[a / b / c / d / e / f / g / gif / h / hr / k / m / o / p / r / s / t / u / v / vg / vm / vmg / vr / vrpg / vst / w / wg] [i / ic] [r9k / s4s / vip / qa] [cm / hm / lgbt / y] [3 / aco / adv / an / bant / biz / cgl / ck / co / diy / fa / fit / gd / hc / his / int / jp / lit / mlp / mu / n / news / out / po / pol / pw / qst / sci / soc / sp / tg / toy / trv / tv / vp / vt / wsg / wsr / x / xs] [Settings] [Search] [Mobile] [Home]
Board
Settings Mobile Home
/lit/ - Literature

[Advertise on 4chan]


Thread archived.
You cannot reply anymore.


[Advertise on 4chan]


File: 612nkFVrbTL.jpg (96 KB, 863x1396)
96 KB
96 KB JPG
Apart from his flimsy grasp of Gödel's proof and hatred of set theory, how accurate are his remarks about the social character of mathematics? I'm getting a master's in Computer Science and decided to give philosophy of mathematics a go. I must say I'm somewhat sympathetic to his position, but find it hard to articulate why or how the rules of mathematical logic (or mathematics in general) can be something other than eternal Platonic truths. I'm not trying to be sassy or looking for a bone to pick on /lit/. If you can help clarify how rule following relates to logical consequence (in the Wittgensteinian sense) or if you have some recommended literature on the topic, please share.
>>
>>18700335
do you know the difference between analytic and synthetic
>>
>>18700335
Well, let me ask you this: if mathematical truths really are logically prior or temporally pre-existent, then why do they seem to be differentiated by scale of consequence and utility? For example, I can give you any number of proof lines (theorems) in algebra that are no more or less true than proof lines of group theory, but Cayley's Theorem is nonetheless clearly distinct in terms of consequences and scope. Is this difference, and the effect that the difference has on the human mind in terms of things like awe or excitement, due to some hierarchy of forms, or due to their recognized utility?

t. formalist btw
>>
>>18700335
>if you have some recommended literature on the topic
start with the greeks
>>
>>18700350
This is a categorical distinction, but its validity ought to be subjected to proof. Anyone who's read any philosophy is able to understand the distinction, but that it holds "against the world" is something that is being challenged by Wittgenstien. So merely asking this stupid rhetorical question isn't doing anything, you're just patting yourself in the back because you can conjure a stupid example like "all bachelors are single" or "triangles have three sides".

>>18700366
Good question. I think my reply would be that something like the Turing-Church thesis (sorry for bringing the subject to computability, I'm not very knowledgeable in algebra but I do get your point) seems to be important for matters that lie more in the sociological side of how the disciplines are conducted than in their inherent properties. When we discover something like Cayley's Theorem or the Church-Turing thesis we've simply found out about a 'space' in this mathematically prior aedifice that allows us, from our human perspective, to access newer stages and describe reality better.

Now, let me be very clear, I'm here acting as a bit of Devil's advocate. Formalism seems nice, but the question still remains. Okay, we've come up with a set of axioms and derived new theorems through the use of logically valid steps. But why do we feel that following a logical rule is somehow "stronger" than following an ethical or legal rule, just to give you an example? Why is it that we can't be formalistic about ethics?
>>
>>18700455
do you know the myth of the given. try not to get so angry
>>
>>18700455
Well to start I dont think formalism has right application outside of mathematics, that its actually a second decision that you make after you decide to do mathematics in the first place.

Mathematics, to me, is a matter of utility primarily. It is not solely about utility, but whatever truth or gnosis or whatever that we get from mathematics is secondary to its ultimate usefulness. The range of that usefulness is quite large, and has gotten far larger over time-- from grain storage and credit and architecture to how we characterize large sections of rational thought. However, that utility element is always there.

The big objection to formalism is, in essence, that formalism says we start nowhere and lead to nowhere, even though the experience of mathematics is very different from this-- it feels like discovery, we get surprised by things, etc etc. I would say in response that this gives formalism more scope than it intends to take for itself. The starting point is not a purely arbitrary set of axioms, it is a set of axioms formulated to solve some particular problem or other. When these are worked out consistently, we are frequently surprised, like with the incommensurability of the irrationals. But then we have to ask ourselves "ok, we constructed these objects with a particular application in mind-- could it be that our construction or our application had obscure consequences that we have only now discovered?" and the answer is typically yes.

