Turing had a strong predeliction for working things out from first
principles, usually in the first instance without consulting any previous
work on the subject, and no doubt it was this habit which gave his work
that characteristically original flavor. I was reminded of a remark which
Beethoven is reputed to have made when he was asked if he had heard
a certain work of Mozart which was attracting much attention. He
replied that he had not, and added “neither shall I do so, lest I forfeit
some of my own originality.”
Turing carried this to extreme lengths and I must confess that at first
I found it rather irritating. He would set me a piece of work and when
I had completed it he would not deign to look at my solution but would
embark on the problem himself; only after having a preliminary trial
on his own was he prepared to read my work. I soon came to see the
advantage of his approach. In the first place he was really not as quick
at grasping other people’s ideas as he was at formulating his own, but
what is more important, he would frequently come up with some
original approach which had escaped me and might well have eluded
him, had he read my account immediately.
The speaker goes on to note that:
His knowledge ranged widely over the whole field of pure and applied mathematics and seemed, as it were, not merely something he had learned from books, but to form an integral part of the man himself. One could scarcely imagine that he would ever “forget” any of it.
I wonder how much Turing’s method of working things out from first principles contributed to the widespread level 3 understanding that the speaker suggests Turing possessed.
Henry Poincare seemed to work this way as well (source):
An interesting aspect of Poincaré′s work is that he tended to develop his results from first principles. For many mathematicians there is a building process with more and more being built on top of the previous work. This was not the way that Poincaré worked and not only his research, but also his lectures and books, were all developed carefully from basics. Perhaps most remarkable of all is the description by Toulouse in [30] of how Poincaré went about writing a paper. Poincaré:-
… does not make an overall plan when he writes a paper. He will normally start without knowing where it will end. … Starting is usually easy. Then the work seems to lead him on without him making a willful effort. At that stage it is difficult to distract him. When he searches, he often writes a formula automatically to awaken some association of ideas. If beginning is painful, Poincaré does not persist but abandons the work.
Poincare also expected the unconscious to keep working on the problem after “he” had stopped:
Toulouse then goes on to describe how Poincaré expected the crucial ideas to come to him when he stopped concentrating on the problem:-
Poincaré proceeds by sudden blows, taking up and abandoning a subject. During intervals he assumes … that his unconscious continues the work of reflection.
To understand something means to assimilate it into an appropriate schema. [...]
Since new experience which fits into an existing schema is so much better remembered, a schema has a highly selective effect on our experience. That which does not fit into it is largely not learnt at all, and what is learnt temporarily is soon forgotten.
This hints that schemata that one forms from reading another’s approach to a problem may inhibit original thought, since assimilation to existing schemata probably occurs mostly automatically (when possible).
In the first place he was really not as quick at grasping other people’s ideas as he was at formulating his own
For some reason this line makes me feel better about the fact that I’m the other way around. I think it’s that the wording implies that these are just two skills, and one can be better at one or the other; it doesn’t make a value judgment.
I’m not sure how useful this advice is for us mere mortals, but this transcript of the 1970 Turing Award lecture has some insight into how Alan Turing worked:
The speaker goes on to note that:
I wonder how much Turing’s method of working things out from first principles contributed to the widespread level 3 understanding that the speaker suggests Turing possessed.
Henry Poincare seemed to work this way as well (source):
Poincare also expected the unconscious to keep working on the problem after “he” had stopped:
This practice makes some sense from a schema perspective of understanding. From The Psychology of Learning Mathematics:
This hints that schemata that one forms from reading another’s approach to a problem may inhibit original thought, since assimilation to existing schemata probably occurs mostly automatically (when possible).
For some reason this line makes me feel better about the fact that I’m the other way around. I think it’s that the wording implies that these are just two skills, and one can be better at one or the other; it doesn’t make a value judgment.