WayfarerJune 02, 2021 at 02:598450 views148 comments
This came up in my youtube feed last week. It's still current, and it seems pretty good to me, but I'm wondering what our resident philosophers of math make of it.
A philosopher of mathematics might consider this a serious issue. However, many mathematicians ignore it. But who knows about the future?
Remember, math is not like constructing a skyscraper, putting in a firm foundation before building the edifice. In math the edifice was largely in place, and the foundations were added afterwards.
It oversimplifies to the point of being terribly misleading. One glaring mistake is not recognizing that undecidability follows immediately from incompleteness.
And the visual gimmicks and props are not helpful.
Reply to TonesInDeepFreeze Well, it is a youtube video, so visual comments and props are the requirements of the media. Sure, a maths textbook would be more accurate, but it will remained forever closed to many, myself included.
//one of the comments: "There is something genuinely reassuring in knowing that nearly 5 million persons have watched this video in just over 1 week..."//
The philosophical implications of Godel's theorems are usually very overblown.
It basically just boils down to how any language capable of formulating e.g. a proof of arithmetic is also capable of formulating self-referential sentences to which there cannot be assigned only one or the other boolean truth value: they must be assigned by the language either neither truth value (so the language is incomplete) or else both truth values (so the language is inconsistent).
It's only in a meta-language, being used to discuss that language as an object itself, that we can say that some such statements (in the object language, about the object language) are true; but in that meta-language we can also prove that those statements are true. The meta-language will itself also be able to formulate statements about itself to which it cannot consistently assign any single truth value, but those statements in turn can only be called "true" in a meta-meta-language, which will also be able to prove those statements (in the first meta-language, about the first meta-language).
There's never a statement in any given language that is both definitely true according to the rules of that language and also not provable in that language, because to be definitely true according to the rules of a language just is to be provable in that language.
If we were to take away anything of philosophical import from Godel, it would be that we should be using either a paraconsistent logic (where statements can be both true and false without explosion) or an intuitionist logic (where statements can be neither true nor false).
There's never a statement in any given language that is both definitely true according to the rules of that language and also not provable in that language, because to be definitely true according to the rules of a language just is to be provable in that language.
What you say just seems wrong for the simple reason that the truth of statements that are not provable cannot be ruled out; we don't know if they are true or not. In other words there can be truths which we cannot determine to be such, or at least it cannot be ruled out that there are.
Thank you. I thought it rather a stylish presentation. I like that guy’s channel.
It is a good channel. Also I think he lives somewhere near me these days because I keep seeing familiar places in the backgrounds of his videos... like in this one, he appears to be hiking near Camino Cielo above Santa Barbara, and in another recent one about soft robots he was at UCSB.
What you say just seems wrong for the simple reason that the truth of statements that are not provable cannot be ruled out; we don't know if they are true or not. In other words there can be truths which we cannot determine to be such, or at least it cannot be ruled out that there are.
I'm not saying that unprovable statements are definitely false, so this is a non-sequitur.
Yeah, I particularly liked his use of Godel flash cards, as it were, to get the point across. Almost no pop explanations ever mention Godel numbers in their presentation of the results.
visual comments and props are the requirements of the media.
Visual gimmicks and props are not required. One can give a talk orally and with supporting text and/or non-gimmicky visuals.
And I don't even object to visuals and props, except my point is that the ones in that video are stupid. The video is a collection of baubles.
And the video is a shallow attempt at entertainment while being not very informative, not clear even as a simplification, and egregiously misleading at certain points.
An example of the stupidity is spending time on Godel's inanition. It has nothing to do with the subject. And the video includes several seconds holding on a cartoon representation of a plate of food, suggesting that is what Godel passed up in refusing to eat. As if we need to be shown what a plate of food looks like. How childishly stupid.
TonesInDeepFreezeJune 02, 2021 at 15:58#5458110 likes
It basically just boils down to how any language capable of formulating e.g. a proof of arithmetic is also capable of formulating self-referential sentences to which there cannot be assigned only one or the other boolean truth value: they must be assigned by the language either neither truth value (so the language is incomplete) or else both truth values (so the language is inconsistent).
That is not at all a reasonable summary of Godel's theorem. Just to start: languages are not what are complete or incomplete, but rather theories are complete or incomplete. Also, it is crucial to understand that Godel's theorem has a purely syntactic part that does not require semantic notions of truth and falsehood.
If we were to take away anything of philosophical import from Godel, it would be that we should be using either a paraconsistent logic (where statements can be both true and false without explosion) or an intuitionist logic (where statements can be neither true nor false).
Paraconsistency is a way out of incompleteness, but not on account of considerations of truth and falsehood but because contradictions are allowed. Again, Godel's theorem has a purely syntactical aspect as well as its semantical implications too.
Intuitionist formal logic is a proper subset of classical formal logic. Intuitionist logic is not a way out of incompleteness.
TonesInDeepFreezeJune 02, 2021 at 16:29#5458260 likes
For a video such as this, the very first words should be:
"I'm going to give you an extremely simplified version of some very complicated mathematics. These simplifications gloss over crucial technical details; thus the simplifications may be misleading if one does not at some point go on to understand the actual mathematics. So, we must be extremely careful not to extrapolate philosophical conclusions from our very cursory treatment of this technical subject."