So I would say that it is less that these truths are eternal and pre-existent, but they are obscured within the idea of the constructed object, and require analytical or synthetic work (in the Kantian sense) to fully enumerate.
>>
>>18700366
What is the underlying structure of reality that seems to "force" some formal systems to be so rich and useful and applicable while other formal systems just seem empty and useless? Can we just choose a formal system and decide it is "the math"? The axiom systems that we use seem to be used because they seem to get at some realities and not the other way around. While this is not fully explained I will remain somewhat a realist about mathematical truths.

Also, the idea that more useful and consequent mathematical truths are the most awe inspiring is absolute delusion. Go to any math department and show the kiddos there the proof that the probability of two positive integers picked at random being coprime is 6/(pi)*(pi) and then show them the demonstration of the mean value theorem. Guess which one will draw onionfaces of blind admiration, the least useful or important one.
>>
>>18700525
That's fair, it isnt a one to one correspondence between useful and awe inspiring.

I would say that the transcendental nature of human life makes some systems richer to us, but Im not sure if thats the same as saying that they are rich in a universal sense. I do need to ponder this some more.
>>
>>18700501
Have you read Carnap? In 'Empiricism, Semantics, and Ontology' he makes a similar point but about scientifical claims.

No matter how much my intuitions seem to lie withing the formalist concept, I still don't see how logical consequence can come about from a purely formalistic perception.
>>
>>18700476
Sure, I've read Sellars. I'm not really angry, I just think it's patronizing to simply ask 'do you know the difference between analytic and synthetic'? Everybody knows that.
>>
>>18700335
Check out other works in the philosophy of mathematics.
>>
>>18701012
Do you have any recommendations?
>>
>>18700416
Please fuck off you're not funny
>>
>>18700335
>his flimsy grasp of Gödel's proof
What makes you think his grasp of Godel's proof is flimsy?
>but find it hard to articulate why or how the rules of mathematical logic (or mathematics in general) can be something other than eternal Platonic truths.
The answer to this question is very simple. Just ask, if they're eternal Platonic truths, then they're truths about what? Sets? Define what sets are then. Nobody can give an actual definition, because there is none. It's all wishful thinking. Eternal truths are statements that describe the fact of some matter. Mathematicians don't even attempt to explain what the matter that these statements describe is. When questioned, they resort to saying it's all just meaningless formula shuffling. That's what mathematics reduces to. If you view mathematics as just formula shuffling, it becomes meaningful but the meaning is not the wishful dreamings in the mathematician's head, but rather the formulas and their shuffling.
>>18700366
Can you explain what you mean by "logically prior"? As for them being temporally prior, it's just the matter of the category of things it deals with. 2+2=4 doesn't become true the moment the first person asserted it. I am the first person in the universe to ever assert the true statement 21235555+ 1= 21235556. Yet it was true before I asserted it. It doesn't even make sense to talk of such a statement as "becoming" true after some time.
>>
>>18700335
Watch Wildberger for the redpill on current mathematics (spoiler: it's basically all fake). Here's a good video for a taste
https://www.youtube.com/watch?v=VUdFdlQNfpg
Then read Wildberger, Zeilberger, Kronecker, Weyl, Brouwer, Bishop, Robinson.
The way we see mathematics today is not at all the way it was viewed by the greats prior to 20th century.
>>
>>18700525
>What is the underlying structure of reality that seems to "force" some formal systems to be so rich and useful and applicable while other formal systems just seem empty and useless?
Structures seeming empty and useless is a fact of our brains, not a fact about the formal systems themselves. The obvious answer is that some formal systems are better suited to accommodate our intuitions about how the abstract world should work.
People ITT need to realize the truth about axiom systems. The meaning of the term axiom system changed drastically after the 20th century. For Euclid it meant a collection of facts that were so obvious that they didn't need proof. But now in set theory this is not the case at all. For contemporary mathematicians an axiom system is a collection of convenient facts that we assume. It's a really sad state of affairs. Such statements would typically need to be proven after they explained what the terms involved actually meant, but instead they turned the tables around, declared the whole of mathematics to be a meaningless symbol game and listed a collection of formulas that you're allowed to start with to play the game. Now there is a meaningful part of this, namely a certain subsection of arithmetic, in which terms can actually be defined carefully and facts about which proven starting from genuine axioms (in the sense of Euclid), i.e. intuitively obvious irreducible facts.
Now anons look at the horrible monstrosity that the mathematicians built and ask "Why is this mathematics and not any kind of formal system?". They have no good answer, because this is not mathematics. One day mathematicians will need to come to terms with this.
The simplest way to see that the whole thing is a pile of bullshit is simply come to a mathematician and ask him questions. Ask him to lay down the discipline from ground up, from the very foundations. You will see what a joke it actually is.
>>
>>18701315
>>18701251
>>18701218
Read
https://prl.ccs.neu.edu/img/sicm.pdf
https://math.stanford.edu/~feferman/papers/CH_is_Indefinite.pdf
https://www.researchgate.net/publication/280387313_Set_theory_Should_you_believe
https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/hersh90.pdf
https://conservancy.umn.edu/bitstream/handle/11299/185661/11_05Edwards.pdf?sequence=1
https://journals.openedition.org/philosophiascientiae/pdf/384

