Von Neumann's critique of abstraction in mathematics

[Anonymous No.11798798]

"It is very hard for any mathematician to believe that mathematics is a purely empirical science or that all mathematical ideas originate in empirical subjects. Let me consider the second half of the statement first. There. are various important parts of modern mathematics in which the empirical origin is untraceable, or, if traceable, so remote that it is clear that the subject has undergone a complete metamorphosis since it was cut off from its empirical roots. The symbolism of algebra was invented for domestic, mathematical use, but it may be reasonably asserted that it had strong empirical ties. However, modem, "abstract" algebra has more and more developed into directions which have even fewer empirical connections. The same may be said about topology. And in all these fields the mathematician's subjective criterion of success, of the worth-whileness of his effort, is very much self-contained and aesthetical and free (or nearly free) of empirical connections. (I will say more about this further on.) In set theory this is still clearer. The "power" and the "ordering" of an infinite set may be the generalizations of finite numerical concepts, but in their infinite form (especially "power") they have hardly any relation to this world. If I did not wish to avoid technicalities, I could document this with numerous set theoretical examples-the problem of the "axiom of choice," the "comparability" of infinite "powers," the "continuum problem," etc. The same remarks apply to much of real function theory and real point-set theory. Two strange examples are given by differential geometry and by group theory: they were certainly conceived as abstract, non-applied disciplines and almost always cultivated in this spirit. After a decade in one case, and a century in the other, they turned out to be very useful in physics. And they are still mostly pursued in the indicated, abstract, non-applied spirit."

[Anonymous No.11798801]

"The examples for all these conditions and their various combinations could be multiplied, but I prefer to turn instead to the first point I indicated above: Is mathematics an empirical science? Or, more precisely: Is mathematics actually practiced in the way in which an empirical science is practiced? Or, more generally: What is the mathematician's normal relationship to his subject? What are his criteria of success, of desirability? What influences, what considerations, control and direct his effort?"

[Anonymous No.11798804]

https://mathshistory.st-andrews.ac.uk/Extras/Von_Neumann_Part_2/

[Anonymous No.11798817]

>>11798798
>empirical
Do you want to start a war or something?
Ive seen people here lose their shit the mere mention of this word.
For them empirical evidence is no evidence.

[Anonymous No.11798996]

>>11798798
von neumann did a lot of set theory and logic in the beginning of his career
i guess that just like many others from his era, he felt disappointed with this stuff

[Anonymous No.11799043]

>>11798798
That's not really a critique. He's just saying math is not an empirical science, which is true; math is just an exercise in logic.

[Anonymous No.11799106]

>>11799043
Logic is empirical in that you can "play the game" in your mind. I can empirically verify the meaning of sentences by uttering in my imagination and seeing how they sound like.

[Anonymous No.11799189]

>>11798996
but wasn't Johnny more of an applied mathematician than a pure mathematician

he's like the anti-grothendieck wasn't he?

[Anonymous No.11799349]

>>11799106
Your imagination is not empirical evidence of anything.

[Anonymous No.11799357]

>>11798798
Von Numale is a total retard, just listen to his interviews on YouTube. Amazing how he conned people like you into thinking he was """"smart"""".

[Anonymous No.11799403]

>>11799349
Learn what empiricism is, kiddo ;^)

[Anonymous No.11800807]

>>11799357
This

[Hugo No.11800824]

>>11799349

Yea I can't see what you are thinking, or seeing even,
most important part here how can we empirically prove that your and my understanding of a "supposedly" not empirical subject are the same,
making this supposedly non empirical subject empirical.

[Anonymous No.11800867]

>>11798817
Why is that so?

[Anonymous No.11800890]

>Von Neumann was born Neumann János Lajos to a wealthy, acculturated and non-observant Jewish family

blah blah words words words who cares what this retard has to say except for what he means - to con you

[Anonymous No.11800893]

>>11798996
Why is it a disappointment? I've heard memes about set theory and the category theory but I have no idea what they mean
t. Engineerlet

[Anonymous No.11800897]

>>11800867
How can you trust your eyes if your eyes aren't real?

[Anonymous No.11800915]

>>11800897

Never got that saying, is that a joke to make fun of silly "profound" sayings?

[Anonymous No.11803004]

>>11799357
OH NO NO NO WE GOT TOO COCKY NEUMANMEN

[Anonymous No.11803026]

>>11800915
It's a quote from Jaden Smith, Will Smith's spoiled son who thinks he's a philosopher because he's de facto famous from his dad and because he has a shitton of followers on Twitter.

[Anonymous No.11803499]

>>11803026

I bet we both understand his frustration, after all, he didn't get his father's height, so he might thirst for some mental height, that's lacking also :(

[Anonymous No.11803510]

>>11798798
who the fuck is this fat retatd and why we should listen to him?

