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.
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.
Sure it is. 1 is the multiplicative identity, the number you start at when you multiply anything. 2^2 is really 1x2x2. 2^1 is 1x2. So 2^0 is… just 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!
Honestly I think it’s misleading to describe it as being “defined” as 1, precisely because it makes it sounds like someone was trying to squeeze the definition into a convenient shape.
I say, rather, that it naturally turns out to be that way because of the nature of the sequence. You can’t really choose anything else
0! = 1 isn’t an exception.
Factorial is one of the solutions of the recurrence relationship f(x+1) = x * f(x). If one states that f(1) = 1, then it only follows from the recurrence that f(0) = 1 too, and in fact f(x) is undefined for negative integers, as it is with any function that has the property.
It would be more of an exception to say f(0) != 1, since it explicitly denies the rule, and instead would need some special case so that its defined in 0.
x factorial is the number of ways you can arrange x different things. There’s only one way to arrange zero things.
I could still debate the proposition that zero things can be arranged in any way.
That sounds like a philosophical position, not a mathematical one.
You are right but that is a dangerous proposal because math is just applied philosophy :)