The genesis of a significant scientific idea is a miracle- the gradual increase in clarity from the fog to the realisation that the truth was as obvious as day. This is a wonderful anti-climax, and anyone who has experienced it will be find Holmes' illustration in "The Dancing Men" uncanny:
Holmes had been seated for some hours in silence with his long, thin back curved over a chemical vessel in which he was brewing a particularly malodorous product. His head was sunk upon his breast, and he looked from my point of view like a strange, lank bird, with dull gray plumage and a black top-knot.
"So, Watson," said he, suddenly, "you do not propose to invest in South African securities?"
I gave a start of astonishment. Accustomed as I was to Holmes's curious faculties, this sudden intrusion into my most intimate thoughts was utterly inexplicable.
"How on earth do you know that?" I asked.
He wheeled round upon his stool, with a steaming test-tube in his hand, and a gleam of amusement in his deep-set eyes.
"Now, Watson, confess yourself utterly taken aback," said he.
"I am."
"I ought to make you sign a paper to that effect."
"Why?"
"Because in five minutes you will say that it is all so absurdly simple."
"I am sure that I shall say nothing of the kind."
"You see, my dear Watson" -- he propped his test-tube in the rack, and began to lecture with the air of a professor addressing his class --"it is not really difficult to construct a series of inferences, each dependent upon its predecessor and each simple in itself. If, after doing so, one simply knocks out all the central inferences and presents one's audience with the starting-point and the conclusion, one may produce a startling, though possibly a meretricious, effect. Now, it was not really difficult, by an inspection of the groove between your left forefinger and thumb, to feel sure that you did NOT propose to invest your small capital in the gold fields."
"I see no connection."
"Very likely not; but I can quickly show you a close connection. Here are the missing links of the very simple chain: 1. You had chalk between your left finger and thumb when you returned from the club last night. 2. You put chalk there when you play billiards, to steady the cue. 3. You never play billiards except with Thurston. 4. You told me, four weeks ago, that Thurston had an option on some South African property which would expire in a month, and which he desired you to share with him. 5. Your check book is locked in my drawer, and you have not asked for the key. 6. You do not propose to invest your money in this manner."
"How absurdly simple!" I cried.
"Quite so!" said he, a little nettled.
Sometimes, however this impression is lost upon the thinker, when the investigation lacks the above brevity.
So, in a slightly neglected tradition among scientists, here is a sampling of the trace of one idea. The actual idea itself, however, will appear in another post.
First, a very rough note in my notebook in 2014 (there are many more, but I can't quite locate them at the moment.) I had begun to ponder the implications of Cantor's notions of "compression":
Later in 2014, a mistake would lead to a spectacular realisation. The mistake is pictured here-
(Slightly better.)
Then, mid 2015. I tried to clarify a lot. I typed out a draft. This is the main part at the head of the draft. There are no mistakes here, and it still means a lot-
Jump to March 2017, and a paper is presented at a conference and published.
Here's the link to the full thing: http://www.grdjournals.com/article?paper_id=GRDCF003017
Introspection will show that it is not much better, in form, than the previous stages. The last theorem is, very simply, wrong, and should be given no credence whatsoever. Time has also proven that it was only a spring-board for a much, much nicer thing. The idea, as of this writing- May 2018- is very, very interesting. It has weaved such a large part of the recreational maths I have occupied myself with, for so long, so beautifully together. In a later post.
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