>The point of view on which I desagree with most mathematicians resides in the basic assertion that mathematics and the natural sciences — which have recently been separated by this name from the remaining sciences, the so-called sciences of the mind (Geisteswissenschaften) — must not only be free of contradiction, but must also result from experience and, what is even more essential, must dispose of a criterium by which one can decide, for each particular case, whether the presented concept is to subsume, or not, under the definition. A definition which does not achieve this, can be advocated by philosophers or logicians, but for us mathematicians, it is a bad nominal definition. It is worthless.
>>
>>18701218
>What makes you think his grasp of Godel's proof is flimsy?
This is a well documented issue, stemming from the fact that he never had access to the actual proof, just some informal prolegomena. I'm trying to find the source but I'm coming up short, so don't take my word for it.

>The answer to this question is very simple. Just ask, if they're eternal Platonic truths, then they're truths about what? Sets?

You're right, I shouldn't have said anything about truth. I accept your premise that truth must range over something (that is, truth must be about some object) and I agree with what you're saying in this paragraph. But let me rephrase what I said before a little bit: what I'm referring to here is that once I know that (P->Q) and I know that P, then I know that P. Modus Ponens seems to exhert a certain "force" on my mind that no other type of rule of scientific description seems to exhert. Modus ponens seems to be stronger than if God himself told me that killing is prohibited. And that's how I feel about the notion of mathematical proof, for example. When I see that nondeterministic finite automata can be reduced to deterministic finite automata, I am satisfied in a way that does not resemble anything else in my life. It seems to be a construction whose falsity is impossible to be claimed.
>>
>>18701315
>The meaning of the term axiom system changed drastically after the 20th century. For Euclid it meant a collection of facts that were so obvious that they didn't need proof. But now in set theory this is not the case at all. For contemporary mathematicians an axiom system is a collection of convenient facts that we assume. It's a really sad state of affairs. Such statements would typically need to be proven after they explained what the terms involved actually meant, but instead they turned the tables around, declared the whole of mathematics to be a meaningless symbol game and listed a collection of formulas that you're allowed to start with to play the game.

I'm the Anon who started the thread, and while I've been advocating for some kind of mathematical realism, I now want to play devil's advocate a bit here. We also don't have any way to satisfy ourselves of self-evident proofs. You're simply saying the foundations of mathematics do not rely on symbolic manipulation, but on a certain epistemological foundation. It smells of kantianism to me, but I don't want to assume anything at this point. So my question to you is this: what do we "get" by assuming mathematical truth is defined by how well mathematical theorems describe reality as 'intuited' from axioms (in your ancient sense)?
>>
>>18701218

>The answer to this question is very simple. Just ask, if they're eternal Platonic truths, then they're truths about what?