First off, thank you for the video. It's uncanny, you know, how you seem to be able to find good quality videos on the www and by quality I'm not referring to the video resolution. You've made what is essentially chance into an art. It must take both intelligence and loads of luck to boot to turn what is essentially a roll of a die into a skill. Kudos! Thanks again.
Last I checked, Godel's incompleteness employs a variation of the liar sentence which, as you know, is "this sentence is false." According to the video, Godel's version of it is, K (for Kürt) = "the sentence with Godel number g is unprovable", the sentence with Godel number g being K itself. Thus, if K's provable, then it's unprovable [inconsistent because of the contradiction] and if K's unprovable then some mathematical truths are unprovable [incomplete].
As you might've already guessed, at the heart of Godel's therems lies the liar paradox. Before I go any further I need to draw your attention to the rather odd fact that Godel and anyone else who uses different versions of the liar sentence for whatever purposes is, all said and done, resorting to a L-I-A-R. Would you or anyone put to service a liar to prove something, anything? Perhaps I'm being too dramatic and perhaps I'm barking up the wrong tree; after all, the word "liar" may have been used just to grab our attention - only for effect, nothing else.
That out of the way, let's revisit K = the sentence with Godel number g is unprovable and the argument presented in the video which hopefully is a variation, salva veritate, of Godel's own.
Argument A [Adele, Godel's wife]
1. K is provable [assume for reductio ad absurdum]
2. If K is provable then K is unprovable [K = the sentence with Godel number g (K itself) is unprovable]
3. K is unprovable [1, 2 Modus Ponens] 4. K is provable and K is unprovable [contradiction] [..Math is inconsistent]
Ergo, 5. K is unprovable [1 - 4 reductio ad absurdum][..Math is incomplete]
A few points that seem worth mentioning.
a) Look at N (Nimbursky, middle name of Godel's wife) = premise 2. If K is provable then K is unprovable [K = the sentence with Godel number g (K itself) is unprovable]. The assumption that has to be made for argument A to do its job of breaking math as it were is that N makes sense, in logical terms, makes sense implies that a truth value can be assigned to it.
The first clue that something's off is that N is a derivative of the liar sentence and we know that the liar sentence doesn't make sense. One could say that the liar sentence is a poisoned well so to speak and every bucket of water, N being one, drawn from it will be lethal or, in this case, highly dubious. Common sense! No?
b) Consider now the fact that argument A is a reductio ad absurdum which, as you know, derives a conclusion and uses that to reject/negate one or more of the assumptions made in the preceding lines of an argument. If you're not familiar, a reductio argument looks like this:
1. p
2. q & ~q [inferred from p]
Ergo,
3. ~p
Now in the argument A, the following assumptions/premises occur
1. K is provable
2. If K is provable then K is unprovable [K = the sentence with Godel number g (K itself) is unprovable]
The two assumptions lead to the contradiction below,
4. K is provable and K is unprovable
We are now justified in rejecting "one" of the premises but it doesn't necessarily have to be the one Godel has rejected which is 1. K is provable. After all, a reductio absurdum doesn't actually identify which premise is false. A reductio ad absurdum is like a detective in faer earlier stages of a murder investigation - fae knows only that someone is the murderer but doesn't know who the murderer is. Thus, I could reject N = 2. If K is provable then K is unprovable [K = the sentence with Godel number g (K itself) is unprovable and if I do that Godel's argument falls apart.
Given premise 2. If K is provable then K is unprovable [K = the sentence with Godel number g (K itself) is unprovable] has highly ignoble origins (the liar sentence), shouldn't we reject it rather than reject 1. K is provable, a perfectly reasonable proposition?
c) There's another issue with statements like N = 2. If K is provable then K is unprovable [K = the sentence with Godel number g (K itself) is unprovable].
Given a proposition P,
1. If ~P then P
2. ~~P or P [from 1 implication]
3. P or P [2 double negation]
4. P [3 tautology]
The statement, If ~P then P can be thought of as P itself, it can be reduced to P. In other words, the conditional if ~P then P is an illusion of sorts because it actually means P
Let's look at the version of the liar sentence that Godel uses which is, if K is provable then K is unprovable.
1. If K is provable then K is unprovable
2. ~K is provable or K is unprovable [from 1 implication]
3. K is unprovable or K is unprovable [from 2, ~K is provable = K is unprovable]
4. K is unprovable [3 tautology]
In essence, 1. K is provable then K is unprovable is logically equivalent to (I've used only equivalence rules of natural deduction), is nothing but, the statement 4. K is unprovable wearing heavy disguise.
What this means is that Godel's argument as presented in the video becomes,
1. K is provable [assume for reductio ad absurdum]
2. K is unprovable [If K is provable then K is unprovable = K is unprovable]
3. K is provable and K is unprovable [1, 2 Conjunction]
Ergo,
4. K is unprovable [1 - 3 reductio ad absurdum]
Did you notice what went wrong? The conclusion, 4. K is unprovable is also a premise 2. K is unprovable. A petitio principii.
I must confess that I didn't. I've not studied symbolic logic, and my mathematics is rudimentary, but I'm interested in why Godel's Theorems 'are unquestionably among the most philosophically important logico-mathematical discoveries ever made' (says this article.) I feel as though I intuitively understand why that is, but when I read up on it, I find it very hard to follow. Which is why I found that video was helpful although as we see, opinions are divided.
