DAILY REMINDER THAT DAVID HILBERT HAS SET BACK MATHEMATICS A

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>>7983869
You're right.

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I already laughed about
>SHILLBERT
yesterday

But let me just post this here, Hilbert has the cutest voice
https://www.youtube.com/watch?v=EbgAu_X2mm4

>>7983963
>tfw no qt 3.14 kraut bf

also
> using 'gestalt' non ironically
> wanting to be taken seriously
nice try shillbert

For too long mathematics has had this narrow minded, parochial outlook where the only realms being considered are the countable(discrete) and the uncountable.

Clearly this is as limited and narrow minded as the Greeks were when they only imagined the countable.

When will we FINALLY make some REAL progress and explore a new space outside of merely the countable and uncountable , the discrete and continuous?

At the moment the math we're doing confined to these two realms , finding more and more arcane trivia to look up is like the Greeks spending their whole time trying to find out with higher and higher values of N whether it is possible to construct a regular n-gon with a compass and straight edge.

We need to move beyond the narrow, parochial world view of there only being the countable or uncountable or else mathematics will simply become more and more inbred.

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>>7984051
>greeks only imagined the countable
>geometry
>countable
>GEOMETRY
>COUNTABLE

op pls

>>7984051
The countable-uncountable dichotomy comes right when you introduce the function concept.

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>>7984084
>in b4 burger

>>7983869
hilberts a homo and so are you

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>>7984166

>KEKEKEKEKKEKEKEKE HOMO KEKEKEKEKEK

suicide is an option

>>7983869
>doesn't know where the algebro geometric proverb "never trust an italian" comes from

>>7984059
greek geometry was based on rational numbers you ignoramus.

>>7984084
how so?
Functions are still only functions on countable or uncountable sets

where is the vision?

are we forever doomed to only do maths in either a countable setting (graphs, combinatorics, algorithms, some number theory, some algrabra) or uncountable settings (analysis, geometry, topology, algebra, algeraic geometry, algebraic topology, and other combinations thereof)

Is this it? are we doomed to forever be in these arenas?
either asking questions about countable sets like graphs and combinatorial structures, or asking questions about sets in continuous spaces that are all just variants on R^n wit different metrics or topologies.

there must be more to it than this.

Mathematics is meant to be about simply coming up with some starting rules, then logically deriving results from those rules. so why can't we come up with any rules or any objects beyond these two countable and uncountable arenas?

>>7984196
>greek geometry was based on rational numbers you ignoramus.


you're adorable

hilbert was a cool guy

So why can't others go the other way, I'm sure Hilbert would redact stuff if he saw current work into real numbers

>>7983869
Rigor wasn't everything to hilbert. He knew how to make physical/intuition based arguments very well and fully encouraged it. It's just that to actually prove something as mathematical fact, you should actually prove it. With how much math has progessed since hilbert, idk how you could argue against him. I'd say the strict bourbakistes (the generation AFTER hilbert aND after the founding bourbakists), who only learned/taught math based on rigor did set math behind though. It would be best if we taught math integrating rigor with physics and other guides of intuition, rather than in a purely "modern" way. Most russian mathematicians are lik this, for example

>>7984561
math is about formalizing your speculation that you see as more structured than the rest of your imagination. Maths and physics have nothing to do with some fantasy of empiricism explored rationally to learn about nature. Math is about to go as wild as possible form empiricirsm while still clinging to some of your inventions of rules of deductive logic.

>>7983869
rigor is for autists

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>>7983963
Lmao, he sounds like an elderly mouse

>>7984196
Let there be a right triangle with two equal sides whose length is one.
What is the length of the third side?

>>7984051
The Lebesgue Sigma Algebra on R and the topological group of ultrafilter on N are two thoroughly studied objects with cardinality aleph_2

If you aren't familiar with either I suggest you read up on them, they are both ubiquitous in good mathematics.

My personal opinion is that cardinalities are without exception the least interesting invariant of an object.

>>7984712

Sssssssssssssssssh don't let Pythagoras hear you

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>>7984196
>how so?
Because the functions from N to {0,1} are already not enumerable. That is to say, the function space N->{0,1}, which is in bijection with the powerset of N, is uncountably infinite.
And in fact, postulating that it's just as big as R (continuum hypothesis) is consistent with any standard set theory.

>so why can't we come up with any rules or any objects beyond these two countable and uncountable arenas?
You can do math that's not sets with cardinalities - then you cannot express those notions. But "the problem" (second order) logic itself is already a type/set theory, so as soon as you make your reasoning modalities strong enough, you get that stuff back.

Besides, uncountable is a catch-all term and there are infinities far beyond the cardinalities of R^n, see e.g.
http://cantorsattic.info/Cantor's_Attic

>I got 23 problems but a bitch aint one.

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hilbert reckted

Cantor:
-Was in a mental hospital
-smeared feces on the walls of his room
-Henri Poincaré referred to his ideas as a "grave disease" infecting the discipline of mathematics,and Leopold Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth." Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory," which he dismissed as "utter nonsense" that is "laughable" and "wrong".


