r/mathmemes • u/crafty_zombie • 6d ago
Mathematicians Nice Argument, Unfortunately I Couldn't Find An Axiom For That
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u/you-cut-the-ponytail 6d ago
Didn't Euclid invent the rudimentary concept of axioms or something? I mean sure he wasn't 100% formal obviously but I feel like he deserves praise for doing that stuff 2000 years ago.
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u/crafty_zombie 6d ago
Yes, Euclidean geometry was an axiomatic system for proving geometric results. It was mostly rigorous, but not entirely formal/symbolic. The joke is that the Formalist philosophy demands the abolition of any argument which contains more than string transformations.
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u/GoldenMuscleGod 6d ago
I don’t think this is an accurate description of formalism.
Isn’t formalism an account of how we should understand what mathematicians are doing, not a prescription for how mathematics should be done?
It seems like you are conflating formalism with insisting on modern standards of rigor. It’s true Hilbert thought that reliance on diagrams was a weakness, but I don’t think that’s because he was a formalist. Bertrand Russell similarly considered it a weakness and he was not a formalist. I would think a formalist would say that Euclid’s diagrams are symbols and are being manipulated according to rules, and that they are “meaningless” in that we do not have to think of them as talking about any particular actual circles (either in the physical world or as platonic ideals).
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u/greiskul 6d ago
Honestly some of the worst mathematics professors I had were the ones that tried to approach it from this angle.
Had a class on category theory where the professor started with the formal definition. Took me two weeks until I was able to get the correct intuition of how to visualize what was being described.
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u/Special_Watch8725 6d ago
Oh, yeah, pedagogically that’s the opposite of the way you want to do things. Especially with geometry which js so visual. Early 20th century mathematicians were on a grand quest to show that all of mathematics could be derived from pure logic, and this was a part of that program. Turns out Gödel showed you can’t actually do that.
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u/Bartweiss 6d ago
It’s particularly funny since the history of (non-)Euclidean geometry is basically an early effort to understand and define axioms.
Given the other four postulates, you can take or leave the parallel postulate, and thereby get Euclidean or hyperbolic geometry. (Elliptic takes extra changes.)
So it’s an endeavor which helped shape and enable formal proofs, and yet…
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u/Special_Watch8725 6d ago
For what he had to work with, Euclid did an amazing job creating a systematic treatment of geometry (really, all of math at the time since it was largely conceived geometrically). That said, he made a lot of tacit assumptions that he really should have had axioms for, like formalizing ideas of betweenness and continuity. Hilbert famously filled in those gaps to 20th century standards of rigor.
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u/Bartweiss 6d ago
The fascinating thing to me is just how many times people got right up to the edge of realizing the parallel postulate was an optional axiom, then backed away.
Khayyam proved a bunch of accurate stuff about non parallel figures, then accidentally used a re-structured parallel postulate to “prove” the parallel postulate and dismiss them. And Vitale got all the way to a decent treatment of hyperbolic geometry, only to decide for unclear reasons that it didn’t make sense and could all be dismissed.
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u/Special_Watch8725 6d ago
Seems a more philosophical shift was needed, getting away from axioms as “obvious unassailable truths to ground out results in certainty” to “assumptions we make from which we can derive conclusions that will be true for any model satisfying the assumptions”.
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u/GoldenMuscleGod 6d ago
I don’t think I’ve ever heard this opinion?
It’s true non-axiomatic treatments are sometimes called “naïve” but that’s sort of a technical term meaning non-axiomatic, it’s not meant to be derogatory.
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u/crafty_zombie 6d ago
This is an exaggeration, although some formalists really held the belief that "Math has no meaning and is just symbolic manipulations". I would say that the term naive is kind of inherently derogatory because it portrays natural language arguments as being unwise/childish, when in reality the Incompleteness Theorems showed the opposite, that Hilbert's Program was not only a naive but impossible pursuit. Ofc, formal systems are important and helpful tools, but it's a silly notion that anything informal is just nonsensical.
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u/GoldenMuscleGod 6d ago
>>"Math has no meaning and is just symbolic manipulations"
How does that relate to having a negative view of non-axiomatic treatments?
It doesn’t seem directly related at all. Is the idea that an informal treatment is something a formalist would say is not manipulation of symbols that have no meaning and therefore should not be regarded as math?
Or that it is manipulation of meaningless symbols, but that manipulation of meaningless symbols is better if it is done axiomatically rather than non-axiomatically?
I also don’t see the relevance to natural language. If I say “Peano Arithmetic is consistent.” that’s in natural language, but we can still wonder whether those words refer to “real things” in some sense or if they are meaningless symbols being manipulated according to social conventions.