Mathematical truths are clearly true about every single object in the real world and also about every single conceivable object. There is no point in narrowing down them to a set of particular objects. If I say "17 is a prime number", I could be a pedantic weirdo and say instead "If I have 17 stuff then I could never divide this stuff into smaller sets, each of the same size, without leaving some remaining stuff", but what if I don't have the 17 stuff? Could it be divided in such a way? I does not seem so, every new set of objects I try to find to use as an example seems to somehow conform to the statement. What if it is imaginary stuff than I can't actualize? it still seems to conform to mathematical law and I can't imagine it being otherwise. I can talk about gigantic prime numbers that have absolutely no relation to physical reality and the math still seems to hold. So, by trying to make the statement about prime numbers more "real", I just made things more complicated, since I am needlessly trying to un-generalize a really really general statement. Why would I do that then? Because some autistic Austrian engineer said so? Fuck off, this is retarded.

>Can you explain what you mean by "logically prior"?

Yes. Logically prior means that in deductive reasoning, you can't get from your first principles to your current conclusion without moving through some other proposition. For example, if I say that triangles have 180 degrees I assumed the 5 postulates of Euclidian geometry, that are in this respect logically prior to the later deduction.
(You could have Googled that).

If you are willing to abandon the Wittgensteinian frame of mind I give you a puzzle: A mathematician may give you a true statement about some visual characteristic of the Mandelbrot Set, you can prove it with some ink and paper and that is it. The statement is clearly true but to verify it empirically it would mean to be acquainted with the entirety of the image of the set, what is impossible. So you have a mathematically true statement that is, in principle, unverifiable by empirical means, but still has bearing on the real world. Every better computer you get and every new rendition of the set would just confirm the mathematical assertion, but you would never exhaust the set and therefore never verify the statement.
>>
>>18701664
Just so you know, I'm the OP and you're not quoting me. So I don't think the guy is assuming a Wittgensteinian point of view. I'm interested in the debate, nonetheless.
>>
>>18701315

>People ITT need to realize the truth about axiom systems. The meaning of the term axiom system changed drastically after the 20th century. For Euclid it meant a collection of facts that were so obvious that they didn't need proof. But now in set theory this is not the case at all.

Agree 100%. But, you should never conflate "this axiom is not obviously true" with "this axiom is not true", most mathematicians would classify AC as a non-obvious true axiom.

>For contemporary mathematicians an axiom system is a collection of convenient facts that we assume.

No, ZFC was not assumed just because of convenience. In this system you can prove theorems that were assumed to be true, for example you can derive arithmetic and analysis from ZFC and all the theorems that were previously demonstrated in those fields. If an axiomatic system did not conform to Euclid's proof of the infinity of primes it would have never been accepted, same with other theorems like Cayley's theorem or quadratic reciprocity. If we had any system that was really useful but missed any of those important theorems then mathematicians would reject it because it does not seem "true". The idea that it is a "true" set of axioms is prior to being convenient, you could say "from the list of set of axioms that seemed to be true, ZFC was the more convenient, thus it was accepted". Now, seeming true and being obviously true are two different things, just because set theory is not obviously true does not mean it does not seem true to mathematicians, the fact that from those axioms you can derive a fuck ton of other theorems that were previously held as true was enough to convince most mathematicians that the axioms were valid.

>t's a really sad state of affairs. Such statements would typically need to be proven after they explained what the terms involved actually meant, but instead they turned the tables around, declared the whole of mathematics to be a meaningless symbol game and listed a collection of formulas that you're allowed to start with to play the game.