Seriously, luck's on your side and/or you know exactly which words to type into the search box if every search you do takes you to high quality material.
I must confess that I didn't. I've not studied symbolic logic, and my mathematics is rudimentary, but I'm interested in why Godel's Theorems 'are unquestionably among the most philosophically important logico-mathematical discoveries ever made' (says this article.) I feel as though I intuitively understand why that is, but when I read up on it, I find it very hard to follow. Which is why I found that video was helpful although as we see, opinions are divided.
Get your hands on an introductory course on logic. It'll take about a month to get a good understanding of basic logic. Some call it, derogatorily I suspect, baby logic but, if you ask me, that's a misnomer. I guarantee that you won't regret it.
Returning to the main point in re Godel's argument, the version in the video, it proceeds as follows:
a) K = the sentence with the Godel number g is unprovable
b). The sentence with the Godel number g is K itself.
Suppose there's a proof for K. It would prove K is unprovable. That's a contradiction: the unprovable is provable.
1. If K is provable then K is unprovable (Godel's key premise)
2. K is provable (assume for reductio ad absurdum)
3. K is unprovable (from 1, 2)
4. K is provable and K is unprovable (2, 3 taken together)
Ergo,
5. K is unprovable (1 to 4 reductio ad absurdum)
The problem is premise 1. If K is provable then K is unprovable is logically equivalent to the statement, K is unprovable. See vide infra,
If K is provable then K is unprovable = K is unprovable or K is unprovable = K is unprovable
In other words, I can substitute "K is unprovable" for "K is provable then K is unprovable" and then Godel's argument becomes,
1. K is unprovable [because, if K is provable then K is unprovable = K is unprovable]
2. K is provable
4. K is provable and K is unprovable (2, 3 taken together)
Ergo, 5. K is unprovable (1 to 4 reductio ad absurdum)
Notice statement 1 (Godel's key premise) = statement 5 (the conclusion). This is, as you already know, a circulus in probando (circular argument).
There's never a statement in any given language that is both definitely true according to the rules of that language and also not provable in that language, because to be definitely true according to the rules of a language just is to be provable in that language.
There may be statements which are definitely true, which really just means true, (since, logically, anything true must definitely be true) even if they cannot be proven, unless by "definitely true' you mean known to be true. I didn't think you meant that, because that seems a silly thing to say, but if you did mean that then what you say is trivially true, has no bearing on Gödel's theorems, and is thus irrelevant.
There's never a statement in any given language that is both definitely true according to the rules of that language and also not provable in that language, because to be definitely true according to the rules of a language just is to be provable in that language.
What you say just seems wrong for the simple reason that the truth of statements that are not provable cannot be ruled out; we don't know if they are true or not. In other words there can be truths which we cannot determine to be such, or at least it cannot be ruled out that there are.
What is mathematical truth is an open question in the philosophy of mathematics that has been much debated over the last 100 years, since Tarsky resurfaced it. This exchange illustrates the traditional axis of controversy between Platonists and formalists, realists and anti-realists.
Reply to Wayfarer I don't mean to be too intrusive but I do want to pick your brain regarding some interesting aspects of Godel's theorems but in a much broader context.
As I mentioned earlier, Godel uses the liar paradox to wit, the sentence L = This sentence is false. Such sentences are referred to as self-referential but that's an incomplete description. There are two characteristics that L has,
1. Self-reference. This sentence is false (not true).
2. Negation that causes, how shall I put it?, tension between what's being negated and what's part and parcel of the self that's being referred to. This sentence is false (not true)
A few things that come to mind:
a) Descartes' cogito argument. A variation of it would be: I do not exist. When one uses the "I", it appears that existence is baked into it. Then comes the negation "do not exist" which denies what the "I" incorporates viz. existence.
b) An interesting but probably nonsubstantive quality of L is that it refers to itself, yes, but, if my English is correct, in the third person ("this") and not "I" (first person). It kinda creeps me out - there's another possibly but not necessarily dangerousagency - the true but hidden liar - who our poorly evolved "spider sense" has detected and that's why we feel more comfortable using "this" and not "I". Warning! I'm prone to flights of fancy but then there's the Cartesian deus deceptor problem we haven't yet solved.
c) What about the Buddhist notion of anatta (non-self)?
If you can correctly extract from the video that Godel's argument is circular, then the video is wrong. We should learn Godel's argument from a carefully written exposition, not from a merely breezy cartoon version.
Get your hands on an introductory course on logic.
What is the introductory course you have taken?
I see you use some sentential logic, but a good understanding of Godel's theorem requires also predicate logic, some set theory, and a first course in mathematical logic.
(By the way, in a conversational context, spelling out and numbering, as you do, such basic sentential logic as applications of modus ponens and conjunction is gratuitous pedanticism that only clutters up whatever it is you mean to say. People don't need to have such basic reasoning annotated for them.)
Godel uses the liar paradox to wit, the sentence L = This sentence is false
That is flat out incorrect. Godel uses an argument analogous to the liar paradox, but not the liar paradox and nothing like "this sentence is false". Rather, it mathematically renders "this sentence is not provable in system P".
If you can correctly extract from the video that Godel's argument is circular, then the video is wrong.
