Zalamea and the Philosophy of Mathematics.

My holiday reading was F. Zalamea’s Synthetic philosophy of contemporary Mathematics, a recent (2012) translation of Zalamea’s 2009 “Filosofía sintética de las matemáticas contemporáneas”, translated by Zachary L. Fraser. I have to admire the translation first: my only other language is German, and I cannot imagine understanding the subtleties of this philosophical  book in anything except my native tongue. It’s readable, though it takes commitment, and some background in Mathematics (I have a degree in it, dating from 1973, but though I am an academic in Computing I really haven’t  studied Mathematics since then). I note that Tzuchien Tho describes the book as “dense bomb of a book” in his Almagestum Contemporarium.

I wish I had read this book earlier. Indeed, I wish it had been translated earlier. Why?

I’ve spent some time trying to understand Category Theory in the last few years, particularly as part of the INBIOSA project, which produced a book. The largest single element in that book  is the INBIOSA white paper, entitled  Stepping Beyond the Newtonian Paradigm in Biology: Towards an Integrable Model of Life: Accelerating Discovery in the Biological Foundations of Science. In this paper we (there’s 17 authors) discuss new ideas that attempt to move understanding of the foundations of biology towards something that might help to bring some mathematical  approach to the functioning of biological systems, towards something that might help explain living material in terms that aren’t just the biochemical equations, diffusion etc. As part of that we were looking for an approach that transcended the logical mathematics, used in what we were aware of the mathematical philosophy. One of our number, Ehresmann, was pointing us towards Category Theory, and certainly I , and presumably others too tried to understand what it was that Category Theory was really bringing to the area.

Now I’ve read Zalamea’s book I have a much better idea, not of the basics of category theory, but of why it was so important. It is a way of expressing how Mathematics works, of how Mathematics can be about Mathematics. Zalamea lights a way towards a new philosophy of Mathematics that brings together the constructive imagination of what he calls eidal Mathematics with the Physically based quiddital Mathematics, and the idea of Mathematics of mathematics in  archeal Mathematics (the italicised terms are Zaladea’s). He sees the recent mathematics of Grothendieck and (many) others as a revolution as important as Einstein’s in Physics, and sees this as requiring a related revolution in Mathematical Philosophy (or perhaps he sees this revolution as actually starting first, as he sees it based in the works of Lautmann who died in 1944, when Grothendieck was only 16).

Be that as it may, I think (and here I am but seeing through a glass darkly) that this different view of Mathematics can underlie a different view of biology. This richer philosophy seems to me to suggest that Mathematics can do more than describe the physical Universe: it can be the engine of that Universe, explaining how it operates. This is nothing new in Physics, but it is something new in biology. Can such a philosophy underlie a change in biology as critical as that of Einstein in Physics? Can it take the reductionist understanding supplied by systems biology, and show how this actually drives the biology? Can it go further, can we use the mathematics of Mathematics to understand how a Universe can become aware of itself? Can such a construction really help us to understand our construction of reality?

I’m back from holiday now. I’m writing this before all the other work that running a University’ Department (well, Division) takes over from trying to think about what really matters. In reality, I’d like to spend a month re-reading Zalamea, and following up more of the references. Then talking to the other authors of the the INBIOSA white paper, and trying to integrate these ideas into it (one month seems rather conservative here). But rather than simply writing it in my notebook, I’m putting it on my blog, so I can try to discuss it openly.

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