Assumps and rigor, ‘sumps and rigor
Go together with utmost vigor.
This I tell you brother
You can’t have one without the other.
Assumps and rigor, ‘sumps and rigor
It’s an institute you can’t figger.
Ask the local gentry
And they will say it’s elementary.
Try, try, try to separate them
It’s an illusion.
Try, try, try and you will only come
To this conclusion.
Assumps and rigor, ‘sumps and rigor
Go together with utmost vigor.
We were told so leery
You can’t have one, you can’t have none,
You can’t have one without the theory.
Assumptions, ‘sumps and rigor
Try, try, try to separate them
It’s an illusion.
Try, try, try and you will only come
To this conclusion . . .
With apologies to Sammy Cahn and Jimmy Van Heusen (and more recently inspired by this post).
Paul Halmos, mathematician (1968):
“Mathematics — this may surprise or shock some — is never deductive in its creation. The mathematician at work makes vague guesses, visualizes broad generalizations, and jumps to unwarranted conclusions. He arranges and rearranges his ideas, and becomes convinced of their truth long before he can write down a logical proof… the deductive stage, writing the results down, and writing its rigorous proof are relatively trivial once the real insight arrives; it is more the draftsman’s work not the architect’s.”
(Obtained from John F Sowa’s slides.)
Then, when acting as draftsman to write the rigorous proof for publication, all the ways he (mostly he in those days) explored to reach the real insight vanish.
Yeah, this is true. Every mathematician should be required to write 2 papers for every proof. One just states the assumptions and the result with a formal proof, the second one should be at least a couple chapters on how the whole thing came about.
“the second one should be at least a couple chapters on how the whole thing came about”
That might be interesting, but would be a tremendous task — would, in many cases, take a book, not just a paper. And would require documentation along the way to the process of finding (creating, figuring out) the proof.
No doubt. It was supposed to be partially tongue in cheek, but I imagine it would be extremely interesting for many important topics if such a thing could be written… but it would also take away from doing more math… so it’s a tradeoff.
This is the reason why remote education does not work. Yes, you can learn things published in the papers, but to _really_ learn the subject you have to attend the same parties as experts do.
Andrew,
You are becoming increasingly arcane. Have you ever considered a sabbatical at Swarthmore College, leading the venerable department of statistics and mathematics in the place of dear heart Gudmund Iversen? Please do not dismiss my entreaties out of hand. Brian Wansink and I breathlessly spoke of you as we polka danced in Phoenix in the summer of 2018.
Best wishes to Martha (Smith), and Peter Shor, Aaron Swartz, and all the others who grace your blog comments.
Best wishes,
Ellie
Assumptions and deductive reasonable are inextrictably intertwined. You can’t have a theorem without “assume a, b and c.” I agree with this part. What i can’t figger is what’s hard to figger about this meta-formula “Theorem=assumptions+deductive reasoning” that i reckon statistics wizard gelman understood when he was harry potter’s age.
Try, to separate them
It’s an illusion.
Try, and you will only come
To this conclusion.
Love and marriage – can’t have one without the other – false. loveless marriages exist. unmarried people love each other. this is propaganda for the quid pro quo of marriage, 80 long years ago – i will share my paycheck if yuo will share my bed.
the conclusion is, that unlike love and marriage which most definitely not entail each other in any way, with “axioms/assumptions” and “deductive reasoning” the marriage is much greater than the sum of its parts.
theorem=axioms+deduction