And the Math Will Set You Free

Abstract

h o l o t r o n i x @ g m a i l . c o m IntroductionHow mathematical is the physical world? Is mathematics essential to describe the universe, or it is just a tool which is not even necessary? There is no unanimously agreed answer to this question. The spectrum of opinions ranges from "mathematics is just a tool helping us to classify our observations" to "the world is nothing but a mathematical structure". Like for any debate, part of the tension between different views is the implicit usage of different definitions. So, to avoid possible confusion, we should define both mathematics and physics.Is mathematics discovered, or invented? Are mathematical structures entities which have their own existence, eternal and unchanging? Then, how can we know them? If we can access them with our thoughts, shouldn't they then be connected to our minds somehow? Can we discover and explore them? Or it is us who invent them? Or maybe we invent the axioms, and then we discover the consequences? Let's try to find out. A universe in a dotThink at a number. Did that number exist before you picked it, or it is you who invented it? Don't rush with the answer, because it is not as easy as it might seem.Consider the most brilliant text ever written. Is it the Bible, or the complete works of Shakespeare, or perhaps the complete collection of arXiv articles? Any of these can be written in a computer. The computer uses a code to assign numbers to letters, for example A → 65, B → 66, ..., Z → 90, a → 97, ..., z → 122. Internally, it represents these numbers in binary form, as strings of 0 and 1, so that for example A becomes 01000001, B becomes 01000010 etc. Let's take any text and assign to it a number between 0 and 1, of the form 0.d 1 d 2 . . ., where d 1 d 2 . . . is the decimal representation of the binary representation of the text.So, for example, the King James Bible starts with 1. In the beginning God created the heaven and the earth. Therefore, its number is 0.192110078742304518747878005390999191523668554050387237..., which obviously is between 0 and 1 (see fig. 1).