguing analysis relevant to the case of the “Lloyd’s building”, and that buttresses the Aesthetic Realist conjecture above, to the effect that it’s not all just a matter of mutually incomparable ‘tastes’, but that there are objectively different levels of aesthetic competence, owing to the importance of informed appreciation.  The analysis appears in a delightful essay, “The Gherkin”, by John D. Barrow, in his book 100 Essential Things You Didn’t Know You Didn’t Know (2008;  the book is better than its title).  The “gherkin” in question is a new building in the City of London, a.k.a. “the Pine Cone” for its pocked appearance:   “Prince Charles sees it as symptomatic of a rash of carbuncular towers on the face of London”.   Yet it won the Stirling Prize for architecture.  What gives?

It turns out the building is, from an engineering standpoint, quite ingeniously designed, and that principally with a view towards eco-friendliness.  Thus it narrows somewhat at the bottom, to defeat the wind-tunnel effect that annoys pedestrians; and tapers again towards the top, which “opens up more of the sky and reduces the dominating effect of the structure because you can’t see the top from close-by on the ground.”   Even the pocks have their point:  “They bring light and natural ventilation deep into the heart of the building”, saving greatly on energy costs.  And more details in this vein.

I’ve never seen it, but it sounds … beautiful …

~

Realism in psychology:

According to his friend and biographer, Freud “had a high and serious respect for the reality of psychological facts.  They were as real and concrete to him  as metals are to a metallurgist.” (Ernest Jones, Freud: Years of Maturity (1955), p. 432)

[continued here]

Further juicy quotations:

As a Platonist, he saw everything on earth as broken arcs, which merely suggested the perfect rounds above.

-- Louis Auchincloss, The Rector of Justin (1964), p. 92

Realists [with a capital R] are not the same thing as ‘realists’ in daily life, who are men who expect neither themselves nor others to be any better than they ought to be, and generally much worse.

-- Ernest Gellner, “The crisis in the humanities” (1964), collected in The Devil in Modern Philosophy (1974), p. 15

To mathematicians who study them, moduli schemes are just as real as the regular objects in the world.

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

Footnote to "The Urysohn Metrization Theorem (continued)"

(Re:  what initially might look like quite wild spaces, which nevertheless are metrizable.)

It turns out they can still be quite wild.  For even among the metrizables, we may find that we have bitten off  more than we can comfortably chew.  Cf. Lynn Loomis and Shlomo Sternberg, Advanced Calculus (1968):

A metric space can be a very odd object.  It may fail to have almost any property one can think of.

Apart from regularity and second-countability, I was about to add;  and yet, reviewing the theorem-statements quoted earlier, not a one states clearly that these two qualities are both sufficient and necessary for metrizability.  The closest we come is with Willard:

The following are equivalent for a T1-space X:

(a)  X is regular and second countable

(b) X is separable and metrizable

which says only that if the space is both metrizable and separable, then the other two qualities follow.

            So:  a fortiori:  having taken what we might have imagined a fairly innocuous step of expanding the field, we are overwhelmed by the abundant results, and quickly seek further constraints to make matters manageable.   There are stronger versions of metrics, such as the norm (homogeneous, translation-invariant metrics) and (stronger still) the inner product, trademark of Hilbert Space. 

            The situation recalls that of post-Chomskyan linguistics, where the giddy sense of unlimited possibilities that came with a naïve conception of generative grammar, soon gave way to a sober search for constraints, lest the theory be devoid of interesting theorems.

[Concluded here.]