I haven't watched the vid, is it any good? Veritasium is usually pretty good but not always. When I saw that he'd done one on this subject my first reaction was, "Not this sh*t again." Then I remembered that every day, there are people hearing about incompleteness for the first time. So it's fine that people are doing new videos on it. On the other hand, it's labeled, "The hole in mathematics," or "The fatal flaw in mathematics," or some such nonsense, and clearly that's giving a lot of people some wrong ideas. After all, computer science doesn't have a "hole" or a "fatal flaw" just because the halting problem is unsolvable, and that amounts to the same thing.
If you watched this vid, can you tell me if it's giving people false ideas? Or is the video accurate and people are getting the false ideas by themselves?
To me, incompleteness is not a hole or a flaw. It's deeply liberating. It shows that mathematics can never be reduced to a mechanical calculation. Mathematical truth will always transcend mere rules.
TonesInDeepFreezeJune 03, 2021 at 18:00#5461480 likes
You mentioned the benefit of a course in logic. In another thread, I have listed what I consider to be the best textbooks leading to the incompleteness thereom. If you like, I can link to that post. And, for a more casual, everyperson read, I highly recommend:
Godel's Theorem: An Incomplete Guide To Its Use And Abuse - Torkel Franzen
It's readable for people with just a modest knowledge of logic and math, authoritative, pays attention to crucial technicalities but not bogged down with them, very nicely written, entertaining and witty too.
TonesInDeepFreezeJune 03, 2021 at 18:01#5461490 likes
I'm interested in why Godel's Theorems 'are unquestionably among the most philosophically important logico-mathematical discoveries ever made' (says this article.)
That article gives an answer. It's a great article. (By the way, Panu Raattkainen is a top notch source on the subject.)
The philosophical implications of Godel's theorems are usually very overblown.
All of the various self-reference paradoxes have always seemed trivial to me, e.g. "This sentence is false." Who cares? Russell's paradox seems just the same, just dolled up in mathematical/logical language. Ditto with Godel's incompleteness theorem. Do these "paradoxes" really have a significant, real-time, practical impact on the effective use of mathematics and computer science in the real world? Or is it only guys who are too smart playing around with trivia as if it mattered?
What is mathematical truth is an open question in the philosophy of mathematics that has been much debated over the last 100 years, since Tarsky resurfaced it. This exchange illustrates the traditional axis of controversy between Platonists and formalists, realists and anti-realists.
I seem to recall reading somewhere that Gödel was a mathematical Platonist. Are you suggesting that Gödel's incompleteness theorem would be trivially true on a formalist understanding of mathematics because to be true in a language just is to be proven in that language?
My disagreement with @Pfhorrest seemed to perhaps hinge on his use of the term "definitely true". He hasn't responded to say whether he would claim that "There's never a statement in any given language that is both true according to the rules of that language and also not provable in that language, because to be true according to the rules of a language just is to be provable in that language." (The statement he made leaving out the word "definite").
Would a formalist allow that there could be mathematical truths that cannot be proven? If so do formalists accept that Gödel has proven that there are such truths? If not should they reject his theorem altogether?
I seem to recall reading somewhere that Gödel was a mathematical Platonist
[quote=Godel and the nature of mathematical truth, Rebecca Goldstein; https://www.edge.org/conversation/rebecca_newberger_goldstein-godel-and-the-nature-of-mathematical-truth ] Gödel was a mathematical realist, a Platonist. He believed that what makes mathematics true is that it's descriptive—not of empirical reality, of course, but of an abstract reality. Mathematical intuition is something analogous to a kind of sense perception. In his essay "What Is Cantor's Continuum Hypothesis?", Gödel wrote that we're not seeing things that just happen to be true, we're seeing things that must be true. The world of abstract entities is a necessary world—that's why we can deduce our descriptions of it through pure reason.[/quote]
I'm interested in why Godel's Theorems 'are unquestionably among the most philosophically important logico-mathematical discoveries ever made' (says this article.)
— Wayfarer
That article gives an answer. It's a great article. (By the way, Panu Raattkainen is a top notch source on the subject.)
I skimmed it. I will go back and read it again. I found the book you mention, it seems eminently readable from the preview, I will add it to my list. Thank you.
All of the various self-reference paradoxes have always seemed trivial to me, e.g. "This sentence is false." Who cares? Russell's paradox seems just the same, just dolled up in mathematical/logical language.
I suspect there's something you're not seeing here. When Bertrand Russell told Gottlieb Frege about the 'barber paradox' it had a momentous impact on Frege's whole life work. I think, in layman's terms, what is at stake is elucidating a set of mathematical and logical principles which are both consistent and complete 'all the way down', so to speak. As the Verisatum video mentions, David Hilbert had said 'we can know, we must know', referring to the 'formalist program', the aim of which was to produce such a complete and consistent set of principles. So, I think paradoxes of self-reference, and later, Godel's theorem, are seen to undermine forever this possibility. 'We don't know, we can't know'. So it has bearing on the limitations of knowledge, as far as I can discern.
I don't mean to be too intrusive but I do want to pick your brain regarding some interesting aspects of Godel's theorems but in a much broader context.
From the intro to Gödel's Theorem: An Incomplete Guide to Its Use and Abuse.
When Bertrand Russell told Gottlieb Frege about the 'barber paradox' it had a momentous impact on Frege's whole life work.
