For example: I don’t believe in the axiom of choice nor in the continuum hypothesis.
Not stuff like “math is useless” or “people hate math because it’s not well taught”, those are opinions about math.
I’ll start: exponentiation should be left-associative, which means a^b should mean b×b×…×b } a times.
I don’t think ‘I don’t believe in the axiom of choice’ is an opinion, it’s kind of a weird statement to make because the axiom exists. You can have an opinion on whether mathematicians should use it given the fact that it’s an unprovable statement, but that’s true for all axioms.
Any math that needs the axiom of choice has no real life application so I do think it’s kind of silly that so much research is done on math that uses it. At that point mathematics basically becomes art but it’s art that’s only understood by some mathematicians so its value is debatable in my opinion. <- I suppose that opinion is controversial among mathematicians.
The exceptions including the number 1. Like it not being a prime number, or being 1 the result of any number to the 0 power. Or 0! equals 1.
I know 1 is a very special number, and I know these things are demonstrable, but something always feels off to me with these rules that include 1.
X^0 and 0! aren’t actually special cases though, you can reach them logically from things which are obvious.
For X^0: you can get from X^(n) to X^(n-1) by dividing by X. That works for all n, so we can say for example that 2³ is 2⁴/2, which is 16/2 which is 8. Similarly, 2¹/2 is 2⁰, but it’s also obviously 1.
The argument for 0! is basically the same. 3! is 1x2x3, and to go to 2! you divide it by 3. You can go from 1! to 0! by dividing 1 by 1.
In both cases the only thing which is special about 1 is that any number divided by itself is 1, just like any number subtracted from itself is 0
The numbers shouldn’t change to make nice patterns, though, rather the patterns that don’t fit the numbers don’t fit them. Sure, the pattern with division of powers wouldn’t be nice, but also 1 multiplied by itself 0 times is not 1, or at least, not only 1.
We make mathematical definitions to do math. We can define 0! any way we want but we defined it to be equal to 1 because it fits in nicely with the way the factorial function works on other numbers.
Literally the only reason why mathematicians define stuff is because it’s easier to work with definitions than to do everything from elementary tools. What the elementary tools are is also subjective. Mathematics isn’t some objective truth, it’s just human made structures that we can expand and better understand through applying logic in the form of proofs. Sometimes we can even apply them to real world situations!
Mathematics isn’t some objective truth
Yes it is. That’s why it’s such a huge part of Physics.
it’s just human made structures
The notation is. The rest is underlying laws of Nature.
Sometimes we can even apply them to real world situations!
Maths whole reason for being is to model real world situations.
Wow you just disproved all of academic mathematical foundations and philosophy! Congrats!
