/sci/ - Science & Math

>>12465249
>mathematicians struggled to model the continuum
>wanted a system of arithmetic to include pi and e, and to make sense of infinite sums
>people start getting angry at analysts for talking bullshit and not making any sense with their "infinites", even prominent geniuses like Abel accuse them of being charlatans and not making any sense.
>analysts start looking for a way to make their theories logically sound
>develop set theory in which asserts that infinity can be completed without actually proving it
>claim it properly explains what they're actually doing when they're manipulating infinities
>actually it doesn't explain shit, people discover paradoxes and logical inconsistencies in this set theory
>best mathematicians at the time like Poincare, Kronecker recognize set theory as the joke that it is.
>logicians upload a "patch" to the set theory where now the word "set" is actually meaningless and all you're doing in this "set theory" is just manipulating formulas according to some predetermined rules
>mathematicians naturally assign semantics to concepts in set theory and discover that obvious bullshit follows from this set theory, like the existence of "arbitrarily large infinites", or numbers that can't be described in any way, and suddenly you can't add rational numbers anymore, not even natural numbers can be added with this "set theory", also you get objects without any size, and you can decompose a sphere into many different spheres of the same kind.
>big battle develops in the foundations over the meaning of set theory and its place in mathematics. Hilbert vs Intuitionists vs other schools somewhere in between. People start asking whether "for all natural numbers n, something is true" is even a meaningful proposition, and what does it take to prove or disprove such propositions.

>>12463756
>mathematicians struggled to model the continuum
>wanted a system of arithmetic to include pi and e, and to make sense of infinite sums
>people start getting angry at analysts for talking bullshit and not making any sense with their "infinites", even prominent geniuses like Abel accuse them of being charlatans and not making any sense.
>analysts start looking for a way to make their theories logically sound
>develop set theory in which asserts that infinity can be completed without actually proving it
>claim it properly explains what they're actually doing when they're manipulating infinities
>actually it doesn't explain shit, people discover paradoxes and logical inconsistencies in this set theory
>best mathematicians at the time like Poincare, Kronecker recognize set theory as the joke that it is.
>logicians upload a "patch" to the set theory where now the word "set" is actually meaningless and all you're doing in this "set theory" is just manipulating formulas according to some predetermined rules
>mathematicians naturally assign semantics to concepts in set theory and discover that obvious bullshit follows from this set theory, like the existence of "arbitrarily large infinites", or numbers that can't be described in any way, and suddenly you can't add rational numbers anymore, not even natural numbers can be added with this "set theory", also you get objects without any size, and you can decompose a sphere into many different spheres of the same kind.
>big battle develops in the foundations over the meaning of set theory and its place in mathematics. Hilbert vs Intuitionists vs other schools somewhere in between. People start asking whether "for all natural numbers n, something is true" is even a meaningful proposition, and what does it take to prove or disprove such propositions.

>>12392904
Merely asserting the existence of the square root of negative one and working as if you've already constructed it and can manipulate it like any other number doesn't make much sense because there is no guarantee that the rules of arithmetic don't break down, i.e. that you don't get any contradictions. That's why mathematicians are actually careful with complex numbers and construct them as pairs of real numbers
(a,b) with the rules of arithmetic
(a,b)+(c,d)=(a+c, b+d),
(a,b)*(c,d)=(ac-bd, ad+bc)
1=(1,0)
0=(0,0).
Then these pairs operate just like you would expect complex numbers to operate and you DEFINE
a+bi = (a,b). Then indeed
i^2= i* i = (0,1)*(0,1)=(0*0-1*1, 0)=(-1, 0 ) = -1.
You can verify that these pairs form an associative, commutative division ring, i.e. a field.
Now with dividing by 0 this doesn't work: the rules of arithmetic break down.
1/0 * 0 =0=1.
There is no way to construct a nice number system the way you do it with complex numbers that can accommodate division by zero, unless the whole number system is just one number x such that x is both 0 and 1, x+x=x, x*x=x. This is called the zero ring.

>>12216143
Your question of what a determinist SHOULD do in order to be logically consistent makes no sense in the first place under determinism, since the word should implies it could be otherwise. But it can't. You're going to do and believe what the universe had decided many years ago before you even existed.

>>12193824
>is this ever more than an abusers cope?
It's only abuse if you're a pussy. And pussies don't belong in maths.

>>12167165
>Give some good evidence that free will exist
Your perception of your own free will is as good evidence as you can get.
Given that there are 0 good reasons to doubt this perception, it's completely rational and justified to believe in free will.
You may think you have reasons, but they're all easily refuted. You can give a reason that convinced you you can't have free will and I'll refute it.

>>12163158
>Sure some hypothetical Laplace's demon might be able to discard the concept of people making choices and controlling their actions and just think of us as lumps of matter obeying the laws of physics
There is 0 reasons to believe this is possible (even theoretically) BTW. This has been a theoretical possibility at the time of Newtonian physics which has been completely deterministic in its description but now we know about quantum mechanics which eliminates all hope of proving determinism through laws of physics.

>questions
Too easy.