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 believe that the polar plot of prime numbers that reveals spirals and rays and when extended out to millions of numbers shows deeper fractals and geometric patterns is a glimpse into the structure of something we haven’t yet discovered.
Maybe it’s the higher dimensional structure of a photon, maybe it’s something we don’t even know about, but the fact that math describes everything in our universe EXCEPT prime numbers sounds like nonsense. There’s something staring us right in the face that we can’t see yet.
1 should be a prime number.
It was once, but got kicked out due to new discoveries and equations I’m not smart/mathematically trained enough to understand.
1 got Pluto-ed?!
I prefer the term “got Ceresed”, but yes. That’s happened.
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.
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^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!
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
Mixed numbers fraction syntax [1] is the dumbest funking thing ever. Juxtaposition of a number in front of any expression implies multiplication! Addition? Fucking addition? What the fuck is wrong with you?
Amen. Pick a lane either they’re both additive or multiplicative. Maybe a different symbol.
Everyone keeps talking about pi r²
This doesn’t make any sense because pies are round. Brownies are square
P vs NP can be solved, and is within P. Good luck proving it though, I’m not smart enough
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.
“Terryology may have some merits and deserves consideration.”
I don’t hold this opinion, but I can guarantee you it’s unpopular.
