This property was generalised by Simeon Ball and Oriol Serra in their paper titled “Punctured Combinatorial Nullstellensatz” where they show that a polynomial which vanishes everywhere on a grid A = A1 x … x An except at some point of a subgrid B = B1 x …. x Bn, then its degree is at least \sum (|Ai| – |Bi|). From this I derived a new generalisation of the Chevalley-Warning theorem which doesn’t seem to follow from the Alon-Furedi bound (hence I didn’t include it in my talk). Nevertheless, it seems to be quite interesting on its own.

]]>Differential and Integral: A constructive introduction to classical analysis, The University of Texas Press, Austin, 1971.

The only constructivism I know, where you actual keep the intermediate value theorem in its usual formulation (and do not add epsilons and deltas into the statement itself).

]]>Thanks for the reference. ]]>

Paul Lorenzen had strange constructivism. He started by writing down words and concatenating these words. So he gets arithmetic in the same way as you are suggesting. Then he goes on and on and reconstructs with these basic operations large parts of mathematics (including ordinal numbers such as $$\omega^{\omega^{\omega}}$$ or all “useful” analysis). The goal of his work was to justify the usual mathematics with these simple operations on words. Are Lorenzen’s ideas similar to the things you/Jack want? Or am I missing some crucial difference? Unfortunately, I do not know how much of his work is available in English.

]]>The construction of a matroid from a polymatroid set function is in the paragraph beginning “In the other direction”.

As to Jack’s comments on matroid theory books, he was very emphatic about that and I felt I couldn’t just pass over it in silence. I hope my ambivalence comes across. But I agree that someone has to do it. Indeed, although Jack’s examples tend to come from vector matroids, he said, more strongly than I would have done, that matroid terminology all derives from graphic matroids.

I don’t think I made as good a job of describing the stuff about matroid subdivisions as I might have done. Both Jack and Alex were very excited about this, and both thanked me for my role as matchmaker at least, if not midwife. But Alex can certainly explain this better than I can. His notes will be (if they are not already) among the course material and are well worth looking at. He might even be tempted to write something for posting here …

]]>Secondly, I think Jack is being unduly harsh on books that “laboriously prove the equivalence” of all the different ways of defining matroids. It may not be his (Jack’s) style, but I still think it is amazing that no matter which aspect of a vector space you try to axiomatise combinatorially (I.e no fields allowed) you end up with exactly the same class of structures. Given this, the author of a definitive introduction to matroids (i.e James) has no choice but to prove these things, laboriously or otherwise. In my opinion, it’s a bit like criticising the author of a Linear Algebra text for showing that “0 is unique” follows from the axioms – it’s not much fun, but somebody has to do it!

]]>What I believe now is stated cleary by Julian Jaynes: you can find this on my page of quotes. We use formal logic in reasoning only as a last resort. Most of the time we do mathematics without having the logical rules uppermost in our minds: we use them unconsciously as a kind of mental hygiene. But we could go back and prove everything formally if challenged (or at least, so we believe).

After my first experiences teaching logic, I used to enjoy provoking the maths and philosophy students by telling them that a proof was anything you could get past a referee. (An important part of a philosopher’s training is to learn to destroy other people’s arguments; you have to have your wits about you to destroy this one.) ]]>