Number Sense

Sunday reflection:

 "[Alain] Connes thinks that expert mathematicians are endowed with a clairvoyance, a flair, a special instinct comparable to the musician's fine-tuned ear or to the wine taster's experienced palate that enables them to directly perceive mathematical objects: 'The evolution of our perception of mathematical reality causes a new sense to develop, which gives us access to a reality that is neither visual, nor auditory, but something else altogether'."  

-- from "The Number Sense" by Stanislas Dehaene

Mathematical Discovery... Deconstructed

This morning's Sunday reflection from Stanislas Dehaene's "Consciousness and the Brain":

"[Jacques] Hadamard deconstructed the process of mathematical discovery into four successive stages: initiation, incubation, illumination, and verification. Initiation covers all the preparatory work, the deliberate conscious exploration of a problem. This frontal attack, unfortunately, often remains fruitless -- but all may not be lost, for it launches the unconscious mind on a quest. The incubation phase -- an invisible brewing period during which the mind remains vaguely preoccupied with the problem but shows no conscious sign of working hard on it -- can start. Incubation would remain undetected, were it not for its effects. Suddenly, after a good night's sleep or a relaxing walk, illumination occurs: the solution appears in all its glory and invades the mathematician's conscious mind. More often than not, it is correct. However, a slow and effortful process of conscious verification is nevertheless required to nail all the details down."

Strogatz, Dehaene in Conversation

Mr. Honner's math site led me, today, to a wonderful Princeton video of psychologist Stanislas Dehaene and mathematician Steven Strogatz in discussion of mathematics, cognition, and teaching (also touches on the relationship between subject-matter interests and neuroscientific disorders). The video is long (79 mins.) and I've only watched part of it thus far (including Strogatz's wonderful initial section starting around the 23.30 mark). Highly recommended if you have the time:

http://hulk03.princeton.edu:8080/WebMedia/flash/lectures/20110427_publect_dehaene_strogatz.shtml

A Demonstration

Try this exercise, I've copied directly from another book:

"Answer the following questions as fast as you can:

-2 + 2 = ?

-4 + 4 = ?

-8 + 8 = ?

-16 + 16 = ?


Now quick! Pick a number between 12 and 5. Got it?
.
.
.
.
.
The number you picked is 7, isn't it?"


...I succumbed to this piece of 'mindreading' when I read it in Stanislas Dehaene's 1997 "The Number Sense: How the Mind Creates Mathematics." Did you?
He calls this a "demonstration of the automaticity of arithmetic memory" and explains it thusly:

"How did I read your mind? The mere presentation of the numbers 12 and 5 seems enough to trigger an unconscious subtraction 12 - 5 = 7. This effect is probably amplified by the initial addition drill, the reversed order of the numbers 12 and 5, and the ambiguous phrase 'between 12 and 5' that may incite you to compute the distance between the two numbers. All these factors conspire to enhance the automatic activation of 12 - 5 up to a point where the result enters consciousness. And you believed that you were exercising your 'free will' when selecting a digit!"

I'm not sure I find Dehaene's explanation completely satisfactory... but, I can't argue with the effect, which I did fall for.