Yes, I was thinking of you when I said above that the idea has been put forward before :)

What I'm suggesting is that the process of *discovering* a deductive proof often (though not always, as pointed out by Timothy Gowers) begins with the mathematician selecting a conjecture, and then making his way backwards trying to figure out what would be necessary to prove it.

Yes, this rings very true, and reminds me of how I much preferred working with sequent calculus rather than with natural deduction when first learning proof-theory -- precisely because you can literally make your way up!

Well, I meant in fact to *contrast* deduction and calculation, which I take to be different things (and this is because I work with a fairly narrow understanding of the concept of deduction). As for the other points you make: that's very much aligned with the points made by Timothy Gowers commenting on my post -- see my update at the end of the post.

How cool :)

Totally, I didn't take this comment to be a criticism of analytic philosophy as such. The 'Let's not go down that road' remark simply means 'Let's approach this with the methodology of mathematics', which is probably a good idea if one is, you know, a mathematician... So I was perhaps a bit 'mean' with my 'why not?' comment at the end. As such, I think this article makes a lot of sense, and the reference to analytic philosophy is made in a rather positive tone.

You are exactly right, Jeff, W's Remarks are an important inspiration for me. Another Wittgensteinian way of cashing out what is going on is that Mochizuki has invented a new language that, for now, only he understands, but there cannot be such thing as a 'private language'.

Probably not as much as they should...