Fair. Mathematics departed from the real world some centuries ago. The objects that mathematicians conceive and create theorems about could never be visualized and mostly are only grasped linguistically. But they still have a bunch of rich inner qualities that are non-trivial and seem real.

https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups

Can I visualize wtf the Theorem about the classification of simple groups is about? Hell no, it is insane. Is this theorem just a bunch of symbolic mumbo jumbo? Does not seem so.
>>
Redpill me on HoTT.b Is it as good alternative to set theory? Is it just a meme?
>>
>>18701921
The people involved with this shit are serious, they are not meme schizos on the internet like most of the "revolutionary theorists" you find out there. But that is as far as I can tell about the stuff, really above my pay grade. Most mathematicians still use set theory tho, and they seem to be happy with that for now.
>>
File: lmao.png (112 KB, 1366x768)
112 KB
112 KB PNG
>>18701371
Yeah, I think I'm good; but on another note, remember to take your meds friend :)
>>
>>18701921
>>18701921
I can't make heads or tails of that shit. It's too fucking hard.
>>
>>18701662
>We also don't have any way to satisfy ourselves of self-evident proofs
What do you mean by this?
>You're simply saying the foundations of mathematics do not rely on symbolic manipulation
Should not rely, if we want those symbols to refer to something. If we're discussing symbol manipulations, then it's ok to talk about symbol manipulations, but when we're talking about something else we should actually explain what that something else is rather than say we're just pretending it's something else and that really it's all the same meaningless formula manipulation, as is done in the case of set theory and the subjects built around it.
>So my question to you is this: what do we "get" by assuming mathematical truth is defined by how well mathematical theorems describe reality as 'intuited' from axioms (in your ancient sense)?
We get mathematics that makes sense, rather than what we have now, which is a confused mess. Note that the reality that mathematics describes is allowed to be abstract (like natural numbers, which are not material objects), provided one is clear and precise in describing what they mean.
And as Kronecker emphasized, to me axioms are secondary, not primary. We don't sit around trying to construct our axioms and then go out to see what follows from them, that's simply not how mathematics is done in practice. We experiment, do interesting things, calculate examples, and then once we want to present the theory we've built up in a concise and logically complete way, or if we want to formalize our proofs for computers to check, we come up with axioms which allow us to capture what we've done in as parsimonious way as possible.
>>
>>18701664
>Mathematical truths are clearly true about every single object in the real world and also about every single conceivable object
Interesting. What does Euclid's infinitude of primes tell me about my shoes?
>What if it is imaginary stuff than I can't actualize?
Provided you have a clear conception and a description of what you're talking about, it's fine if you can't actualize it. I state some facts about every sequence of 1's and 0's of length 1000, even it's physically impossible for me to list all of them and thus definitively verify the fact empirically. But that doesn't matter, since I know precisely what is meant by a sequence of 1's and 0's of length 1000. For example, I can say that every such sequence has either 500 ones or 500 zeros, and know this to be true without writing down every such sequence and checking by hand.
>Logically prior means that in deductive reasoning, you can't get from your first principles to your current conclusion without moving through some other proposition
That can't be true, since you can get to every proposition you can get to through some other proposition. By that logic, nothing is logically prior.
Typically, logically prior is used as "X is logically prior to Y", but you omit Y, that's why I asked. Logically prior to what?
>For example, if I say that triangles have 180 degrees I assumed the 5 postulates of Euclidian geometry, that are in this respect logically prior to the later deduction.
If you say that triangles have 180 degrees that's not a deduction but an assertion. And I don't see how asserting such a thing assumes in any way Euclid's postulates.
It makes sense to talk about the exposition of some topic being logically prior to some other topic, but not of mathematical statements or objects themselves, or as you said truth. Are prime numbers logically prior to even numbers? Is Euclid's theorem of infinitude of primes logically prior to Hall's theorem in graph theory?
> A mathematician may give you a true statement about some visual characteristic of the Mandelbrot Set, you can prove it with some ink and paper and that is it. The statement is clearly true but to verify it empirically it would mean to be acquainted with the entirety of the image of the set, what is impossible.
