Setting out your stall
An example to all graduate students, Jeff Giansiracusa has on his website a research statement and a research proposal to search for a functorial relationship between Poisson manifolds and von Neumann algebras. I like this point from the latter, which allows for a certain hedging of bets if the desired functors don't appear:
Incomplete analogies have played, and will continue to play a prominent role in mathematics.This is reminiscent of the observation Weil makes in the first column of page 6 of this letter to his sister that when an analogy is fully worked out, the 'majestic beauty' of the resulting theory 'can no longer excite us'. I wonder what's happening with the analogy between 3-manifolds and algebraic number fields. In the addendum to TWF218, Kevin Buzzard suggests that genuine analogies are to be distinguished from two instances of the same thing. But then maybe it's just a question of time.
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I was a bit miffed by Kevin Buzzard's remark in week218 where he said
A word on analogies.If you want to say that the p-adic integers are analogous to the formal power series ring C[[z-a]] (call it C[[z]] for simplicity) then in fact some people would say that this was not an analogy---this was simply two instances of the same thing, namely a complete discrete valuation ring.
Mainly it was just my wounded pride: of course I knew that formal power series and p-adic integers are both examples of discrete valuation rings; I was trying to lure people into understanding the relationship between these entities which led people to invent the concept of discrete valuation ring.
But, more importantly, I don't think an analogy ceases to be an analogy once one has formalized it. It may lose its charm, as Weil points out:
The day dawns when the illusion vanishes; intuition turns to certitude; the twin theories reveal their common source before disappearing; as the Gita teaches us, knowledge and indifference are attained at the same moment. Metaphysics has become mathematics, ready to form the material for a treatise whose icy beauty no longer has the power to move us.
But, the first formalization of an analogy is not always the best or final one! And, one can rarely appreciate the formalization of an analogy without experiencing the analogy "in the raw" first. So, it's important to break through the icy crust of formalism and recall the mysterious analogy that led people to it in the first place.
This is related to Grothendieck's comments on "innocence", which I mentioned elsewhere.
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