Does Mathematics provide truth?
If it ain’t broke, don’t fix it. Unfortunately, Mathematics, as you know it, is broke. Sorry.
‘But, why should I care!?’ I hear you say
After all, even if I misunderstand how Mathematics reaches its ‘truths’, it is still ‘workable’, right? Come on, my iPhone turns on, planes stay in the sky… Yet, it was ‘workable’ for hunter gatherers to assume the world was flat. It was ‘workable’ to assume the dogmas which were held before the enlightenment. We need to have an honest examination of the foundations of Mathematics — just as failing to examine previous ‘workable’ truths would have stymied progress, likewise it is harmful to never question Mathematics.
What do you want from me?
I want to persuade you that Mathematics, although it is one of the best sources of human knowledge and deduction, has less secure foundations than you think.
Why is this?
Mathematics, you see, is more like a Science than you’d expect. To show something is true you need to have a set of axioms and an agreed burden of proof. However, in Science, you have ‘workable’ theories, not end truths. Mathematics claims that 2+2=4, and that this is more than a theory.
To explain what axioms and burden of proof are, here is an example. I ask you if the bird is blue. We assume lots of shared knowledge about birds, eyesight, the world: these are the axioms for our discussion. If, for example, I deny the existence of birds, this discussion won’t get very far! Likewise, if you deny the axioms of mathematics, proving mathematical statements will be impossible. There is also the burden of proof. In this case, if we both look at the bird and see that it is blue, the matter is settled, i.e. that seeing it is blue is enough evidence to convince us the bird is blue. Clearly, we could have extra doubts — perhaps our eyes are faulty, or the bird is a weird sort of drone. Yet, in normal life we ignore more outlandish possibilities.
I am going to use a famous example in maths to highlight some of the issues at play here. I am going to show why traditionally Mathematics is viewed as certain knowledge, but at the end show why it is not.
As simple as ABC
For other mathematicians to check your work they need examples to get used to the notation and reasoning, rather than dealing with a whole raft of new ideas at once. The burden of proof in Mathematics is just other Mathematicians carefully checking your working, so examples are crucial. The first empirical aspect creeps in here.
How can you be sure the other Mathematicians spot every potential mistake? In practice, this results in them ‘testing’ out the theory with examples of things they are already know. So, it’s already looking a bit like a Science, with new theories being built off and tested on the results of older results.
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 of new concepts and notation developed or provide examples. Because his previous work has 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.
Yeah. But it’s still not a Science
Humans really struggle with abstract concepts. Hence the need for examples and a semblance of empiricism. However, Mathematics seems 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 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.
The Mathematician it seems, only uses examples to aid with abstract deductive statements, but is fundamentally not being empirical.
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
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.
In the 19th century this happened. Mathematicians found that their current axioms and burden of proof was leading to contradictions in places. So they upped their rigour both in the foundations of maths and the burden of proof demanded.
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. However, the choosing of the axioms and what burden of proof is needed is more Scientific than you’d think. Mathematicians have in the past adapted these to prevent contradictions, just as Scientists adapt theories to new empirical data.
And if Mathematical knowledge is less secure than we thought, the whole Universe remains more of a mystery also.
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(望月 新一 Shinichi Mochizuki Mochizuki Shin'ichi, born March 29, 1969) is a Japanese mathematician working in number…en.wikipedia.org