If I may interrupt for a second, Bertrand Russell did not approve of re-stating his paradox as the barber paradox:
[quote=Russell]You can modify its form; some forms of modification are valid and some are not. I once had a form suggested to me which was not valid, namely the question whether the barber shaves himself or not. You can define the barber as "one who shaves all those, and those only, who do not shave themselves". The question is, does the barber shave himself? In this form the contradiction is not very difficult to solve. But in our previous form I think it is clear that you can only get around it by observing that the whole question whether a class is or is not a member of itself is nonsense, i.e. that no class either is or is not a member of itself, and that it is not even true to say that, because the whole form of words is just noise without meaning.[/quote]
Whilst the equation is true according to the rules of the math. The rules of the math cannot prove the equation true. To prove the equation true we need to look outside the rules of the math.
TonesInDeepFreezeJune 04, 2021 at 01:10#5462960 likes
"self-reference" used pejoratively in reference to Godel's theorem is a red herring. The self-reference is seen by looking outside the object language. The theorem can be proven in finitistic combinatorial arithmetic. The proof methods are no more suspect than those of proof in finitistic combinatorial arithmetic.
Do these "paradoxes" really have a significant, real-time, practical impact on the effective use of mathematics and computer science in the real world?
The proof of the incompleteness theorem does not rely on paradox. Anyway, it's pretty rare for the various non-foundational branches of mathematics, especially applied mathematics to be concerned with the incompleteness theorem. But there are important mathematical questions that are elucidated by the incompleteness theorem, including "There is no general method for deciding whether or not a given Diophantine equation has a solution." That settled a question that even a student of high school algebra might wonder about. Basic mathematical curiosity alone leads to the question whether there is a mechanical procedure to determine whether any given Diophantine equation has a solution. And there are other answers in mathematics that incompleteness elucidates. And the methods and context of the incompleteness theorem led to the earliest developments in computability and recursion theory, as those even became branches of mathematics in light of the techniques and context of the incompleteness proof. And, for philosophy of mathematics, Godel's theorem is a central concern. Perhaps most saliently is that (put roughly) incompleteness settles that Hilbert's hope for axioms that would settle all mathematical questions cannot be achieved.
TonesInDeepFreezeJune 04, 2021 at 01:13#5462990 likes
Reply to TonesInDeepFreeze I thought that was the point I was trying to make in quoting from his book, but thanks for spelling it out to clear up any ambiguity.
TonesInDeepFreezeJune 04, 2021 at 01:32#5463040 likes
Are you [SophistiCat] suggesting that Gödel's incompleteness theorem would be trivially true on a formalist understanding of mathematics because to be true in a language just is to be proven in that language?
Aside from whatever SophistiCat might say, it is not the case that formalism regards the incompleteness theorem in that way.
(1) Sentences are not true in a language. They are true or false in a model for a language.
(2) Sentences are not proven just in a language, but rather in a system of axioms and rules of inference.
(3) There are different versions of formalism, and it is not the case that in general formalism regards truth to be just provability.
(4) Godel's theorem in its bare form is about provability and does not need to mention the relationship between truth and proof (though a corollary does pertain to truth). The theorem is about provability, which is syntactical. Even if we had no particular notion of truth in mind, Godels' theorem goes right ahead to show that for systems of a certain kind, there are sentences in the language for the system such that neither the sentence not its negation are provable in the system.
Would a formalist allow that there could be mathematical truths that cannot be proven?
Any formal statement can be proven in some system or another. The incompleteness theorem, where it bears on truth, gives us that for a given system S of a certain kind, there are true arithmetical statements that are not provable in S. But those statements are provable in other systems. Even if a statement is arithmetically false, there are systems that prove the statement. Even if a statement if logically false, there are system that prove that statement (though, of course, those systems are inconsistent). So, if one is self-admittedly speaking only quite loosely to say "there are truths that cannot be proven" then we must regard that as standing for the more careful, "for a given system S of a certain kind, there are arithmetic truths that are nor provable in S".
TonesInDeepFreezeJune 04, 2021 at 01:34#5463050 likes
I thought that was the point I was trying to make in quoting from his book, but thanks for spelling it out to clear up any ambiguity.
Yes, your quote from the book is well taken. And it is clear in the context of the book, but some people might not realize that context, so I just wanted to underline it.
TonesInDeepFreezeJune 04, 2021 at 01:45#5463090 likes
Would you say the below is a fair description of what Gödel is saying? Whilst the equation is true according to the rules of the math. The rules of the math cannot prove the equation true. To prove the equation true we need to look outside the rules of the math.
No.
(1) The Godel sentence is not an equation.
(2) "rules of math" is unclear.
(3) We don't look outside the "rules of math" even given a reasonable understanding of what 'the rules of math" might mean.
(4) A correct way to say it this:
For a given system S of a certain kind, there are statements that are arithmetically true but that are not provable in S.
"of a certain kind" can be rendered as "that is recursively axiomatizable, consistent, and and extension of Robinsion arithmetic"
"arithmetically true" can be rendered as "true in the standard model of the language of first order Peano arithmetic):
So the statement is not provable in the system. But when we look at the standard model, we see that the statement is true. Godel didn't himself refer to models, but it's the way we would formalize it now. And if we don't want to be so pedantic to formalize with models, we can say that we see that the statement is true by "outside the system" just looking at the way the statement was formulated and how it relates to ordinary arithmetic.
That's pretty close without splitting hairs technically, as we would split those hairs in a more formal treatment.
