April 2, 2011

Mathematics, Logic, and Computation

This time I intend to explain how I think about "mathematics" and "logic".

I have no problem speaking of "mathematical truths", like "the truth that every natural number has a unique factorisation into prime numbers". I also have nothing against speaking of "mathematical objects", like "an n-dimensional space of constant curvature k". Furthermore, I am happy to speak of "ideas" of mathematical objects, like "the idea of a triangle". In fact, I think Plato deserves a lot of credit for introducing "the world of ideas/forms".

Yet, I also embrace some aspects of what might be considered the opposite of Platonism: Constructivism. While Plato thought of mathematical objects as "eternal", constructivists say that mathematical objects are "constructed", where that construction is something that happens in time.

So how can mathematical objects be timeless and yet be constructed in time? Think of fairy tales: Writing a fairy tale is something that takes place in time, but the fairy tale itself need have nothing to do with our time. In a similar way, one can construct a mathematical object that has nothing to do with our time.

But there are further twists to this. The fairy tale does have its own, "internal" time (inside the fairy tale, one thing happens first, then another happens, and so on), and in the same way a mathematical object may have its own, internal causal structure. To take a somewhat extreme example, the object could be a "space-time".

For another twist, imagine a fairy tale that includes a person who writes fairy tales and constructs mathematical objects. In a way, the "construction" of mathematical objects can happen "outside time".

Although I have now spoken of people and fairy tales, I may as well speak of computations. For example, just as one fairy tale can be written inside another fairy tale, so one computation may be realised within another computation. Mathematical objects may be timeless, but they certainly arise in computations, and you can make beautiful pieces of mathematics come out of computations (look at some fractals if you are unconvinced).

But what are those timeless "mathematical objects" and "mathematical truths"? Read my blog post Objects as Information to get part of the answer. I think of mathematical objects (and mathematical truths) as "pieces of information" that can be stored in computer memories and used in computations.

But what exactly do the pieces of information in question look like? I will return to this in future blog posts. Suffice it to say for now that if you catch a mathematician talking about a mathematical object you can ask the mathematician for "axioms" describing the mathematical object. Those axioms give you a good starting point when you want a piece of information that completely captures the mathematical object. As you interview the mathematician you may, however, find that the mathematician lacks a clear idea of what exactly the mathematical object is, and so you should also look at how it is used and what intentions the mathematician has when using it (I am reminded of my blog post Meaning and Goal-Directedness). Looking at how objects are used is also a key to identifying mathematical objects inside computations: Look at the functional role played by information at various points in the computation.

Leaving to future blog posts the question of what exactly mathematical objects look like, let us now instead think about mathematical truths. What on earth does it mean that one mathematical truth "follows from" another mathematical truth? To take a simple example, by what right can one say that "from x=y and yz it follows that xz"? Well, "x=y" and "yz" are pieces of information (see my post Objects as Information), and the question is whether these two pieces of information when combined somehow contain the information that "xz". Now, "=" and "≠" can each be thought of as being (more or less identical to) a piece of information, and these are contained in "x=y" and "yz" respectively (what this means in practise is that you can ask a person who tells you that "x=y" what he/she means by "=", and likewise for "yz"). From these pieces of information you will get the information that "from x=y and yz it follows that xz", at least unless someone is using the signs "=" and "≠" in a very non-standard way.

So the "follows from" of logic is really the "is contained in" of pieces of information. Obtaining new pieces of information from old ones is a hallmark not only of "logic" but also of "computation", and I would say "logic" is really about computation/causality.

In a simple computation things proceed one step at a time ─ having performed step n you proceed to use the result of that step to perform step n+1 ─ but in general things can be much more complicated, not just with computers but also in mathematics and in physics. It may make little sense to speak of the cause for something. Rather, truths may be "over-determined", "true for more than one reason".

Well, this was at least an introduction to how I think about "mathematics" and "logic". In future posts I will discuss "symmetry", "definability", "identity", "truth values", "relations", "parts", "ideas of objects", "variable objects", and other stuff which, while central to mathematics and logic, would have made this post way too long.