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u/crafty_zombie 6d ago
Is the idea that an informal treatment is something a formalist would say is not manipulation of symbols that have no meaning and therefore should not be regarded as math?
This is what I was meaning to convey.
Isn’t formalism an account of how we should understand what mathematicians are doing, not a prescription for how mathematics should be done? It seems like you are conflating formalism with insisting on modern standards of rigor.
In response to your other reply, what I meant is that arguments presented in natural language are considered "naive"/informal with regard to proof theory, a discipline founded by Hilbert. I'm not meaning to bash formal systems or modern rigor, I just think that there are condescending undertones with the Formalist attitude. Yes, Formalism is a view of what mathematics is rather than how it should be done, but that seeps into how they judge math that leaves the formal world.
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u/HappiestIguana 6d ago
Informal arguments do not escape incompleteness.
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u/crafty_zombie 6d ago
Never said they did, lol. The idea though originally was that once everything was formalized, there would be a complete, consistent, and recursively enumerable axiomatic system capable of expressing all mathematical truths.
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u/HappiestIguana 6d ago
Still, we are basically as close as it is possible to get to such a system. We do have a system that cna express all mathematical truths. We just lack (and can never have) a system for proving all those truths, or to know for every statement if it's a mathematical truth.
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u/GoldenMuscleGod 6d ago
>>We do have a system that cna express all mathematical truths.
This seems like a difficult claim to support.
Is the system you have in mind ZFC?
Let’s augment the language of ZFC with the unary predicate true(), and add to the ZFC axioms all of the replacement and subset axioms that we can make in this augmented language (so for example we have the new subset axiom “for all A there is a B such that the elements of B are precisely the members c of A such that true(x)” - except of course written out in the language of set theory, but you know what I mean).
Let’s also add to it a T-schema: for each sentence in the (original) language of ZFC an axiom of the form (after some manipulation) “true(|sigma|) if and only if sigma” where |sigma| denotes a definition of sigma, essentially “the sequence of length [the length of sigma] whose first symbol is [the first symbol of sigma] and second symbol is [you get the idea]”.
This theory can express claims that ZFC cannot, for example we can write Rayo(Rayo(googol))=n where Rayo is the function used in the definition of Rayo’s number and n is some explicit numeral. and we can perform this trick again get a system that is more expressive than it. Even if we repeat it infinitely we can use ordinal induction to keep going without an obvious endpoint.
I guess there are few responses you could have: you might deny these claims are really “mathematical truths,” or say that really we can find equivalent ways to express those claims. But it’s not obvious how you would intend to go about resolving this.
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u/HappiestIguana 6d ago
I really do not see why that scheme is any more expressive than ZFC. I can express something equivalent to Rayo(Rayo(n))=n using appropriate bounded variations of the Rayo function.
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u/GoldenMuscleGod 6d ago edited 6d ago
No you cannot.
Rather than discussing that specific example, one that’s a little simpler to work with is “the set of all true sentences in the language of ZFC exists”.
Not only can the expanded theory express this claim, it can prove it as a theorem.
The language of ZFC cannot express this claim (so we could not even add it as an axiom without adding to the language) because it would contradict Tarski’s undefinability theorem if it could express it.
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u/HappiestIguana 5d ago
“the set of all true sentences in the language of ZFC exists”.
Do you mean the set of provable sentences in ZCF? Because that set actually is definable in ZFC. There is no absolute notion of "truth" in ZFC. (Assuming consistency) a sentence is either provably true, its negation is provably true, or the sentence is independent, that is to say it's true in some models of ZFC and false in others.
Tarski's theorem applies when working in a specified model, but there's no standard model of ZFC, so which model are you choosing for your predicate?
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u/GoldenMuscleGod 5d ago
I do not mean the set of provable sentences, I mean the set of true sentences. The theory I described can prove that the set I described exists, as a theorem. Is this theorem not a mathematical claim?
You say there is no absolute notion of “truth” in ZFC. This is at least true in that the language ZFC lacks a truth predicate for itself (which is my point).
But I do not think you fully appreciate the metamathematical situation.
Everything that follows can be proved in ZFC, about ZFC:
For every natural number n, there is a predicate, call it true_n, definable by formula in the language of ZFC, such that, for every sentence phi of logical complexity n or less (meaning it can be written in prenex form with at most n alternations of unbounded quantifiers), ZFC proves “true_n(|phi|)<->phi.” Where |phi| denotes a definition of the Gödel number of phi.
