wofsy said:
The idea of a duality between zero and infinity is meaningless to me. There are many infinities that are non-commensurate, in fact there is no largest one, so which one is dual to zero - whatever dual means? For instance,is the infinity of the continuum dual to zero - or how about the set of all sbsets of the continuum?
This is the thought I have: fundamentals always come in complementary pairs. This is a fact of logic. So uniqueness in a monadic sense is impossible. Candidate fundamentals like zero must have a matching concept that is asymmetric, so that together they are a broken symmetry resulting from the breaking of a deeper symmetry.
Now one answer might be that the duality is of the form of the binary code - counting reduced to its actual simplest form. The 1 and the 0. A something and a nothing. The many, all other number and pattern, can then be constructed from these atomistic components.
So forget numberlines, this is a dimensionless approach. All you need is pure absence/pure presence, and dimensionality itself can be constructed in the information theoretic view.
But binary code is a symmetric kind of "symmetry-breaking" and so unnatural. We really should prefer a stable asymmety.
A second way of finding the duality here would be to contrast the continuity of the numberline with the discreteness of its points. This is a classic dichotomy - discrete~continuous, familiar from Zeno's paradoxes.
So this would mean we could treat any chosen number as being dual to the line on which it lives. The relationship is a broken symmetry, asymmetric. But I don't see that zero, as just another point in this system, would be uniquely dual in that sense. Rather all points would be dual to the whole of the line.
What I was arguing before was that zero might be properly dual in an infinite dimensional number realm. On a 1D line, points just sort of slide around as nothing intersects at a point to single it out as unique. But add further dimensions and yes, an origin is being fixed at a place. The zero becomes the only point dual to the whole of the space arguably.