From a short dialog today in my combox as an aside to our discussion of Natures lack of determinism and any consequences on discussions of free will.

So you think the universe is not continuous because irrational numbers are not real? Do you think that differentiability is a useful concept but doesn’t really apply to reality? Why then Wigner’s “unreasonable success of mathematics” if there is no underlying reality to those mathematical concepts (like pi).I wasn’t clear. Pi does not exist in the real world. It’s not that we can’t measure pi exactly, but that it’s literally impossible for it to exist, exactly. How could you have a circle in the real world whose radius or circumference is an irrational number? You couldn’t. So pi, and math generally, is just an elegant approximation of reality.

This is worth a little elaboration. Continuity, mathematically speaking is all “about” that dense uncountable set of irrational numbers. Differentiability likewise requires not just continuity but that the manifold in question be “smooth.” Pi as was noted in a following reply is not limited to the ratio of circumference and diameter but crops of in a myriad of places. My interlocutor JA offers that just like that ratio for pi, all these others are “idealizations” and don’t reflect any reality.

When we make mathematical models of the Universe in Physics the common way of approaching these models is to assume that our measurements are inexact and that many of these models are closer to what is “really” being measured than our inexact measurements. When pi appears in descriptions of electron orbits we think that this value pi is “real” and the measurements of electron energy levels which depend on fundamental constants like pi and Planck’s constant and the electron mass are approximate. Someday we expect that we will arrive a a theory in which Planck’s constant and the electron mass like pi fall out as consequences of a mathematical understanding so that just like circumference/diameter all these numbers will be arrived at via fundamental relationships.

Or take the continuity/differentiability matter, which by the by depends as noted above on irrational numbers as well. Early astronomers like Galileo and Kepler took very imprecise measurements to deduce some relationships to describe motion. Newton and a host of later mathematicians went to work with this erecting an elaborate and very beautiful framework which today are known the Hamiltonian and Lagrangian descriptions of classical mechanics. These equations then can be pressed into service many many orders of magnitude past their original measurements without requiring modification and allow for example cis-lunar docking of spacecraft. These descriptions as well drive our methods and intuitions in the quantum (very short distance or high energy) regions and the relativistic ones as well. One suggestions as to why the mathematics of continuous differentiable manifolds is so important and successful at describing nature is that this description of nature (as continuous and differentiable) is accurate, that is it reflects reality.

Current Physics understands a number of fundamental particles to be “point-like”, that is to say that their best description physically speaking is as a “point.” A point in space is commonly thought to be an idealized mathematical concept. There is no “such thing” as a real “point.” Small dots or specks of dust are used to illustrate for the imagination what something approaching a point might be as a learning aid. However quarks and electrons, for example (and setting aside String theory for now) are described in the theory which we use today that best describes nature, the Standard Model, are point-like objects. Our best description of these (real) things is as a point (and it might be added that protons, neutrons, and baseballs are not point-like in our best descriptions). My eldest daughter recoiled when she heard my description of an electron as “point-like.” The principal problem for her was that electrons could not be point-like and massive. Yet mass is just a property. Like spin and charge, mass is just a numerical value assigned to that point-like object which affects how it interacts with other objects.

That being said, which is more real? The inexact measurement values or theoretical value which they approach? If the things you see with your eyes and other perceptive senses are seeing things which you believe to be real, then I offer that these concepts, pi, continuity, and point-like electrons represent our best understanding of what that reality “really” is. They are as real as the chair you sit upon for they are fundamental pieces of our understanding of how that chair is best described. If the chair is real then there are only two possibilities. Either our current (Standard Model) as our best description of that said chair reflects reality (in which case pi, irrational numbers and so on are also real) or there exists a future theoretical model (consistent with our current measurements) will replace it. If that future theory also has properties like continuity and constants (some irrational like pi) arise naturally in that future (correct) theory then … aren’t irrational numbers therefore real? How could it not be so?

Filed under: Mark O.Science

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