While Hilbert described Cantor's work as:-

...the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.

>>7984059
There are only countable many objects that can be constructed with just a ruler and a straight edge.

>>7987408
so why don't you give us some information about *why* set theory is so bad rather than a bunch of ad hominem shit?
Complex numbers were ignored for centuries until someone found out they are actually useful. So the fact that real mathematicians objected Cantor's constructs proves nothing

>>7983869
You're giving him too much credit. For the most part the only people who really worry about rigor (to the extent that Hilbert advocated) are logicians. For a working mathematician "rigor" is making sure you never divide by zero. Hell, most mathematicians probably don't even notice when they use the axiom of choice.

If you're ever at a point in a proof where you have to resort to model theory and/or axiomatic set theory then your name is Bertrand Russell and you're trying to prove 1 + 1 = 2. That's the only time a "mathematician" worries about rigor.

>>7983869
>>7984051
>>7984196
>>7985936
>>7985985
this confuses the engineer

>>7983940
NOT A PARADOX REEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE

>tfw mathematicians have still failed to identify what numbers are

>>7989638
>tfw we can do math without having to know what numbers are

>>7989421
This is incredibly, unbelievably false. I hope people don't actually believe this bullshit.

Rigor is ridiculously necessary and it's what we, as mathematicians, continuously worry about as we write a proof. We go through several rewrites before the final product is presented.

>>7987366
[math]e^{\ln 2}[/math]

>>7983869
It's more of the state of mathematics education and culture, where the person and the creative process are ignored in favor of proof and results

>>7989643
>Mathematics is simply fiction with certain constraints

>>7989716
I haven't heard of this problem before, so here I am thinking about it trying to determine if there is ever such a case without thinking of a = e and b = ln(2). Does the problem say e and logarithms don't count?

>>7985936
This is not necessarily true, you need a weak form of GCH to conclude that.

>>7985985
That's not the continuum hypothesis you cretin.

>>7989716
Except, the question is is a^b transcendental a=/= 0 and a=/= 1 and b is irrational.

>>7989797
wot?

a=e is irrational. b=ln2 is irrational. a^b=2 is rational.

being transcendental implies being irrational... lol

>>7989680
That's not the type of rigor that OP is referring to. My "divide by zero" example was chosen for explanatory purposes, but I am aware that rigor is important in a proof. My point is that the standard for rigor in a mathematicians sense was pretty much the same before and after Hilbert. Hilbert (unless I'm mistaken) advocated for rigor at the foundational level, and I'd be surprised if Hilbert ever claimed that mathematicians were regularly making mistakes in their papers.

>>7989790
That's fair enough, I usually assume ZF+GCH

Regardless they are both greater cardinality than [math]|\mathbb{R}|[/math] which is what the person I was replying to seemed to be interested in.

>>7984051
You have the right sentiment, but the language you are using is probably not correct. The "realms" we have yet to imagine/discover probably have very little to do with the cardinality of the things we're working with.

Btw, one reason the continuum is useful is that our physical world seems to well described by real numbers.

>>7989871

That's the reason why you think the cardinality of an object is not interesting.

>>7989871
>I usually assume ZF+GCH
why would you assume this

Is it possible to do mathematics without logic ?

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>>7990033

>>7990051
you know that mathematicians do not learn logic, right ?

>>7990033
Every academic discipline requires logic. But none of them require "logic" in the logicians sense

spoilers unless you're a logicians /spoiler

>>7990076
u wot? Sophomore year I had a class that was specifically an intro to logic. The whole point was to teach logic and the symbols used (v for 'or', the implication ->, iff <->, etc.) and then teach the basics for building proofs (types of proofs, then very basic number theory, set theory and abstract algebra). I've heard similar things from friends at other schools too. Then again, mathematics is logic, just applied to numbers.

>>7990237
Yeah no.

>>7984265
But isn't his claim largely true?

>>7984196
> R^n wit different metrics or topologies.
>metrics or topologies
>metrics
>or topologies

dont think you understand either of those words

>>7987366
pretty sure that's not Hilbert's 7th problem family

>>7983869
Honestly OP has a point, he's wrong, but he has a point.

>>7984051
>I don't know ordinal numbers

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>>7983869
>MOCHIZUKI

>>7990566
>In an isosceles triangle, if the ratio of the base angle to the angle at the vertex is algebraic but not rational, is then the ratio between base and side always transcendental?
>Is [math]a^b[/math] always transcendental, for algebraic [math]a \not\in \{0,1\}[/math] and irrational algebraic [math]b[/math]?

>>7987366
Troll image. Hilbert's 7th problem has to do with transcendental numbers.

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http://euclid.colorado.edu/~tubbs/courses/Chapter%20One.pdf

>>7984712
go to bed Hippasus

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>>7993036
he sleeps with the fishes

boump

>>7989638
A natural number is a transitive set linearly ordered by the membership relation, and which also has no proper injection into itself.

is there numbers without set theory?