/sci/ - Science & Math » Thread #13817535

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Has modern mathematics gone crazy?

Anonymous Thu 04 Nov 05:50:24 2021 No.13817535 View Reply Original Report

Quoted By: >>13817537 >>13817545 >>13818696 >>13819830 >>13819849 >>13820658

https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers

This does't make sense.
No computer nor human operates on natural numbers that way.

Anonymous

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Anonymous Thu 04 Nov 2021 05:52:26 No.13817537 Report

Quoted By: >>13819769

>>13817535
I do

Anonymous

Anonymous Thu 04 Nov 2021 06:00:13 No.13817545 Report

Quoted By: >>13817553

>>13817535
the point isn’t that these ‘are’ the natural numbers, it’s that
1. these sets form a structure isomorphic to the natural numbers, and you can define addition and everything on them, so ZFC has a way to simulate the natural numbers if you want to… a bit like how a Linux install can simulate a Windows machine using Wine, if you want to play games
2. the definition is actually robust and forms the basis for definingthe ordinals, which is useful for many things including transfinite induction

Anonymous

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Anonymous Thu 04 Nov 2021 06:09:46 No.13817553 Report

Quoted By: >>13817675 >>13820658

>>13817545
>the point isn’t that these ‘are’ the natural numbers, it’s that
Natural numbers are a requisite for propositional logic which then upon further development goes to define ZF.

Anonymous

Anonymous Thu 04 Nov 2021 06:43:36 No.13817588 Report

Quoted By: >>13817614

OP is correct. Students are taught the "dynamic types" method instead of the "static types" method. What OP is looking for is the first few pages of A Shorter Model Theory by Hodges. However this is too advanced for OP, so OP has to tough it out before moving onto something that corresponds with actual usage outside mathematics.

Anonymous

Anonymous Thu 04 Nov 2021 07:05:22 No.13817614 Report

Quoted By: >>13819751

>>13817588
>Students are taught the "dynamic types" method instead of the "static types" method.
Where can I read more about this in relation to Logic or Set Theory?
>What OP is looking for is the first few pages of A Shorter Model Theory by Hodges.
Could you please elaborate a bit more on this?

Anonymous

Anonymous Thu 04 Nov 2021 07:48:02 No.13817675 Report

Quoted By: >>13817690

>>13817553
>Natural numbers are a requisite for propositional logic
What? Any examples of when?

Anonymous

Anonymous Thu 04 Nov 2021 08:01:54 No.13817690 Report

Quoted By: >>13817714 >>13820658

>>13817675
>What? Any examples of when?
>a countably infinite set of elements called proposition symbols
https://en.wikipedia.org/wiki/Propositional_calculus#Generic_description_of_a_propositional_calculus
Countably infinite means that they are indexed by $\mathbb{N}$ .
Another way of saying this is
>a set of elements $\{p_{0}, p_{1}, ...\}$

Anonymous

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Anonymous Thu 04 Nov 2021 08:22:53 No.13817714 Report

Quoted By: >>13817907 >>13820658

>>13817690
You can simulate the propositional calculus directly as sets, not that differently from how you simulate the natural numbers. Or yes, you can simulate the naturals and then use some kind of coding to simulate propositional logic inside of them. In either case the point is not so much that these are the real definitions or the best definitions, but that ZFC is able to reason about these objects through the isomorphic copies we can construct.

Anonymous

Anonymous Thu 04 Nov 2021 10:17:36 No.13817907 Report

Quoted By: >>13818022

>>13817714
>you can simulate the naturals and then use some kind of coding to simulate propositional logic inside of them.
Could you elaborate on this?

Anonymous

Anonymous Thu 04 Nov 2021 11:11:44 No.13818022 Report

Quoted By:

>>13817907
>simulate the naturals
OP’s encoding (aka von Neumann numerals)
>simulate propositional logic within the naturals
This could mean more than one thing and there are different ways of doing it. None of them are that simple. Godel’s original proof of Incompleteness finds a way to encode logical statements as numbers, and then evaluate them with a formula about numbers. The MRDP theorem lets you represent predicate logic using systems of diophantine equations.

Anonymous

Anonymous Thu 04 Nov 2021 15:48:32 No.13818696 Report

Quoted By: >>13819713

>>13817535
Now encode some operations like addition, subtraction, multiplication, and encode more mathematical objects like reals, vectors, etc.. Now make sure that you have no name conflict in your encodings. Congrats you have generated like 10000000000 pages of unreadable brackets for like 2% of math. Whoever came up with this shit is retarded and the fact that they didn't see immediately how impractical it was and yet people still tried to use it just goes to show that mathematicians are not sending their best.

