After a conversation about mathematical creativity yesterday, I went back to Hadamard’s remarkable book The Psychology of Invention in the Mathematical Field (published in 1945, and re-issued in 1996 with a new preface and the snappier title The Mind of the Mathematician by Princeton University Press).
Hadamard’s main thesis in the book is that there are four stages to mathematical discovery, which might be called preparation, incubation, illumination, and verification. I have discussed this here, but on another reading, I was struck by various other things which I hadn’t remembered. Here are a few.
Hadamard’s mistakes
Hadamard admits that mathematicians (including himself) make mistakes. This is itself an interesting fact which any theory of mathematical creativity must account for. Actually his mistakes don’t really seem so drastic to me, but they do concern cases where the process of discovery didn’t work as it should.
He describes the case of a formula which he proved (but did not publish) at the beginning of his career. Later he worked on Picard’s theorem, and failed to realise that his formula would have led to a significant advance. He did it the hard way, and the approach was pointed out later by Jensen, who had rediscovered the formula.
Another instance was his failing to realise that two very significant results about convergence of power series (for example, if the coefficients are real and positive, then there is a singularity where the circle of convergence meets the positive real axis) followed immediately from the results in his thesis.
Ferrol’s testimony
He quotes at length from a letter from the mental calculator Ferrol to Möbius, which fits in strikingly with the theory proposed by Julian Jaynes in his book The Origin of Consciousness in the Breakdown of the Bicameral Mind.
If I was asked a question, rather a difficult one by itself, the result immediately proceeded from my sensibility without my knowing at the first moment how I had obtained it; starting from the result, I then sought the way to be followed for this purpose. That intuitive conception which, curiously enough, has never been shaken by an error, has developed more and more as needs increased. Even now, I have often the sensation of somebody beside me whispering the right way to find the desired result; it concerns some ways where few people have entered before me and which I should certainly not have found if I had sought for them by myself.
It often seems to me, especially when I am alone, that I find myself in another world. Ideas of numbers seem to live. Suddenly, questions of any kind rise before my eyes with their answers.
Ferrol, by the way, could do algebraic as well as arithmetic calculations.
Do mathematicians think in words?
I should preface this discussion with Hadamard’s statement about how proponents of a view different from his can be so certain of themselves:
This is [an] instance of the double fact (1) that the psychology of different individuals may differ in some essential points; (2) that, if so, it may be almost impossible for the one to conceive the state of mind of the other.
The view that thought is impossible without language is associated with the philologist Max Müller, who said,
How do we know that there is a sky and that it is blue? Should we know of a sky if we had no name for it?
Various thinkers, including Hegel, have agreed. However, an impressive list of names is ranged on the other side, including Francis Galton (Charles Darwin’s cousin and pioneering statistician among other accomplishments), Wolfgang Köhler (the psychologist who wrote The Mentality of Apes), Alfred Binet (psychologist, who tested this by experimenting on his daughters), and the vast majority of mathematicians including, very strongly, Hadamard himself.
Here is Hadamard’s commentary on his mental imagery when he goes through Euclid’s proof that there are infinitely many primes. I think this is quite close to my musings about induction in connection with Mathematical Structures (see the LMS–Gresham lecture). Here he is considering the proof that there is a prime greater than 11.
Steps in the proof | My mental pictures |
---|---|
I consider all the primes from 2 to 11, say 2, 3, 5, 7, 11. | I see a confused mass. |
I form their product 2×3×5×7×11 = N. |
N being a rather large number, I imagine a point rather remote from the confused mass. |
I increase the product by 1, say N+1. | I see a second point a little beyond the first. |
That number, if not a prime, must admit of a prime divisor, which is the required number. | I see a point somewhere between the confused mass and the first point. |
Another thing that Max Müller said was that “mythology is a disease of language”. This so irritated J. R. R. Tolkien that he turned it around and said, more than half seriously, “language is a disease of mythology”. I am guessing that Tolkien would have been on Hadamard’s side in the debate about words and thought.
Logic v intuition
Hadamard gives several good examples. If you throw a ball, which then moves under the influence of gravity alone, common sense suggests that it will move in the vertical plane through its initial direction. But how is this proved? The first proof a student meets involves solving the equations of motion and observing that the motion lies in a plane; common sense has disappeared. It can be recovered by a different proof involving the (considerably more difficult) theorem that equations of this type, with given initial conditions, have a unique solution. If the ball were to move out of the plane to the right, there would be a different solution (obtained by reflection) where it moved out to the left. You could say that the ball has no reason to prefer left or right, and so stays in the plane; this common-sense argument is justified by the mathematical proof.
