Friday, July 31, 2009

Hadamard: The Mathematician's Mind

I've been meaning to discuss the book The Psychology of Mathematical Invention in the Mathematical Field, by Jacques Hadamard. It was inspired by a lecture of Poincare to the French Psychological Society entitled "Mathematical Creation". Although I always considered Hadamard a 19th century figure (born 1865), the book was published in 1945 and he lived until 1963!

Here is the Amazon description:

Fifty years ago when Jacques Hadamard set out to explore how mathematicians invent new ideas, he considered the creative experiences of some of the greatest thinkers of his generation, such as George Polya, Claude Lévi-Strauss, and Albert Einstein. It appeared that inspiration could strike anytime, particularly after an individual had worked hard on a problem for days and then turned attention to another activity. In exploring this phenomenon, Hadamard produced one of the most famous and cogent cases for the existence of unconscious mental processes in mathematical invention and other forms of creativity. Written before the explosion of research in computers and cognitive science, his book, originally titled The Psychology of Invention in the Mathematical Field, remains an important tool for exploring the increasingly complex problem of mental life.

The roots of creativity for Hadamard lie not in consciousness, but in the long unconscious work of incubation, and in the unconscious aesthetic selection of ideas that thereby pass into consciousness. His discussion of this process comprises a wide range of topics, including the use of mental images or symbols, visualized or auditory words, "meaningless" words, logic, and intuition. Among the important documents collected is a letter from Albert Einstein analyzing his own mechanism of thought.


Hadamard emphasizes a four step process of invention, consisting of

Preparation: conscious effort attacking the problem, including analysis of various methods and approaches; the outcome often appears fruitless

Incubation: a period of subconscious effort while the conscious mind is occupied with other matters

Illumination: the solution forms in the conscious mind

Verification: the solution is verified through conscious effort

This description agrees with my own experience. It seems that certain activities like walking are good for Incubation, and the shower is a good time for Illumination. Over the past weeks I've been deliberately trying to follow this method in my own research, with modest success :^) I noticed that it is also useful for overcoming writer's block -- put aside the paragraph you are struggling with and walk around for five or ten minutes before resuming. The walk often allows a small conceptual reordering, enough to proceed.

Incidentally, I often found my father and other professors and scientists I grew up around to be absent minded. I was never that way, and therefore wondered if I was fundamentally not of the scientific type (a smart kid, but not nerdy enough!). However, what I've found is that a lifetime of working on abstract problems has now given me the ability to call forth a problem of my choice and work with it in my head, leading to the tendency to occasionally detach from my surroundings. So, I appear (e.g., to my wife and kids!) to be the scientist type, but as a consequence of developing a particular functional ability rather than due to a specific predilection or tendency.

This paper contains a nice discussion of Hadamard's essay, and describes a modern study using Hadamard's original questionnaire. I liked these comments on the role of talking about problems:

... it is also clear from the mathematicians' responses that, while working in the initiation phase, they have a much higher regard for transmission of mathematical knowledge through talking than through reading. ...

Jerry: I assimilate the work of others best through personal contact and being able to question them directly. [..] In this question and answer mode, I often get good ideas too. In this sense, the two modes are almost indistinguishable.

George: I get most of my real mathematical input live, from (good) lectures or one-on-one discussions. I think most mathematicians do. I look at papers only after I have had some overall idea of a problem and then I do not look at details.

9 comments:

gaddeswarup said...

I had this experience recently. In retirement now, I felt the urge to write a short book on algebraic topolgy showing the power of abstract thinking. I left out the section on abelian groups and went ahead up to two thirds of the material that I wanted to write. I did not have suitable algebra books at home but thought about the fundamental theorem on finitely generated abelian groups off and on. Then one day, a simple idea occured to me but even after three days I could not make it work. On the third night after I dozed off, I kept dreaming about a small point 'why don't you try this?'. I woke up at 2 am and wrote it up. Then after a day, it turned to be incomplete but with a bit more effort, it gave a complete argument.
Strangely, a friend has been writing a book on algebraic toplogy and he too left out this topic. The ensuing correspondence is posted in my blog:
Kalyan Mukherjea

mock turtle said...

often i have gone to bed having worked a problem much of the day

upon waking a solution occurs to me

Sam said...

One of the biggest hurdles to making mathematical advancements or scientific inventions are PhD programs. A person's IQ starts to decline in one's early 20's (note most huge intellectual breakthroughs happen relatively early in a man's life), so that's when the actual work should be done instead of prep for the work, which is what a PhD consists of.

- "Shawn"

Don said...

I'm not competent to talk about math. I never liked it and found it boring. I did like discussing the big issues that math can engender, but not actually working on problems. But, while at Cal as an undergraduate and graduate student, I sat in on some high level math and physics courses. I generally found myself, after a bit, and with help from others, able to ask pertinent questions and follow the course. That's why this point interested me:

"... it is also clear from the mathematicians' responses that, while working in the initiation phase, they have a much higher regard for transmission of mathematical knowledge through talking than through reading. ...

Jerry: I assimilate the work of others best through personal contact and being able to question them directly. [..] In this question and answer mode, I often get good ideas too. In this sense, the two modes are almost indistinguishable.

George: I get most of my real mathematical input live, from (good) lectures or one-on-one discussions. I think most mathematicians do. I look at papers only after I have had some overall idea of a problem and then I do not look at details."

The learning of math is often portrayed as reading books and papers and solving problems. If it were taught in a more conversational manner, as it often is at the higher levels, someone like myself might have found it more interesting. The lower level courses seemed tedious and mechanical.

"This description agrees with my own experience. It seems that certain activities like walking are good for Incubation, and the shower is a good time for Illumination. Over the past weeks I've been deliberately trying to follow this method in my own research, with modest success :^) I noticed that it is also useful for overcoming writer's block -- put aside the paragraph you are struggling with and walk around for five or ten minutes before resuming. The walk often allows a small conceptual reordering, enough to proceed."

Oddly, this sounds how I write my novels or think about philosophy. I comment on blogs using a more immediate take approach, which works better for me when commenting on others ideas.

Don the libertarian Democrat

Steve Sailer said...

I often have one good idea immediately upon waking. Usually, however, it's just a minor good idea, such as how to rephrase something I wrote before bed for more impact.

Still, I usually won't have as good an idea as that for several more hours in the morning (I'm a night person).

Dave Bacon said...

The "absent minded" professor. I think xkcd hit it for me http://xkcd.com/602/

anon said...

Yes. The absent minded professor is made not born.

Anna Haynes said...

On the effects of walking - I find yard work is equally or more efficacious, it keeps the pesky part of my brain occupied so the thinking can proceed.

I'm also wondering if this finding might be relevant -
During exercise, the human brain shifts into high gear on 'alternative energy'(lactate) - maybe a brain that's burning a little lactate runs differently enough to evade the stumbling blocks that impeded a brain on pure glucose.

LaurentMelchiorTellier said...

I try to walk most of the day, if I can, it just disposes me to focus on the task at hand. I'm walking right now as I type this, at ca. 1.5 km/hr. It's not an increase in mental ability for me, it's more of a shift in emotional predisposition.

I find that the effect is more pronounced with social intelligence. My best conversations with people are while walking, sparring, building, etc. Maybe it's something about sharing a (minor, logistical) goal.

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