[Anonymous No.11804690]

>>11798798
>>11798801
I love reading about this shit. Math has a pretty complicated relationship with reality

[Anonymous No.11804715]

>>11799189
He did turn himself into an applied mathematician, but he was a very capable pure mathematician as well. His career kinda makes the distinction a bit dubious.

[Anonymous No.11804721]

>>11798798
Honestly this quote is totally retarded.

>which the empirical origin is untraceable, or, if traceable, so remote that it is clear that the subject has undergone a complete metamorphosis

Wishy washy subjective bollocks. "So remote" -> "/complete/ metamorphasis". Not a logical deduction. The transformation of states in this scenario is thresholded by his feelings, not logic.

It is clear that maths is a ruleset paired to our empirical observations of reality. You can build a skyscraper as high as you like; at no point will any floor ever decouple from the foundations.

Was this dude supposed to be intelligent, or something? What an idiot.

[Anonymous No.11804784]

Even the most basic fundamentals are flawed - you can't even divide by the very first two consecutive numbers used to construct the entire rest of math, yet that is never addressed as a flaw and everyone just pretends that it's not there or worse, that it's "undefined" which is the most antiscientific cope that humanity has ever come up with. In order to solidify the cope, we invent detached abstract concepts which still produce paradoxes anyway even in their own abstract realms, and people STILL buy all of it and pretend that there's no issue whatsoever. In fact they've bought into it so hardcore that they willingly dedicate their entire lives to studying this flawed subject as if it's the undeniable objective reality and therefor the best investment of one's time. This blind belief in modern day math as the ultimate language of the Universe will be equally humiliating to humanity two centuries from now like geocentrism was two centuries ago, and everyone who stands against it will be called a dunning-kruger crackpot which is the modern version of a heretic who refuses to blindly follow the logically-inconsistent scientism of modern society, built not on facts but on the desire to not have the degree you just wasted a decade on drop in value. So the burden gets unloaded onto the next generation, but it too wants to selfishly climb the academic ladder and not oppose it so it too refuses to oppose it, and we end up in a cycle where no one ever bothers pointing out the delusional tangent that the academic establishment has taken on and everyone just deals with it, producing papers about absolute bullshit that has no connection to reality nor will ever have a connection to reality. Basically the equivalent to releasing papers in astrophysics while using geocentrism as your undeniable foundation.

[Anonymous No.11805245]

>>11798798
lmao what an idiot

[Anonymous No.11805289]

>>11800890
He actually started lectures with
>The goys have proven the following theorem...

Anyway, he's right about the developments but any judgement is a normative and subjective one.
The thread sort of lacks concrete substance. Of course, all of this was (and to an extent is) highly debated.

[Anonymous No.11805787]

>>11804721
this

[Anonymous No.11806009]

>>11800890
brainlet

[Anonymous No.11806274]

>>11804721
Based

[Anonymous No.11806597]

bump

[Anonymous No.11807970]

bump

[Anonymous No.11808156]

>>11804721
Perhaps a better way to fit what he's saying in to your analogy: our team of experts set out to develop the logic systems to facilitate the creation of a great skyscraper that will tower above all others. Now that many generations have passed, we have more or less come to a halt in progress on the 63rd floor because our current experts have become enamored with the esoteric intricacies of interior design, and oh how we could sure use a lovely jacuzzi tub here for when we get tired of building skyscrapers. These are obviously non-essential things for building the core structure, they just fluff up the quality, but not in the way the original intentions had planned on.

[Anonymous No.11808187]

>>11800867
>>11798817
Because empiricism *does not get you truth*.
And that is the difference between the natural science and mathematics, one can get to truth, the other can not.

[Anonymous No.11808254]

>>11804721
Unconsidered variables outside of your scope maybe catastrophically dynamic and so in time can have a big impact on a theory or a real life construct.

I got logic out of his quote. Basically unaccounted for things can have consequences.

You are just a self-pleasing dick quad.

[Anonymous No.11808334]

>>11808156
>>11808254
Von Numale Copers detected.

>11808254
He said 306 words and all you erroneously gleaned was just "take things into account". Kek.

He didn't say that, laddie. He started out with the implicit assertion (implicit as a cop out, so his intellectual betters can't BTFO Numale before he excuses himself with "just a joke bro") that mathematics isn't appended with the due consideration of its empirical roots (provably wrong). He then went on to say that was a good thing (he provided examples of abstract subjects becoming useful). At no point did he say "unaccounted for things can have consequences".

>11808156
>expanding my analogy in an incorrect way
The Skyscraper was mentioned purely for the inter-floor dependency and how each floor is connected to the foundation. Mr Numale said
>which the empirical origin is untraceable
Which is provably wrong within his context of "modern mathematics".

[Anonymous No.11808859]

Mathematicians who agree that it is important for mathematicians to stay in tune with empirical realities:
>Von Neumann
>Lebesgue
>Kolmogorov
>Atiyah
>Arnol'd
>Mazur
>&c.