Several mathematicians who exhibited a high degree of intuition (Fermat, Riemann and Galois) are discussed in some detail.
Hadamard takes Felix Klein to task for saying, in the face of evidence, that “It would seem as if a strong naive space intuition were an attribute of the Teutonic race, while the critical, purely logical sense is more developed in the Latin and Hebrew races.” Indeed, shortly afterwards, the physicist Duhem made the opposite (and equally unsupported) claim that German scientists, especially mathematicians, are lacking in intuition, or even deliberately set it aside. By looking at Weierstrass and Riemann, Bertrand and Hermite, Poincaré refuted both views. Mathematicians need both logic and intuition, even if they possess them in varying degrees (and intuition, especially, tends to escape our efforts to define it).
Impact
Hadamard gives several examples relevant to the nonsense du jour about impact.
- The Greeks investigated properties of the ellipse with no thought of application. Millennia later, Kepler was able to use their results to identify planetary orbits as ellipses; this led Newton to the theory of universal gravitation.
- We have helium, essential for non-flammable lighter-than-air travel, as a result of two pieces of curiosity-driven research; the wish to know the composition of the sun; and increasing precision in measuring the density of nitrogen by Rayleigh, Ramsay and others.
- Bernoulli invented calculus of variations with a frivolous example (the curve of fastest descent). Volterra widened the concept to think of spaces of functions (and functionals on them) as analogous to number systems (and functions on them). Then, in quantum theory, measureable physical quantities turn out to be functionals.
He gives a lovely analogy in relation to his own work. He describes a piece of research to a physicist friend, who asked to what it applied. On being told that it had no applications (as yet), the friend compared Hadamard to
a painter who would begin by painting a landscape without leaving his studio and only then start on a walk to find in nature some landscape suiting his picture. This argument seemed to be correct, but as a matter of fact, I was right in not worrying about applications: they did come afterwards.
I’m not sure I understand how “common sense has disappeared”. Intuition a.k.a. common sense tells me that there can only be one solution, so if I find one and it lies in a plane … (but yes, if I imagine a solution not lying in the plane, then I can see it’s not unique and therefore counter-intuitive).
I paraphrased what Hadamard actually said; it is certainly my fault rather than his.
I think a clearer example is in my post about the commutative law,
https://cameroncounts.wordpress.com/2011/09/21/the-commutative-law/
which has the same pattern: a commonsense proof by dot diagrams; a completely unintuitive proof from Peano’s axioms; and a proof from the Zermelo-Fraenkel axioms which is much more complicated but formalises the commonsense proof.
In a line of thinking that goes back through Peirce to Aristotle, intellectual creation and discovery are treated under the heading of apagoge (abductive or retroductive reasoning). Here is a memorable passage from Peirce’s essay titled “The Neglected Argument for the Reality of God”.
“The whole series of mental performances between the notice of the wonderful phenomenon and the acceptance of the hypothesis, during which the usually docile understanding seems to hold the bit between its teeth and to have us at its mercy,—the search for pertinent circumstances and the laying hold of them, sometimes without our cognizance, the scrutiny of them, the dark laboring, the bursting out of the startling conjecture, the remarking of its smooth fitting to the anomaly, as it is turned back and forth like a key in a lock, and the final estimation of its Plausibility,—I reckon as composing the First Stage of Inquiry. Its characteristic formula of reasoning I term Retroduction, i.e. reasoning from consequent to antecedent. In one respect the designation seems inappropriate; for in most instances where conjecture mounts the high peaks of Plausibility,—and is really most worthy of confidence,—the inquirer is unable definitely to formulate just what the explained wonder is; or can only do so in the light of the hypothesis. In short, it is a form of Argument rather than of Argumentation.
“Retroduction does not afford security. The hypothesis must be tested. This testing, to be logically valid, must honestly start, not as Retroduction starts, with scrutiny of the phenomena, but with examination of the hypothesis, and a muster of all sorts of conditional experiential consequences which would follow from its truth. This constitutes the Second Stage of Inquiry. For its characteristic form of reasoning our language has, for two centuries, been happily provided with the name Deduction.”
I’m not competence to judge, but science and theology/metaphysics should be separated.
Logic should be the key for almost everything. Nonetheless, Logic should be supported by the laws of Dialectic. Since dialectic and logic are complementary.
Consequently, basic concepts like fairness or intuitiveness can be checked & supported by logic.