Is Mathematics Empirical?

No.

Well, it’s a little more complicated.

The ‘burden of proof’ in Mathematics certainly has an empirical aspect. However, the method is not.

Is it true?

Burden of proof is how much you need to do to show something is true. In practice, this often translates to:

How much do I need to do to persuade others this is true

As if these are experienced Mathematicians, you’d hope they might spot your mistakes – just as you’d be able to spot the mistakes in a young child’s addition.

When faced with new concepts and notation, the effect can be dazzling. Imagine someone has given you a line of French to translate and you are a beginner. What are the odds you get it correct if there are strange words and grammar you don’t understand? So the Mathematician needs examples. This not only familiarises her with the notation and how to manipulate it, but by proving smaller steps with examples gives her confidence in the method.

As simple as ABC

I really must stress how important this ’empirical aspect’ is. Recently a Japanese Mathematician claimed to have solved a very difficult problem called the ‘ABC conjecture’. However, the Mathematician failed to explain the whole raft on new concepts and notation developed or provide examples. Because his previous work was been so careful, his proof is taken seriously, but he has developed so many new ideas that it is near impossible to verify. To give you a taster, he has developed something called ‘Inter-universal Teichmuller Theory’. There have been several workshops on his ideas, which left everyone baffled.

But…

Humans really struggle with abstract concepts. Hence the need for examples and a semblance of empiricism. However, Mathematics is substantially different! It is a method of proof where you start out with a set of axioms and then find the implications. In Physics you make a ‘best guess’ about what is true and see whether it matches up to what happens. And then you have to substantially change your theory again and again. A physicist does not prove results for ‘n-dimensions’. That is because ‘n-dimensions’ (as opposed to a mere 3 or 4) is not that relevant to the world. And frankly, n dimensions is an absurdly abstract concept! There is something different about the two approaches. A Mathematician is much clearer about her axioms and then acts like a deduction machine. A physicist shows that a certain theory fits observations, and then he runs with it. The physicist sees how far he can run with it, then makes changes if it goes wrong.

Imagine a leaking boat

The Physicist patches up a leaking boat and is content to float until the world reveals the next leak. The Mathematician prides herself on a slick, painstakingly crafted vessel where she knows the position of every plank of wood and water molecule – there will be no leaks. The ‘Ideal’ Mathematician wouldn’t need concrete examples to understand abstract concepts. The Ideal Physicist needs empirical data and experiments!

When a Mathmatician’s boat leaks

Yet –

Both when it comes to the axioms in Mathematics and burden of proof, there’s cause for concern.

When choosing abstract axioms and an acceptable burden of proof, a Mathematician does seem to take a Physicist’s approach. She creates some axioms and a burden of proof and sees how it runs. If she runs into contradiction, either the burden of proof was too slack or the axioms are wrong. Thus, once the burden of proof and axioms are accepted in Mathematics, it is substantially different to Physics, which is much more lax about radically changing its models.

You might also be interested in…

https://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/

http://theconversation.com/a-purported-new-mathematics-proof-is-impenetrable-now-what-52491

http:// https://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/106658#106658

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