Shitters who think "dood le making up abstruse abstract shit is, like, the thing that makes pure mathematics UNIQUE and AMAZING!!!"
>4chinners
>redditors

[Anonymous No.11808896]

>>11808187
I don't understand. Can't the truth be verified by observation?

[Anonymous No.11808949]

>>11808859
That's not what people are saying here, nor what Neumann is quoted as saying. You literally got it completely wrong.

>Two strange examples are given by differential geometry and by group theory: they were certainly conceived as abstract, non-applied disciplines and almost always cultivated in this spirit. After a decade in one case, and a century in the other, they turned out to be very useful in physics. And they are still mostly pursued in the indicated, abstract, non-applied spirit.

[Anonymous No.11809076]

>>11799357
His Double Commutant theorem is all I need to know of his genius. Sounding smart and being smart are two different things. I would expect the people of \sci to be more interested in the latter.

[Anonymous No.11809111]

>>11808187
>>11808896
People who are obsessed over mathematics don't understand that maths are just a language with arbitrary rules that work.
Maths can't lead to any truth because philosophically is unable to do so.

We established those rules and then follow that to its logical conclusion.
This fact can't be hard to some people that think about math as magic and completely separate from reality as some sort of way to fin God or something like that.

[Anonymous No.11809128]

>>11809111
I dont know what you think truth is, but the process of "following those rules to its logical conclusion" is the truth mathematicians are referring to. When you prove a theorem, you know it's true. Almost no one is delusion enough to think this is leading to some fictitious cosmic "truth" or something. The process of math is in contrast to the process of science, where you build a model, collect evidence which either disproves the model or tells you the model covers the cases you studied. In that scenario you are only ever certain of when the models break.

[Anonymous No.11809176]

>>11809128
Funny how theorems that do not have proofs can be true as well

[Anonymous No.11809208]

>>11809076
What's wrong with the bicommutant theorem?

[Anonymous No.11809680]

>>11809111
Well this was written in a time before everyone had a computer, or had to work with computers in some way, shape, or form. We now know that math, much like a computer program, must be entirely detached from whatever physical process it is modeling (or is being applied to). They obviously knew about computers (von neuman certainly did), and on some level they did understand all this, but they didn't have the same visceral images of purely logical systems in their faces all the time, like we have now. They didn't grow up spending half of their waking time in virtual worlds.

I think on some level, to them, the distinction between real and abstract was mostly a theoretical one, whereas for us it is second nature. We start learning this the moment we are old enough to put our fingers on a touchpad.

[Anonymous No.11811973]

>>11804784
Dividing by zero makes your calculations end up completely false which is a grand if not the only reason why it is unused. Sincerely, do you have schizophrenia?

[Anonymous No.11812165]

>>11811973
You literally have division by zero in some structures. Look up wheel theory

[Anonymous No.11812183]

>>11808187
Sabine called
Its for you

[Anonymous No.11812188]

>>11809111
This.
math is for describing the world around us
not proving anything

[Anonymous No.11812195]

>>11808859
>4chinners
>redditors
And dont forget modern academia.

[Anonymous No.11812413]

>>11809176
>theorems that do not have proofs
That's an oxymoron, you retarded cunt.

[Anonymous No.11812421]

>>11808949
self owning here this wasn’t his point he was merely asserting that in some instances mathematics can provide tools for empirical science well after being established as abstract fields of study. The discussion in the essay is clearly against abstract math for its own sake.

[Anonymous No.11812442]

>>11812421
>>11808949
>>11808859
You're all fucking idiots. There is not a single word in OP's paragraph which references his own opinion on the matter. Learn to read.

[Anonymous No.11812455]

>>11812442
I read the essay itself evil nigger faggot.

[Anonymous No.11812651]

>>11804721
I bet this guy thinks roads are safe spaces and cars are an oppressive agent of the patriarchy, shouldn't you be writing shit on signs instead of on /sci/?

[Anonymous No.11812666]

>>11812413
>That's an oxymoron
can you prove it? hint: you actually can

[Anonymous No.11812675]

What the fuck are numbers even? I feel like I'm attending a sermon each time I do math, like I'm just have to trust that 2+2 equals 4.

[Anonymous No.11813290]

>>11809208
I never said anything was wrong with it. It's my favourite theorem. What I was saying is it's evidence enough for me that hes a genius. Theres plethora more too

[Anonymous No.11813296]

>>11809176
What? A theorem without a proof isnt a theorem.

[Anonymous No.11814047]

>>11812413
>>11813296
>what are axioms?

[Anonymous No.11814309]

>>11814047
Loosely, an assumption, i.e. not a theorem.

[Anonymous No.11814417]

>>11814309
And theorems are made up of axioms?