I arrived at Keele University already knowing something about formal mathematical reasoning, already with some kind of a bee in my bonnet about proof. I know this because when I started doing mathematics properly (in the second year, since Keele had a foundation year before one could specialise), I did some of my exercises with fairly detailed formal-ish proofs. Presumably this didn't last very long since there isn't much university mathematics one can hope to do that way, but my notes are now lost.
The first ideas I can recollect about the foundations of mathematics concerned Principia Mathematica. I acquired the paperback of the Principia ``to *56'' and devoted some time to it. It seemed to me a bit unclear and I thought it would be useful to define the logical system formally. This was my first grapple with a kind of regress problem, and I decided that one should define the logical system using the very simplest language in which it could possibly be done, so that there was no real danger of that language being misunderstood.
That language seemed to me to be the language of post productions, of which I had learnt from Marvin Minsky's book ``Finite and Infinite Machines''. For a while this was to be a dissertation for the philosophy department, but eventually I decided that it wasn't going to work out.
It was purely about syntax, and at this time I had no idea of semantics. It doesn't any longer seem to me problematic to define the syntax as precisely as we wish, and its clear that the way to do that is not to chose the simplest possible notation for the purpose.
By the time I got to the end of the degree I had some definite ideas for research relating to the foundations of mathematics. One was that I wanted to work on computerised formalisation of mathematics. The kind of thing which was actually being done round about then in Edinburgh (except that their logic LCF was a bit to weak, though it would eventually be traded up to HOL). The other was to work on support for use of computable reals. I don't believe that I had any original ideas on how these were to be done.
I did half a degree in philosophy, part of which was mathematical logic and the philosophy of mathematics. I was not a good philosophy student, I was more interested in working things out for myself than in studying the works of other philosophers, and quite unsuited to the latter. In the examinations one was expected I think even when asked a question about some particular problem without reference to any other philosopher, to show some knowledge of what other philosophers had said about the problem. I think the model answers were demonstrations of scholarly knowledge peppered with small amounts of original analysis. However, I didn't have the scholarly knowledge and I was pretty slow. I liked to think about the problem posed and come up with a position on the spot, with no more reference to other philosophers than could be avoided.
I recall an obscure conversation with Jonathan Dancy on a bus during the period after taking the exams but before getting the result. He said that for the philosophy I had been on the list of those considered for viva (meaning that I was on a borderline), but that they had decided against giving me one. He said he had argued that the style of the essays should be taken into account, evidently he thought this a strong point, but there just wasn't enough content so they wouldn't have it. I think Dancy thought he was telling me something from which I could conclude that I would be getting a first overall (to get a first in a joint honours you needed a first in one subject but only an upper second in the other) and he must have heard enough about my maths to think that I would have no doubt about a first in mathematics.
I digress, mainly to suggest that originality is in my blood. Not in the laudatory sense of having meritorious ideas which no-one else has had. In the sense of ignoring what has been done before, and thinking things out from scratch. Creating some solution of my own, not necessarily new to the world, not necessarily of outstanding merit. Originality as a bad habit, or a pathology, like autism.
As far as ideas from this time which still have something for me, I can remember the following:
Roger Bishop Jones 2016-01-07