22/7 cheers?
22/7 cheers?
This knowledge is needed in what way?
What difference does it make how humans have defined the numbers they invented?
Are we saying they could not have been defined differently?
Are we saying that this system humans invented is universal and not arbitrary and contingent?
That is an interesting question about numbers.
I know I am supposed to be in awe of these schemes humans have put together and they have a logical consistency which is good and of course numbers have given us the modern world but what are numbers is not a question that has been answered in this thread. It has not been addressed.
How some numbers have been defined, sometimes after the fact, after they have existed for a long time, has been addressed.
The philosophy of mathematics is so much more interesting than mathematics.
https://en.wikipedia.org/wiki/Philosophy_of_mathematics
And some will say this is the philosophy section.
Last edited by untermensche; 03-17-2018 at 02:02 PM.
Can you give me some papers in the philosophy of mathematics that particularly interest you? I'd be happy to discuss them in a thread. Hilbert has a nice paper talking about why the concept of infinite doesn't exist in reality. You might enjoy that paper. On the basis of it, he embarked on a serious effort to put together metamathematical proofs in his (sadly impossible) goal to demonstrate that ZF set theory is a conservative extension of primitive recursive arithmetic. Your probably won't like those papers, since those are actual mathematics.
Which might pose a problem if you want to pursue your interest here. If you pick up an anthology on the philosophy of mathematics today, virtually all of the articles will be written by philosophers with proven competence in mathematics and logic. If you scroll down the wiki page to the part talking about the 20th century, you will see it mentioning specifically formal logic and set theory, all developed by mathematicians and mathematically competent philosophers. It mentions "formalism", which is mostly associated with the greatest mathematician of the century,Hilbert; "intuitionism", developed by another working mathematician Brouwer, and "logicism", mostly associated with Frege and Russell, both of whom worked on mathematical formalisations of the concept of number. Moving into the middle century, we get told about the introduction of Category Theory, which is again, just more mathematics. Indeed, category theory is an outgrowth of the sorts of algebraic approach that characterises the OP, and category theorists are usually great at telling you all sorts of funky things about these algebraic constructions.
By the way, the early attempts to define real numbers back in the 19th century, exactly the sort of thing that beero1000 is celebrating, were often dismissed by other mathematicians at the time as mere philosophy.
You can't pretend to be interested in contemporary philosophy of mathematics without also being interested in the sorts of constructions going on in the OP.
Can you tell me what the OP has to do with philosophy?
Here's a very simple question.
What is "one"?
Philosophically.
And I do not need any papers to tell me invented concepts like infinity can't possibly be real. It's like thinking Pi is something real or imaginary numbers are real or points are real or lines are real. There is no reason to assume it is something that could possibly be real.
It is not a tough question or one anybody should have difficulty with.
It takes a lot of delusions to even think it could possibly exist as something real.
Last edited by untermensche; 03-17-2018 at 06:14 PM.
The terminal object in Finord?
Well, that's a definition that a category theorist might give, which was mentioned in the wiki article you linked. Did you read it?Philosophically.
Okay.And I do not need any papers to tell me invented concepts like infinity can't possibly be real.
I would really love to link On the Infinite, but I can't find the damn thing. I know it's in "From Brouwer to Hilbert", but I have no idea why this paper from the 1920s is not freely available online in the 21st century. I almost want to go on a rant, but will save it for now.
I think it's been 10 years since I read it, so I can't vouch for the accuracy of this synopsis, but here goes:
After the crises of early 20th century mathematics, showing that cavalier arguments involving the infinite end up in contradiction, Hilbert wants to say that the "infinite" should become just a theoretical tool, and that we must always remain faithful to the true ground of mathematics. He uses, as an analogy, the "points at infinity" that appear in projective geometry. These points at infinity are weird, but useful, and can always be analysed away. They're, to use Hilbert's term, "ideal elements". He wants to use this idea of weird, useful, but analysable away, as a means to tame the infinite and set theory, reducing it to a tool with which to make the real claims of mathematics.
Those real, meaningful claims of mathematics occur only in the most simple claims of arithmetic: they're concrete claims about concrete marks on paper, like the claim that 2 + 3 = 5. Hilbert goes further than this and says that this claim is the following:
II + III = IIIII
where "+" is just concatenation of marks. This is, ultimately, all that mathematics is. It's simple arithmetic, where arithmetic reduces to concatenations and simple substitutions of marks.
This renders a huge amount of mathematics, on the face of it, effectively meaningless. Everything else that mathematicians are doing is unreal. Even the very basic claims of arithmetic like
m + n = n + m
are literally meaningless. Such claims only acquire meaning once you plug in concrete terms, at which point the equation becomes verifiable by actually concatenating the marks and then comparing the marks on each side.
Hilbert believed that everything else you do over the top of this had to be, in principle, analysable away, reducing any statement involving variables, existentials and universals to simple procedures which reduce to claims about concrete equations about marks. However crazy it sounds, he was convinced there was a proof that this could be done for the entirety of set theory, and he relied heavily on his experience with metamathematics which had already shown that arbitrary logical systems could be boiled down to mucking around with concrete numbers.