Discovery vs. Invention, again

In our essay on Hadamard,  we remarked (as did he)  that his title The Psychology of Invention in the Mathematical Field would more felicitously have been : The Psychology of Discovery in the Mathematical Field -- and this, for reasons internal to mathematics.

Another consideration, unrelated to math per se, lies at quite another angle to all that.  Namely, that the psychology of someone who discovers (or who purports to), may differ widely from that of one who invents.

Whoso wishes to learn more about the Creation, wherewith he was confronted when (all unbidden) he first came into our sphere here below,  keeps his eyes and mind wide-open, as he walks through the world.  Whereas the inventor could be shut-up in his Fortress of Solitude, his basement tinkering-shop. 

Sometimes these cognitive stances are difficult to separate, as in experimental physics (discovering phenomena,  vs. designing experiments and inventing your own equipment).  In biology, the split can be stark.

The Naturalist observes, and notes, and ponders, and sleeps and dreams upon it, until a pattern becomes clear -- a pattern  in whose creation, he had no hand.

The Biological Blasphemer, by contrast,  one such as Dr. Frankenstein, or ‘reassignment’ surgeons, or those latter-day famuli of Dr. Moreau, 

busily sticking mouse-genes into Drosophila,

or pig-livers into people,

or cockroach-souls into orangutans,

or attempting to get a scorpion to mate with a Lamb --

these, indeed, do not discover what already lies (in some bioPlatonic World of Forms) beyond the creation of His hand,  but merely the extramarital whelping of their own: 

they rather denature Nature, until,  with Satanic originality, they reign alone, unchallenged, in a godless cosmos,  surrounded by devilkins of their own devising.

[Update 25 February 2014]  In only the latest development along these lines -- for new ones are announced almost daily, a drumbeat that deadens sensibilities -- there’s this:

http://www.nytimes.com/2014/02/26/health/fda-meeting-considers-controversial-fertility-procedure.html

Already, Heather has two mommies;  now she can have a daddy and two mommies right from conception, three people contributing genes to the gallimaufry.  In this way, Inventors hope to come up with a child born with three eyes, each of a different color.

~

These thoughts (or fulminations)  were prompted by re-reading the essay by C.S. Lewis, “Christianity and Literature”, reprinted in Rehabilitations (1939), and Christian Reflections (1967).  From  p.7 of the latter collection:

 

If I have read the New Testament aright, it leaves no room for ‘creativeness’ … Pride [in particular, that of the self-regarding artist] does not only go before a fall, but is a fall -- a fall of the creature’s attention  from what is better, God, to what is worse, itself.

-- Yet I must refrain from quoting further, since, once begun, I must needs quote the whole of it.   Therefore, reader, rather lay this present trifle aside, and study that essay for yourself.

Lewis did not live to witness  the iridescent developments of our (post)modern arts scene;  yet I doubt if these would have moved him to revise his views.  Contemporary artists (“artists” -- I would place the word in indignant quotes, save that, by now, the notion is past protecting), ever striving for something truly new,  have managed to come up with some strikingly original ideas, such as: 

* sticking a crucifix into a beaker of piss;

* masturbating onstage;  or

* nailing their penis to a plank of wood.

(Would we were making those up;  not so.)

Yet what must be their chagrin, to learn, that they thereby have done no more than recycle the shenanigans that began back in the nineteen-teens, with Dada, in Paris and Zurich (though, then, served up  with less frenzy  and more wit) !

~

[Boring and unimportant, merely-personal footnote:

It is psychologically odd that, while the pursuits I favor  are generally those that do not involve getting shot at, or tramping through mosquito-infested marsh, or fiddling with explosives in the lab,   yet my attitude to mathematics  resembles more that of the naturalist -- or even a lepidopterist -- than that of a rocket scientist.

(My own such propensities would fall well beneath the notice of Hadamard;  here I address only my fellow oligophreniacs.)  ]

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.

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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.

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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.

~

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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