Reply to TonesInDeepFreeze The thing is, though, that these kinds of ideas tend to filter through into popular culture, one way or another. There’s a profusion of ideas from current physics that have done so - Schrödinger’s cat, many worlds, the multiverse. Of course it’s true that to really understand those concepts requires, if not a degree in mathematical physics, at least some quite extensive reading and reflection.
But then, on the other hand, we have many popular intellectuals and scientists proselytising science as a world-view, telling us that science understands the world better than we do ourselves. I think that’s why Godel’s theorem has been seized on - rightly or wrongly - as a foil against the proselytising scientists - the Hawkings and De Grasse Tysons of the world who seem to claim ‘secret knowledge’ that none of us can access without years of study, but who also evince little insight into classical humanism and philosophy proper.
Anyway, I suppose to answer my own objection, that is the rationale behind books such as Franzen’s, which at least enable the educated layman to better consider the issue on its own terms.
TonesInDeepFreezeJune 04, 2021 at 01:50#5463120 likes
I don't take exception to the Goldstein quote. But her book about incompleteness needs to be read critically. As I recall (though I can't cite specifics from memory) she gets too casual sometimes to the point of misstating certain key points.
From the intro to Gödel's Theorem: An Incomplete Guide to Its Use and Abuse.
You might find the answers in there.
Thank You Wayfarer
[quote=Wikipedia]Relationship with the liar paradox [of Godel's Incompleteness Theorems]
Gödel specifically cites Richard's paradox and the liar paradox as semantical analogues to his syntactical incompleteness result in the introductory section of "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I". The liar paradox is the sentence "This sentence is false." An analysis of the liar sentence shows that it cannot be true (for then, as it asserts, it is false), nor can it be false (for then, it is true). A Gödel sentence G for a system F makes a similar assertion to the liar sentence, but with truth replaced by provability: G says "G is not provable in the system F." The analysis of the truth and provability of G is a formalized version of the analysis of the truth of the liar sentence.[/quote]
Please carry out your own investigations into the issue if you're interested of course. Good luck.
TonesInDeepFreezeJune 04, 2021 at 02:03#5463160 likes
Wikipedia:Gödel specifically cites Richard's paradox and the liar paradox as semantical analogues to his syntactical incompleteness result
I suspect there's something you're not seeing here.
I think you're exactly right. That was the point of my post. It's not that I think they're wrong and I'm right. I just don't get it. I'm hoping someone will answer my questions - why do these seemingly trivial paradoxes and inconsistencies matter so much? Where do they meet the world?
When Bertrand Russell told Gottlieb Frege about the 'barber paradox' it had a momentous impact on Frege's whole life work. I think, in layman's terms, what is at stake is elucidating a set of mathematical and logical principles which are both consistent and complete 'all the way down', so to speak.
But why, in any practical sense, does that matter? Does it make mathematics less useful or effective in any significant way? It seems Platonic. Forms existing without relation to physical reality.
Basic mathematical curiosity alone leads to the question whether there is a mechanical procedure to determine whether any given Diophantine equation has a solution. And there are other answers in mathematics that incompleteness elucidates.
Basic mathematical curiosity is a pretty good reason to study something. I don't have any problem with that.
I just want to make it clear - I don't doubt the results of these brilliant mathematicians work. I'm not like one of those relativity deniers who think that I can see something that mathematicians and scientists have worked on for centuries.
TonesInDeepFreezeJune 04, 2021 at 02:44#5463330 likes
But why, in any practical sense, does that matter? Does it make mathematics less useful or effective in any significant way? It seems Platonic. Forms existing without relation to physical reality.
Isn't it all to do with the foundations of mathematics and logic? 'Foundations' suggests to me something real, a system of thought that can be anchored against a, or the, absolute. Both Frege and Russell were attempting that in different ways. There's an SEP article on Russell's paradox here. So it may not matter in the practical sense of you and I carrying on with our lives, but it is a philosophical issue of great significance.
Even as regards to 'Platonic forms' - if they're real, not in the sense of being only 'in someone's mind', then that's significant. Because it suggests that 'what is real' extends well beyond what is, well, materially existent. If forms, and numbers, are real, then they're real in a different sense to the objects of physics, no matter how subtle. That's what fascinates me about platonism.
There's an SEP article on Russell's paradox here. So it may not matter in the practical sense of you and I carrying on with our lives, but it is a philosophical issue of great significance.
I resist philosophical labels, but I've come to the conclusion that I probably am a pragmatist. I have a strong resistance to philosophical issues that don't have practical consequences. As I said, let me spend some time reading.
Then I'll come back later and demonstrate more of my mathematical ignorance.
I too will plead, as at this time, especially as I am rusty in the subject, I wouldn't be able to marshal enough knowledge to explicate the details of the mathematics mentioned there.
Here's a more direct reference to Godel's incompleteness theorem vis-à-vis the Liar paradox.
[quote=Wikipedia]Gödel's incompleteness theorems are two fundamental theorems of mathematical logic which state inherent limitations of sufficiently powerful axiomatic systems for mathematics. The theorems were proven by Kurt Gödel in 1931, and are important in the philosophy of mathematics. Roughly speaking, in proving the first incompleteness theorem, Gödel used a modified version of the liar paradox, replacing "this sentence is false" with "this sentence is not provable", called the "Gödel sentence G"[/quote]
TonesInDeepFreezeJune 04, 2021 at 03:19#5463480 likes
Wikipedia:Gödel used a modified version of the liar paradox, replacing "this sentence is false" with "this sentence is not provable"
Yes, there is an analogy and similarity, but also the modification used makes a great difference too.