I'm sorry but this doesn't make any sense. What kind of statement are we talking about? Visual characteristics exist in the mind, so if the statement is about a visual characteristic then I would be able to verify it by looking at it, no?
>>
>>18702366
Prime numbers have applications in cryptography you pseud.
>>
>>18701839
>But, you should never conflate "this axiom is not obviously true" with "this axiom is not true", most mathematicians would classify AC as a non-obvious true axiom.
Do YOU classify AC as a non-obviously true axiom? If so, how do you justify it?
>No, ZFC was not assumed just because of convenience
But it was. I suggest you read up on history of set theory and analysis. It was invented to justify the nonrigorous arguments in analysis. Rudin even gave a lecture on it
https://www.youtube.com/watch?v=hBcWRZMP6xs
It was too hard to justify what they were doing, so instead they decided they won't even try to and just listed a couple of convenient facts which now constitute the ZFC, without ever justifying them or even explaining what the terms involved mean.
>The idea that it is a "true" set of axioms is prior to being convenient, you could say "from the list of set of axioms that seemed to be true, ZFC was the more convenient, thus it was accepted"
Again, if they cared about truth, they would try really hard to be logically complete and properly explain what they mean by the terms involved. They did none of those things. They just assumed ZFC as a list of convenient facts and moved on, checking a bit that they don't result in an immediate contradictions like the previous attempts (part of the system being convenient).
As for the system being true by nature of the consequences, you should note that things like Cayley's theorem (at least for finite groups) and quadratic reciprocity follow from much much simpler systems. And in quadratic reciprocity the terms are clearly defined so you don't need to rely on any kind of axiomatic system to explain the theorem and prove it. What's more, a lot of the consequences of ZFC have been deemed as intuitively false, most notably Banach-Tarski type paradoxes and the well-ordering principle. This seems to go against the view that ZFC was justified by nature of its consequences. Most likely it was justified by its convenience to analysis and the fact that we didn't immediately find it contradictory.
>Mathematics departed from the real world some centuries ago
One century ago, to be precise.
>The objects that mathematicians conceive and create theorems about could never be visualized and mostly are only grasped linguistically
I have no problem with that. As long as you can grasp what it is, you don't need to be able to visualize it. Classification of finite simple groups I consider a meaningful statement about definite (albeit abstract) things. This is not the case with most things in set theory and the subjects heavily relying on it like analysis, probability and geometry.
According to modern mathematics, the continuum hypothesis (CH) is as meaningful of a statement as classification of FSG's. But this is obviously false. We now know (due to results from Godel, Cohen, Solovay, Levy) CH is basically meaningless. It's ridiculous to place it in the same league as for example the twin prime conjecture.
>>
>>18702375
Yes, I know. What's your point? And why do you call me a pseud?
>>
>>18701959
>The people involved with this shit are serious, they are not meme schizos
Voevodsky gave a lecture about the serious possibility that PA is inconsistent.
https://www.youtube.com/watch?v=O45LaFsaqMA
According to the standards of the arrogant uneducated midwit infinitists in this thread like >>18702046
, he would be a schizo.
>>
>>18701068
Cantor, Frege, Hilbert, Godel, Brouwer, Field, etc.
>>
mathematicians have the best social character
>>
>>18702925
Brouwer is a shitty mentalist. Sorry, but intuitionism as argued by Heyting and Brouwer is for pseuds.
>>
>>18703046
You need to start liking your enemies, or you won't make it far in philosophy.
>>
>>18700335
That book is absolute schizolit
>>
>>18700335
Currently I am reading Everet W. Beth's book "The Foundations of Mathematics", which deals with this topic from not only a rigorous mathematical perspective but also the history of the science of logic and how that affected our view of logic as a whole, in addition to various different logics. I'm only on page 126 out of 600 but it is a very good read. If you can find it for a reasonable price I'd recommend it. Good thread BTW OP. The Foundations of Logic/Mathematics/Computer Science are the most underrated topics of study imo and the most important for understanding the universe as a whole.
>>
>>18700335
I kinda quite agree with the problem of mathematician insecure about "hidden" contradiction. That is not the real issue when it comes to normal life mathematics, and we can just remove it by adjusting some axioms.
Although I agree on this, the book is still a fuckton of schizo rambling
>>
>>18703170
>Good thread BTW OP.
>The Foundations of Logic/Mathematics/Computer Science are the most important for understanding the universe

