Last week, I posted video of my talk to the undergraduate math students on truth, provability and the fabric of the universe — and heard from several readers who requested that I post it in a non-flash format.

My readers’ wish is my command. Here is the file for download in an m4v format.

This is relatively low resolution. The best viewing is still the flash version here.

Over at Less Wrong, the estimable Eliezer Yudkowsky attempts to account for the meaning of statements in arithmetic and the ontological status of numbers. I started to post a comment, but it got long enough that I’ve turned my comment into a blog post. I’ve tried to summarize my understanding of Yudkowsky’s position along the way, but of course it’s possible I’ve gotten something wrong.

It’s worth noting that every single point below is something I’ve blogged about before. At the moment I’m too lazy to insert links to all those earlier blog posts, but I might come back and put the links in later. In any event, I think this post stands alone. Because it got long, I’ve inserted section numbers for the convenience of commenters who might want to refer to particular passages.

1. Yudkowsky poses, in essence, the following question:

Yudkowsky phrases the question a little differently. What he actually asks is:

This, I think, threatens to confuse the issue. It’s important to distinguish between the numeral “2”, which is a formal symbol designed to be manipulated according to formal rules, and the noun “two”, which appears to name something, namely a particular number. Because Yudkowsky is asking about meaning and truth, I presume it is the noun, and not the symbol, that he intends to mention. So I’ll stick with my version, and translate his remarks accordingly.

2. Yudkowsky provisionally offers the following answer:

Continue reading ‘Accounting for Numbers’

The (really really) big news in the math world today is that Shin Mochizuki has (plausibly) claimed to have solved the ABC problem, which in turn suffices to settle many of the most vexing outstanding problems in arithmetic. Mochizuki’s work rests on so many radically new ideas that it will take the experts a long time to digest. I, who am not an expert, will surely die with only a vague sense of the argument. But based on my extremely limited (and possibly mistaken) understanding, it appears that Mochizuki’s breakthrough depends at least partly on his willingness to abandon the usual axioms for the foundations of mathematics and replace them with new axioms. (See, for example, the first page of these notes from one of Mochizuki’s lectures. You can find other related notes here.)

That’s interesting for a lot of reasons, but the one that’s most topical for The Big Questions is this: No mathematician would consider rejecting Mochizuki’s proof just because it relies on new axiomatic foundations. That’s because mathematicians (or at least the sort of mathematicians who study arithmetic) don’t particularly care about axioms; they care about truth.

There’s a widespread misconception that arithmetic is about “what can be derived from the axioms”, which is a lot like saying that astronomy is about “what can be discovered through telescopes”. Axiomatic systems, like telescopes, are investigative tools, which we are free to jettison when better tools come along. The blather of thoughtless imbeciles notwithstanding, what really matters is the fundamental object of study, whether it’s the system of natural numbers or the planet Jupiter.

Mathematicians care about what’s true, not about what’s provable; if a truth isn’t provable, we’re fine with changing the rules of the game to make it provable.

Continue reading ‘Simple as ABC’

In the comments section of Bob Murphy’s blog, I was asked (in effect) why I insist on the objective reality of the natural numbers (that is, the counting numbers 0,1,2,3…) but not of, say, the real numbers (that is, the numbers we use to represent lengths — and that are themselves represented by possibly infinite decimal expansions).

There seem to be two kinds of people in the world: Those with enough techncal backgroud that they already know the answer, and those with less technical background, who have no hope — at least without a lot of work — of grasping the answer. I’m going to attempt to bridge that gap here. That means I’m going to throw a certain amount of precision to the winds, in hopes of being comprehensible to a wider audience.

Continue reading ‘The Number Devil’

The Intelligent Design folk tell you that complexity requires a designer.

The Richard Dawkins crowd tell you that complexity must evolve from simplicity.

I claim they’re both wrong, because the natural numbers, together with the operations of arithmetic, are fantastically complex, but were neither created nor evolved.

I’ve made this argument multiple times, in The Big Questions, on this blog, and elsewhere. Today, I aim to explain a little more deeply why I say that the natural numbers are fantastically complex.

Continue reading ‘Non-Simple Arithmetic’