mathematics

[ algebraic topology | differential topology | fractal geometry | mathematical logic | set-theory | combinatorics | graph-theory | knot-theory | category-theory | sheaf-theory | topos-theory ]


NOTGrothendieckRelating the theory of oppositions to general mathematics rather than to logic alone is one of the “trademarks” or specific flavours of OG (whereas, for instance, our friends of LG work out more specifically, and with great achievements, the logical aspect of oppositions). For sure, this does not help OG in seducing logicians, computer scientists or analytical philosophers (who at present day are those who could support and sponsor it, and they don’t yet do it), but we think that underestimating (or hiding) the mathematical aspects of oppositions would harm the global theory of oppositions in the long run, as it has always happened to do so far (cf. the section “ancestors” in this website, particularly those on Sesmat, Blanché and Sauriol). So we hold this point and don’t give it up: “amicus Plato, sed magis amica veritas“. The idea, for short, is that: (1) the realm of oppositions is very quickly subject to complexity growth (which is what mathematical logic, in its comforting prima facie simplicity, is generally brandished against) and (2) that in the realm of oppositions one can observe – descriptively, before than deductively – the emergence, as patterns, of several interesting (although prima facie mysterious) mathematical characterisations. This point is highly contentious with respect to logic (and analytical philosophy) in so far logicians tend to reduce mathematical practice to establishing deductions, whereas the history of mathematics clearly shows that real mathematical discovery also has a strong dimension of empirical description (of new realities explored or produced) which cannot be from the beginning simply deductive and axiomatically perfect. With that respect – that is: wishing to protect the mathematical side of the enterprise – in this section we try to organise some ideas on this. Bearing in mind that this page, as several others of this site, is a work in progress, we can say (modulo the necessary future updatings) that so far at least eleven branches of general mathematics seem to be popping their nose in between oppositions. These are:

As for many other parts of mathematics, it might be expected that much more will appear in the future if the relevant studies are carried on rigorously. We will do our best for that aim. Any remark (especially, but not only, if helping with technical points) is warmly welcome.

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