Reply to TheMadFool Yes. Again, note the 'roughly speaking'. I think that is what our expert contributors are emphasising. But I do agree there is a broader, underlying point. It all seems to revolve around the issue of self-reference or reflexivity.
I can't recall learning of anything in classical analysis (complex or real) affected by incompleteness, but that doesn't mean much since I have been out to pasture for many years and my memory is imperfect. If one of you comes across something please post. :chin:
Correct but it's not just self-reference, it's also negation of some kind. The self-referential sentence, "I exist" doesn't create problems like the self-referential negation, "I don't exist."
Let's look at the liar sentence L = this sentence (L itself) is false.
According to most books, the logic is as below,
Option 1
1. If L is true then L is false (the liar sentence)
2. L is true (assume)
3. L is false (1, 2 modus ponens)
4. L is true and L is false (2, 3 together, contradiction)
Ergo,
5. L is false (2 - 4 reductio ad absurdum)
Option 2
6. If L is false then L is true (the liar sentence)
7. L is false (assume)
8. L is true (6, 7 modus ponens)
9. L is true and L is false (7, 8 together, contradiction)
Ergo,
10. L is true (7 - 9 reductio ad absurdum)
Now, what's interesting is,
1. If L is true then L is false = L is false or L is false = L is false
That means the argument in option 1 becomes,
11. L is false = If L is true then L is false
12. L is true (assume)
13. L is true and L is false (11, 12 together, contradiction)
Ergo,
14. L is false (12 - 13 reductio ad absurdum)
However, notice line 11 (premise) = line 14 (conclusion). In other words, what was to be proved was assumed beforehand among the premises. The argument is circular which simply means 14. L is false is unwarranted.
Similarly, revisiting option 2, the statement 6. If L is false then L is true = L is true or L is true = L is true
The argument for option 2 then becomes,
15. L is true = If L is false then L is true
16. L is false (assume)
17. L is true and L is false (15, 16 together, contradiction)
Ergo,
18. L is true (16 - 17 reductio ad absurdum)
Notice here again that line 15 (premise) = line 18 (conclusion). Put simply, the conclusion has been assumed in the premises. Circular argument, which means we're not justified in concluding 18. L is true.
What does this all mean? We can arbitrarily assign a truth value to the liar sentence (true/false) but that's where it all stops - all logic beyond that is going to be circular and useless. Since the alleged contradiction of the liar paradox can only occur after an inference which begins with an assumption of a truth value for the liar sentence, and since, as explained above, all such arguments are circular, we're no longer justified to infer anything at all (that includes any further truth value for the liar sentence) from the initial assigned truth value for the liar paradox. Ergo, there being no inferrable truth value, there can be no contradiction. In short, the liar paradox doesn't entail a contradiction at all.
TonesInDeepFreezeJune 04, 2021 at 06:00#5463960 likes
"[...] a self-referential sentence which “says of itself” [...] Such figures of speech may be heuristically useful, but they are also easily misleading and suggest too much." - https://plato.stanford.edu/entries/goedel-incompleteness/#DiaSelRef
Read more there for more explanation.
On the other hand, see page 44 of Franzen's'Godel's Theorem: An Incomplete Guide To Its Use And Abuse' for a different take on the matter.
In any case, in whatever sense we may reasonably say the proof uses "self-reference", the proof is not vitiated by it, as the proof can be carried out in finitistic combinatorial arithmetic.
TonesInDeepFreezeJune 04, 2021 at 06:10#5463980 likes
If L is true then L is false = L is false or L is false = L is false
The equal sign there and the ones following it are not syntactical. Perhaps you mean the biconditional. If so, rewrite to see whether your argument still holds up,
My disagreement with Pfhorrest seemed to perhaps hinge on his use of the term "definitely true". He hasn't responded to say whether he would claim that "There's never a statement in any given language that is both true according to the rules of that language and also not provable in that language, because to be true according to the rules of a language just is to be provable in that language." (The statement he made leaving out the word "definite").
Leaving out the "definitely"s completely changes the meaning, so no, I wouldn't claim that modified sentence.
We cannot know for sure ("definitely", or "certainly") that some proposition is true, without in the process having proven it, so we cannot know for sure that any given proposition is true-but-unprovable, because to be sure of the first part we would have to violate that second part.
It might in some principled way remain the case that something or another could be true but not provable, but we could never say for sure that we had an example of that, because in somehow saying for sure that something was true, we just would be proving it.
TonesInDeepFreezeJune 04, 2021 at 14:49#5465190 likes
we cannot know for sure that any given proposition is true-but-unprovable, because to be sure of the first part we would have to violate that second part.
It is unprovable in the system being discussed. It is provably true in the mathematics used to discuss that system.
It might in some principled way remain the case that something or another could be true but not provable, but we could never say for sure that we had an example of that
We have a sure example. It's as sure as finitistic combinatorial arithmetic. It's a quite complicated calculation, but it is still a finite calculation.
I too will plead, as at this time, especially as I am rusty in the subject, I wouldn't be able to marshal enough knowledge to explicate the details of the mathematics mentioned there.