Bro why
Wittgenstein's whole point is that thinking foundation of mathematics as very important topic of mathematics(or universe) is in nature very wrong
>>
>>18700335
Why the fuck Wittgenstein rejects a lot of concept of infinity? He agreed on countable infinity exists but after aleph_1 he rejects it and that makes construction of real number impossible
>>
>>18700335
mathematics has no "social" character and wittgenstein doesnt say that either. mathematics is the ultimate reality, the being, the fundamental principle to which all reality , including ideas and emotions, can broken down.
>>
>>18703240
No u
>>
>>18703243
cope, mathlet.
>>
>>18703240
Can you break down what you just said into mathematics for me? Thanks.
>>
File: download (13).jpg (8 KB, 183x275)
8 KB
8 KB JPG
>>18703224
The real numbers never made sense. They don't even have a foundation lmao.
>>
>>18700335
>decided to give philosophy of mathematics a go.
Bro... nobody love Wittgenstein at that branch..
>>
>>18703250
I am literally a master in mathematics
>>
>>18703251
not interested in doning that.
>>
>>18702366
>Interesting. What does Euclid's infinitude of primes tell me about my shoes?
This is dumb, will not respond.

Provided you have a clear conception and a description of what you're talking about, it's fine if you can't actualize it. I state some facts about every sequence of 1's and 0's of length 1000, even it's physically impossible for me to list all of them and thus definitively verify the fact empirically. But that doesn't matter, since I know precisely what is meant by a sequence of 1's and 0's of length 1000. For example, I can say that every such sequence has either 500 ones or 500 zeros, and know this to be true without writing down every such sequence and checking by hand.
Sequences of 0s and 1s are ok then, because we clearly defined the objects of study. But you don't think most modern mathematics is of such type, most mathematics is thus concerned with ill-defined objects. The general praxis in mathematics is to accept anything defined in set-theoretical language as a valid object of study, sets being used as the basis of everything since they are incredibly general. Sets are ill-defined by design, since we need some very basic building block, the same way a language needs letters and sounds. I would never lose sleep over the fact that no poet can give me a good definition of the letter A, but I get that some people may have problem with the set thing, it just feels "to loose". I will assume this is your problem then, the set thing. And more specifically modern set theory.

>That can't be true, since you can get to every proposition you can get to through some other proposition. By that logic, nothing is logically prior.
Typically, logically prior is used as "X is logically prior to Y", but you omit Y, that's why I asked. Logically prior to what?
Lots mathematical proofs use the law of excluded middle, so we say that to prove X we assumed the law of excluded middle that is thus logically prior to the theorem. This is the use of the term. You could use the term in a very vague way like "the law of non-contradiction is prior to all of mathematics" and I see no problem with it. I also see no problem stating "the law of non-contradiction is a logic prior" since for any math or logic you try to create you'll need this law. But at this point I am just being a pedantic asshole, I get what you said, it is fair.
>>
>>18703275
>I am literally a master in mathematics
>>
>>18703224
Because he was against mathematics needing a foundation whatsoever, so he goes against set theory which basically started this craze.
>>
>>18703240
>mathematics has no "social" character and wittgenstein doesnt say that either
Lol what a fucking idiot
>>
>>18702366
If you say that triangles have 180 degrees that's not a deduction but an assertion. And I don't see how asserting such a thing assumes in any way Euclid's postulates.
The assertion only becomes true when it is given a context, the Euclidian postulates determine a given geometry in which the statement is true. It is like saying "the bird is blue". For this assertion to be true you will need lots of priors "there are birds", "it is possible to a bird to be blue", etc...

>It makes sense to talk about the exposition of some topic being logically prior to some other topic, but not of mathematical statements or objects themselves, or as you said truth. Are prime numbers logically prior to even numbers? Is Euclid's theorem of infinitude of primes logically prior to Hall's theorem in graph theory?
Clearly you should not use logically prior in those contexts. But some stuff seem to be logically prior to all of this: "the law of non-contradiction is logically prior to Euclid's theorem of infinitude of primes and Hall's marriage theorem. Just because not all theorems A and B can be put in a relation ARB of logical priority it does not mean that the term is non-sense, this would be a fallacy.

ignore the Mandelbrot example, it is a refutation of verificationism, not of your position, I misunderstood you there.
>>
>>18702467
>Do YOU classify AC as a non-obviously true axiom? If so, how do you justify it?
Yes. Find me a cartesian product of two non-empty sets that is an empty set and I'll change my mind.

