Calculus without limits

# Do we need infinity?

In the new Netflix documentary “a trip to infinity” the question of quantum space comes up. It is interesting to see Brian Green (the famous TV star covering about quantum gravity, big bang, fabric of the cosmos elegant universe or string theory) now seems have been converted to the “finite discrete” side (advocated by Carlo Rovelli, an other host to “a trip to infinity”.

One should add that there is absolutely no experimental evidence for a discrete space time. The speculation that space or time are discrete are speculations like the guess of Democritus (based on empirical inability to crush sand arbitrary much). We have not been able to perform any experiment showing some kind of granular space time structure. It is likely by analogy from matter, were atoms emerged and by looking at the mathematics of quantum mechanics.

The question whether space or time have some discreteness might never be settled simply because even if experiments would indicate a Planck scale discreteness, it could happen that on a scale like $10^{-100}$ the continuum again would rule. Better than asking whether space or time is continuous or not is the question, whether it is continuous in a particular scale.

I myself am interested in the much more reasonable question “do we need infinity?” This is what this blog essentially is all about. If we look at a mathematical structure which needs infinity, we can ask ourselfs from a pure engineering or artistic point of view, whether it is not more practical (this is the engineering part) or more elegant (this is the artistic part) to forget about infinity. The netflix movie features quite a few mathematicians and none actually made the point for infinity. Only Steven Strogatz told that he feels comfortable with infinity. Actually, this is the feeling of almost everybody, from kids to students to professional mathematicians. We all feel that a circle is more natural than a polygon. If you look at a polygon, you have to specify a number, you have less symmetry. On the other hand, if you look at planar geometry in which circles are an important part, we can do all what Euclid did using finitely many symbols in a symbolic way. If we give an equation for a circle, then this is a finite algebraic expression. We do not need involve infinity. Every single theorem in Euclid can be verified in a finite formal world. The question whether the actual space is infinite never really comes up.