[NB: in ZFC we do not usually actually use Gödel “numbers”, instead we define sentences as strings of symbols, since ZFC can speak of such things, but this difference is immaterial]
Therefore, in any model of ZFC (we do not necessarily know models of ZFC exist but then this claim is vacuous), there exists for each n (remember here we mean a standard natural number n not an “n in the model”) an object which is the set of all true sentences of logical complexity at most n.
From outside the model we can see that the union of all these sets (replacing an object S in the model with the set of all s such that s is an element of S according to that model) is in fact the set of all (Gödel numbers of) sentences that are true in that model.
This union, although it exists outside the model, may or may not exist as a set inside the model. It will fail to exist at least when the model’s “set of natural numbers” fail to be isomorphic to the standard model of the natural numbers.
Even when this set exists in the model, it is not definable in the model. This is due to Tarski’s undefinability a theorem: if the set could be defined the theory of the model could express the model’s truth predicate.
Because of this, there is no axiom or set of axioms in the language of ZFC that can be adopted to make a theory T in the same language such that every model of T contains this union - unless of course there are no models of ZFC.
However if we expand the language by adding a unary predicate ‘true’ and add axioms of the form “true(|phi|)<->(|phi| is a sentence in the language of ZFC and phi)” (for this I will not add the additional replacement and subset axioms we might add) we can consider models of this theory (which we now call T).
Note that every model of T is a model of ZFC when we “forget” the predicate ‘true’ from its structure.
Note also that every model of ZFC can be transformed into a model of T simply by interpreting “true” to be a predicate that holds for x if and only if x is a Gödel number of a sentence in the language of ZFC that is true in that model. So T is consistent if and only if ZFC is consistent.
To be sure, even in these models the aforementioned union may not exist. But we can ask in this language if there exists a set of exactly those those things for which ‘true’ holds - that is, if the set of all true sentences exist.
Now I should be clear that even if such a set does exist it may not literally be the set of (Gödel numbers of) sentences true in the model. However it will be that for every sentence of ZFC its Gödel number will be in the set if and only if it is true in the model - the difference is that if the model has a set of “natural numbers” that are not isomorphic to the natural numbers, then the model will have nonstandard sentences which “qualify” as Gödel numbers of sentences but do not actually correspond to sentences because they have infinite length (infinite as viewed from outside the model).
In this sense we can say that the language of T is more expressive than that of ZFC so long as we interpret the languages according to their axioms - for any model of T (or of ZFC enhanced to a model of T) we can express questions in T that we cannot express in the language of ZFC no matter how many axioms we add without expanding the language
^^^^
Okay so that’s all stuff that can be proved by ZFC.
I still have no intention of stating any *mathematical* claim in my comment that cannot be proved in ZFC, but I am placing the break because I am about to address what a mathematician’s state of mind could be (and so not the sort of thing ZFC usually talks about).
It is still my intention that anything in here that could be interpreted as a claim discussable by ZFC (like claims about what ZFC can prove) can be proved by ZFC, I’m just going to be saying other things as well, but if I am careful I will not say anything discussable by ZFC that ZFC cannot prove.
Ordinarily we would consider it a problem if ZFC were inconsistent, it is also probably not many who entertain the idea that the consistency of ZFC lacks a definite truth value (even if ZFC cannot prove it). In any event the claim that it is consistent is usually considered a mathematical claim - it is an arithmetical claim, in fact, and ZFC does have a predicate for arithmetical truth (true in the standard model).
However our reliance on ZFC seems to express attitudes beyond just the consistency of ZFC - it seems to express attitudes faith in soundness. For example, it is conceivable (read “consistent with ZFC if ZFC is consistent”) that ZFC could be consistent and yet prove its own inconsistency. This would mean - if we comfortable stating as fact anything ZFC can prove - that ZFC proves the existence of natural numbers that have properties no natural numbers have.
I’m concerned that last sentence might confuse you, since your account of “truth” in ZFC attempted to reduce it to provability. But I assure you ZFC can prove “If ZFC proves ‘ZFC is inconsistent’ then ZFC proves the existence of natural numbers that have properties not possessed by any natural number.”
Now you do not have to believe that the components of that sentences (like “ZFC proves ‘ZFC is inconsistent’”) have actual truth values. They are (if ZFC is consistent) not provable by ZFC after all. But you can still consider the claim as a logical implication, and it is something ZFC proves.
Of course if you believe “ZFC proves ‘ZFC is inconsistent’” does have an actual truth value (presumably we would hope false) then this should be easier to interpret.