Anonymous

Anonymous Thu 04 Nov 2021 21:05:00 No.13819713 Report

Quoted By: >>13819818

>>13818696
The sole point is that the implementation exists.

Anonymous

Anonymous Thu 04 Nov 2021 21:16:38 No.13819751 Report

Quoted By:

>>13817614
this is the difference between adding constants with explicit definitions (ordinary teaching way) vs. adding constants that aren't defined (the axiom that defines their value is omitted) and instead assuming axioms according to usage, for example adding two constants, 1,2 and a binary operation + and assuming 1 + 1 = 2 as an axiom
I'm pretty sure you actually can't write down how to add constants to the language without first defining summation as calculating the aggregation of quantities.
What I'm talking about is more like a trick whereby you can construct a theory that "doesn't know what addition means" because the concept of aggregating quantities is imposed by some rule or definition
there aren't two methods in reality, just one method plus the concept of a simulated or emulated virtual learning

Anonymous

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Anonymous Thu 04 Nov 2021 21:20:40 No.13819769 Report

Quoted By:

>>13817537
I take you at your word.

Anonymous

Anonymous Thu 04 Nov 2021 21:29:38 No.13819818 Report

Quoted By: >>13819944 >>13820481

>>13819713
Bro it doesn't scale at all and it immediately doesn't make sense. 0 = the empty set? NO!

Again try to define another object. You will immediately run into the problem of name clashes with all of your integer. So now you have to take your object with some kind of identifier. But now you have to invent the integers before you invent the integers so you can put some kind of integer tag on all of your integers so you know they are integers.

So then you write pages and pages of unreadable brackets to define something like integers and addition on them using only brackets and an inclusion operator and you realize that it took you forever and you wrote so much and you did so little. Why?

OP is correct. The mathematicians failed completely on this one but they don't want to give it up for some reason.

Anonymous

Anonymous Thu 04 Nov 2021 21:32:46 No.13819830 Report

Quoted By:

>>13817535
ZF set theory is dumb formalism that follows Russell's interpretation of the Barber Paradox and its equivalents. Mathematics is a field that should be reserved for geniuses, and when that doesn't happen this is what you get. And now it's stagnated because genius status has been replaced academic and prestige status, so in order to change it the official path is eating all this garbage until you get a PhD and then pray to God to not be cancelled until you manage to fix the mistakes. Perelman abandoned this pyramid scheme for a reason. But things are changing thanks to the internet.

Anonymous

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Anonymous Thu 04 Nov 2021 21:33:14 No.13819831 Report

Quoted By: >>13819839

>it's a "/sci/ doesn't understand baby mathematical logic" episode
kek

Anonymous

Anonymous Thu 04 Nov 2021 21:34:32 No.13819839 Report

Quoted By: >>13819872

>>13819831
Where is your explanation of this baby mathematical logic?

Anonymous

Anonymous Thu 04 Nov 2021 21:35:40 No.13819843 Report

Quoted By:

Children should be taught large cardinal axioms first, then we can move on to counting numbers, addition, subtraction etc.

This is the only way for math to advance.

Anonymous

Anonymous Thu 04 Nov 2021 21:38:14 No.13819849 Report

Quoted By:

>>13817535
>No computer nor human operates on natural numbers that way.
uuh lisp machines and BASED richard stallman operate that way

Anonymous

Anonymous Thu 04 Nov 2021 21:45:53 No.13819872 Report

Quoted By:

>>13819839
There is at least one idiot who is confusing objective language with metalanguage.

Anonymous

Anonymous Thu 04 Nov 2021 22:13:30 No.13819944 Report

Quoted By:

>>13819818
>Bro it doesn't scale at all and it immediately doesn't make sense. 0 = the empty set? NO!
And bro, this doesn't matter. This is just to make sure that everything can be done in principle using explicit calculus with sets in ZFC, nothing else. Nobody back then had computers to program to check proofs or had to worry about making formal languages usable for humans so that they could write computer verified proofs, so they didnt worry at all

Anonymous

Anonymous Fri 05 Nov 2021 01:47:34 No.13820481 Report

Quoted By:

>>13819818
your brain is the empty set

Anonymous

Anonymous Fri 05 Nov 2021 02:49:20 No.13820658 Report

Quoted By:

>>13817535
convince me that this autism in mathematics is not retarded. the empty set is just chosen because of the axiom that says the empty set exists, anything else would fit the bill there, what's the fucking point of this definition? there's no useful result out of this.
>>13817690
>>13817553
based.
the importance is structure, using sets is not illustrative in any way
>>13817714
>You can simulate the propositional calculus directly as sets
quite the achievement there. fuck set theory. do something useful neetmathician.
I'm graduating with a thesis on stats or some kind of algorithm. fuck this axiomatic autism

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