> It was invented to justify the nonrigorous arguments in analysis. Rudin even gave a lecture on it
https://www.youtube.com/watch?v=hBcWRZMP6xs [Embed]
This seems cool, will watch.

>hey just assumed ZFC as a list of convenient facts and moved on
That is not all there is to ZFC. It checked every single theorem previously held as true by prior mathematics. It seemed to have no contradictions. It is really general, so as to allow for the construction of a bunch of new mathematical objects that can be studied.

>you should note that things like Cayley's theorem (at least for finite groups) and quadratic reciprocity follow from much much simpler systems. And in quadratic reciprocity the terms are clearly defined so you don't need to rely on any kind of axiomatic system to explain the theorem and prove it.
Fair. I agree with this. ZFC is overkill for those more simple theorems. But they still would rely on axioms, albeit axioms on the sense that the Greeks used for this term(obvious truths no one would doubt) instead of the more modern notion.

>What's more, a lot of the consequences of ZFC have been deemed as intuitively false, most notably Banach-Tarski type paradoxes and the well-ordering principle.
There is no paradox, you can just multiply balls like that, one day we will be able to use this phenomena to our advantage, we are just to primitive for that right now.
>>
>>18703205
Wittgenstein was wrong about everything tho.
>>
>>18703205
Who cares what Wittgenstein says about mathematics? He's some pseud that can't understand basic the Dedekind construction of the reals for no good reason. Frankly, I don't think people that don't understand math should be trying to investigate its foundations rigorously.

>>18703258
>What are Dedekind cuts?
>>
>>18703440
>YoU cAn JuSt MuLtIpLy BaLlS lIkE tHaT
Give me a physical proof that how is paradoxical decomposition possible such as Banach-Tarski and many of Hausdorff Paradoxes
>>
>>18703565
Does it seriously matter if mathematics corresponds exactly to physical reality? Why?

The whole point is to form precise objects of thought and discover their relations. Probably not all of those objects and relations exist in the physical world. Who cares?
>>
>>18703576
Wait
I thought you are other guy
Nevermind
>>
>>18703565
who the fuck ever said that the balls are "physical"? Holy fuck dude, they are just hypothetical balls. I can imagine in my head 1 ball being multiplied into two balls out of nothing. I was just joking there. The point is that the Banach-Tarski theorem never postulates that the balls are physical in any way so to think there is a paradox there is to simply insert a context that isn't there to begin with.
>>
>>18703594
Mathematics deals with hypothetical objects all the time. It could easily be defined as "the science of hypothetical objects". Why would Banach-Tarski's theorem bother a sane person is beyond me, if you further assumed that it says stuff about physical balls then the problem is with you.
>>
>>18703613
btw, I know that a lot of weirdos in academia are spooked by Banach-Tarski and that a lot of brainlets call it a paradox. But I hope my fellow /lit/izens don't fall for this cheap sophistry against our beautiful Axiom of Choice.
>>
>>18703626
Quick question-- precisely what sort of metric space does B-T assume this is being worked out in? The wikipedia article says three dimensions, but that seems less plausible if its R^3 with the distance norm.
>>
>>18703594
Bro I was joking too
Anyway, "cartesian product of two non-empty sets that is an empty set" would be the most ridiculous thing of AC. I saw very ridiculous conclusion that includes size of two sets, that was equivalent to CH.
Do you know the most ridiculous conclusion of we reject axiom of infinity?
>>
>>18703649
R^3 defined as usual allows you to get 3 balls from one. What should obviously be the case, since I can imagine in my head a fucking red ball being multiplied into 3 out of nothing and this imagining of mine does not seem to have collapsed my brain into a black hole.

>>18703659
No idea. But is it anything drastic?
>>
>>18703626
I don't want to say all mathematician who questions AC weirdos.
There's a lot of example but those set theorists who did Godel's program should be mentioned a lot. Although they failed.



Delete Post: [File Only] Style:
[Disable Mobile View / Use Desktop Site]

[Enable Mobile View / Use Mobile Site]

All trademarks and copyrights on this page are owned by their respective parties. Images uploaded are the responsibility of the Poster. Comments are owned by the Poster.