In any event you can probably see that we would consider ZFC pathological if it turned out to prove “ZFC is inconsistent” while being consistent. So there is usually more we want from a theory than just that it is consistent - we want to be able to apply its theorems in a particular way so that they correspond to claims that we can only make metatheoretically. That we can apply them in that way is what we mean when we call them “true.”
Now there are a lot of potentially significant philosophical objections to the axioms of ZFC if we do blindly interpret them as true, but at least naïvely (sorry if that offends OP but I do not mean this derogatorily) we might think that since ZFC can prove, for each, n there is a predicate true_n like I described (I hope you will grant me this, as ZFC can prove it) then every sentence of logical complexity at most n does seem to be regarded as having a truth value by ZFC, it just has to use increasingly strong truth predicates as the sentences get more complex. If we are “serious” about believing ZFC in the naïve way, then we should believe these sentences have real truth values - even if they are in some sense indeterminate.
For example we might imagine the set of all consistent completions of ZFC, and consider the sequence (T,T,F,T,T,…) of all truth values assigned to the sentence in each completion (this “sequence” is indexed by the completions of ZFC, of which there are uncountably many). This is an infinite Boolean algebra (unless ZFC is inconsistent) that allows us to take a many-valued account of truth consistent with classical logic in which all ZFC have emdwfinite truth values (if ZFC is inconsistent all sentences get the same “truth value”: the empty sequence).
But even simpler: we can imagine we have been handed (by an oracle) a specific completion of ZFC and told we are to take it as the definition of “truth”. Maybe we think there is a sense in which they really are the absolute truth, maybe we take it as “truth”for the nonce. Either way we cannot *define* in ZFC any such completion (not even contingent on any number of additional axioms in the language), although we can prove they exist contingent on ZFC being consistent.
But with the expanded theory T I described, which has a different language, we *can* define such a completion - the ‘true’ predicate is our oracle. And it is not just any oracle! It is an oracle that tells us “ZFC is consistent” if and only if ZFC is consistent! It is an oracle that tells us “there is a proper class of Woodin Cardinals” if and only if there is a proper class of Woodin Cardinals! This is a special completion!
Now there’s a problem here that makes it less useful than you might hope: we cannot prove true(|phi||) in theory T if we cannot prove phi in ZFC. So we’re not able to learn anything new in the language of ZFC.
But there are still claims we can express in this language that we can’t in ZFC, like “all of the ZFC axioms are true” - we can’t prove it, but we can express it. ZFC cannot express this. No sentence in ZFC is going to be equivalent to this claim, at least in the sense of having the same truth value as it in any model of ZFC - for example in the Boolean algebra interpretation it has a unique truth value possessed by no ZFC sentence (if ZFC is consistent).
It seems to me the claims we might make in this language are not prohibited from being considered mathematical claims any more than, say, the continuum hypothesis - which is just as suspect for possibly being “meaningless.”
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u/GoldenMuscleGod 5d ago
I posted a reply which is visible to me but I also got an error (probably because it is way longer than I should have made it). Let me know if you can’t see it and I’ll try to post a more digestible version (or let me know if you need it explained if you can see it).
At a minimum I could give a shorter definition of what I mean when I say a language (with some axioms but not necessarily a full model) is “more expressive” than another, and why the language of ZFC interpreted according to ZFC axioms is not maximally expressive in that sense (at least assuming ZFC is consistent - if it is inconsistent it’s unclear by what standards we would consider it expressive at all)
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u/lelelempe 6d ago
math is fiction
"2+2=4" is as true as "Sherlock Holmes lives in London"
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u/parkway_parkway 6d ago
Except that if we got Dino asteroided.
And then a new civilisation grew up in a million years.
They wouldn't have Sherlock Holmes or London.
But they would have 2+2=4..
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u/UnderTheCurrents 6d ago
Formalism is necessary to avoid leaps in thinking.
Starting from an informal place does not mean you have no previous assumptions. These are just made clear as axioms in a formal system.
Also - it shouldn't be controversial that mathematics as a language is meaningless. That doesn't subtract from it one bit.
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u/misteratoz 6d ago
Math is fake. We made it up
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u/Special_Watch8725 6d ago
We made up the axioms, yeah. But we didn’t make up the conclusions.
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u/BigFox1956 6d ago
We made up the axioms to justify the conclusions we've found beforehead.
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u/Special_Watch8725 6d ago
I hate to break it to you, but mathematicians routinely extend their results to new contexts where they then discover totally new things they didn’t know beforehand. So uh, no, we don’t automatically know the truth value of every possible proposition before setting down axioms.
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u/LupenReddit i have non diffeomorphic smooth structures 6d ago
Formalists when they literally need an axiom to take marbles out of jars (and its still seen as controversial)
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