The last Riemann post raised again the issue of defining numbers in terms of diagrams. In logos theory one cannot simply pull infinite sets, or real numbers, out of a hat. The rabbit prefers to live in a burrow, where any number represents an enormous variety of objects.

The Euler characteristic of a category was one natural way of assigning rational numbers to finite type diagrams. What about irrational numbers? Legend has it that Pythagoras treated Hippasus rather badly (perhaps by killing him) after Hippasus demonstrated that the square root of 2 was irrational. A proof begins by assuming that $\sqrt{2}$ is rational and hence expressible as an indecomposable ratio of two integers $a$ and $b$, and then derives a contradiction. This relies on the concept of primeness for the ordinals. From the classical assumption that, if P is false, (not P) must be true it is deduced that $\sqrt{2}$ is irrational, the only alternative to being rational. Ah, hang on a minute. In logos theory we don’t want to assume that complement is an involution, so the proof doesn’t quite work, but there clearly needs to be more than one kind of number. The geometric definition of $\sqrt{2}$ uses two squares of side length 1, which are both cut in half and glued together as shown. This assumes that the correct units for area are $L^{2} = L.L$ where $L$ is a unit of length, but of course $L$ could be any unit. It seems there are an awful lot of assumptions being made to define irrational numbers. In three dimensions in M Theory the complement is a triality. Thus we should probably assign a different numerical status to $\sqrt{2}$ and $\sqrt{3}$. And only when we get to weak $\omega$ categories do we get all the reals.

If we tried to measure a real edge of a square of unit area we could never show it was exactly $\sqrt{2}$ because that would require subdividing the interval further and further until the atomic, or subatomic, structure of the environment altered our sense of the edge, or altered the edge itself so that there was no longer anything to measure.

On the other hand, zeta values show up as soon as one starts thinking about knots and QFT, although the zeta values are treated as abstract basis elements for an algebra over the rational numbers. It is convenient to assign real number values to them in order to evaluate physical quantities, but this comes from the rules of ordinary complex analysis, which we would like to move beyond.

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## Matti Pitkanen said,

July 29, 2007 @ 2:41 am

Thanks for a nice example about logs theoretic reasoning! This kind of example teaches much more than a list of formal axioms.

By the way, I read for some time ago the claim that Hippasus was not even temporary of Pythagoras. I am disappointed: it was so nice story;-).

There are indeed two manners to see numbers. The physicist’s way is rather primitive but practical: number has just real magnitude and representation involves topology via limiting procedure.

Number theorist sees their anatomy and in this framework sqrt(2) is whose square produces 2: could one define sqrt(2) operationally using realizing squaring operation and its inverse physically without bringing in topology and limiting procedure? What comes in mind that p-adic length scale hypothesis gives fundamental length scales as proportional to sqrt(p):s.

The values of Zeta (zeta(n) are used as basis in advanced calculations of Feynman diagrams: this is a really elegant manner to do precise numerics by bringing the magnitudes in only at the last step.

And could one efine roots of unity “operationally” via corresponding symmetries, which would be exact symmetries of dark matter at corresponding level of hierarchy (that is those of field body consisting of magnetic flux tubes related by Z_n rotational symmetry ). Nature would have done what is impossible to us! Exact symmetry would play a key role also in the physical representation of numbers.