On 15/01/2024 01:54, Charles Haynes via Silklist wrote:
Ah, ok. How much of the theoretical foundations of math are you familiar with? The 9 axioms of ZFC are the things that underly math that no one can prove. Sort of by definition. Which is one way if getting around the "faith" argument in math. Those axioms are "definitional" if you like rather than "taken on faith" but whatever you call them they're things everyone who uses math accepts as true - but can't possibly prove.
I have a minor quibble with this framing!"no one can prove" kind of implies it's a mystery, perhaps something that can be observed but nobody has figured out how to explain it, like the behaviour of fundamental particles in physics or something.
As I see it, mathematical axioms like those of ZFC are engineering choices mathematicians have made to create a useful system. Those particular axioms have been chosen because they create a system that is useful for solving lots of problems. And the fact they "can't be proved" just means that they're the chosen axioms of that system. You might choose a different set of axioms and create a system in which those 9 statements which are axioms in ZFC *can* be proved as non-axiomatic statements, but that different set of axioms will probably be more complicated and messy; if it was simpler, it would have replaced the ZFC axioms.
It's like, people often assume that maths proves that 1+1=2 is some kind of fundamental fact. I think that's not quite right. I think mathematicians have created a system in which 1+1=2 and that's popular because you can use it to count sheep and stuff, but there's nothing fundamental about it. It's a purely engineering task to design a system that is useful for the problems you face.
There is another arithmetic system (called GF(2^n)) in which 1+1=0, which isn't so useful for keeping track of sheep, but turns out to be really handy for cryptography and error detection/correction codes, so is popular in those circles. You can't say one is "more true" than the other in any absolute sense, though.
Is it confusing that "1+1=2" and "1+1=0" are both True Statements of Maths? Yeah, but only because the notation is abbreviated and depends on context. To write them out in full, you'd have to say something like "The natural number 1, when number-added to the natural number 1, is equivalent to the natural number 2" and "the Galois field element 1, when Galois-field-added to the Galois field element 1, is equivalent to the Galois field element 0". That's just a bit verbose, so mathematicians instead say "Ok, we're using Galois fields here. ...blah blah... 1+1=0 ...blah blah..."
So perhaps maths is "real" when that particular maths we're talking about happens to match the structure of whatever rules underlie the universe, and maths that doesn't happen to match the structure of anything we've found yet might not be real. I'm not aware of anything in physics that works like Galois fields, but I'm not a physicist, so perhaps they're not particularly "real" even though they're useful in computing. We can give something artificial meaning, without nature having done so.
-- Alaric Snell-Pym (M0KTN neé M7KIT) http://www.snell-pym.org.uk/alaric/
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