I'm back and I've read the texts you referenced. They were very interesting but not very helpful in answering my questions. I don't need to go any further with this except to say I was surprised to see that some of the philosophical claims associated with Godel's theorum are similar to the mystical/philosophical claims associated with quantum mechanics. These in particular struck me:
proves that Mechanism is false, that is, that minds cannot be explained as machines.
They all insist that Gödel’s theorems imply that the human mind infinitely surpasses the power of any finite machine or formal system.
I'm going to leave it here. I appreciate your help. This was fun.
It is unprovable in the system being discussed. It is provably true in the mathematics used to discuss that system.
Right, which is what I said a few posts back in that exchange. It's only in the meta-language, not the object language, that we can assign a truth value to the proposition, and in the meta-language we are also able to prove that proposition. The meta-language will itself also contain propositions which it cannot prove, and to which it cannot assign truth values; and in yet another language being used to discuss that first meta-language as an object language, a truth-value can be assigned to the unprovable statements of the first meta-language, but in that even higher-order language those statements can also be proved. You never have both a definite truth value and unprovability on the same level. "True but unprovable" only works when you mix levels: it's true, according to the higher-level system we're using to discuss the lower-level system, and unprovable, according to that lower-level system; but that lower-level system has no idea whether or not it's true (because it's unprovable), and in the higher-level system it's provable (which is how it can be known true).
Leaving out the "definitely"s completely changes the meaning, so no, I wouldn't claim that modified sentence.
We cannot know for sure ("definitely", or "certainly") that some proposition is true, without in the process having proven it, so we cannot know for sure that any given proposition is true-but-unprovable, because to be sure of the first part we would have to violate that second part.
It might in some principled way remain the case that something or another could be true but not provable, but we could never say for sure that we had an example of that, because in somehow saying for sure that something was true, we just would be proving it.
I agree that we cannot know for sure that some proposition is true without having proven it, and thus it would seem to follow that we cannot know that any given proposition is "true-but-unprovable". But the truth of any proposition is not dependent on our having proven it, but only our knowing of the truth of it is.
If I understood mathematics much better than I do I might be able to offer an opinion as to whether Godel has proved that there are true propositions within mathematics that cannot be mathematically proven.
Perhaps the answer to that last question is no] it seems: is this from the SEP is correct and if I have correctly interpreted its meaning:
"A common misunderstanding is to interpret Gödel’s first theorem as showing that there are truths that cannot be proved. This is, however, incorrect, for the incompleteness theorem does not deal with provability in any absolute sense, but only concerns derivability in some particular formal system or another. For any statement A unprovable in a particular formal system F, there are, trivially, other formal systems in which A is provable (take A as an axiom). On the other hand, there is the extremely powerful standard axiom system of Zermelo-Fraenkel set theory (denoted as ZF, or, with the axiom of choice, ZFC; see the section on the axioms of ZFC in the entry on set theory), which is more than sufficient for the derivation of all ordinary mathematics. Now there are, by Gödel’s first theorem, arithmetical truths that are not provable even in ZFC. Proving them would thus require a formal system that incorporates methods going beyond ZFC. There is thus a sense in which such truths are not provable using today’s “ordinary” mathematical methods and axioms, nor can they be proved in a way that mathematicians would today regard as unproblematic and conclusive.".
Is there any proof that such a "formal system that incorporates methods going beyond ZFC." will or even could be found? In light of all this what you said about there always being a meta-level wherein the unprovable truths within a system can be proven seems questionable.
Even if there were always a meta-level capable of (non-trivially) proving any true mathematical proposition, then there would still seem to be a kind of infinite regress of incomplete systems. Would this
fact render all such proofs non-exhaustive and/ or trivial I wonder (in my mathematical ignorance)?
TonesInDeepFreezeJune 05, 2021 at 03:30#5467300 likes
The general idea you expressed is okay, but I suggest some clarifications and context (much of which you likely know already).
We are concerned not just with the object-language and a meta-language, but the object-theory and a meta-theory.
With a meta-theory, there a models of the object-theory. Per those models, sentences of the object-language have truth values. So the Godel-sentence is not provable in the object-language but it in a meta-theory, we prove that the Godel-sentence is true in the standard model for the language of arithmetic. Also, as you touched on, in the meta-theory, we prove the embedding of the Godel-sentence into the language of the meta-theory (which is tantamount to proving that the sentence is true in the standard model). That is a formal account of the matter. And in a more modern context than Godel's own context, if we want to be formal, then that is the account we most likely would adopt.
Godel himself did not refer to models. Godel's account is that the Godel-sentence is true per arithmetic, without having to specify a formal notion of 'truth'. And we should find this instructive. It seems to me that for sentences of arithmetic, especially ones for which a computation exists to determine whether it holds or not, we are on quite firm ground "epistemologically" to say, without quibbling about formality, that the sentence is true when we can compute that it does hold.
that lower-level system has no idea whether or not it's true (because it's unprovable)
There is an even more fundamental reason that the object-theory does not yield a determination of truth. That is that the object-language does not have a truth predicate. There's a subtle difference: A theorem of the object-theory is true in any model of the theory, so in that sense one would say that the object-theory does determine the truth of certain sentences. But the object-theory does not itself have a theorem that the sentence is true in models of the theory (or else, the object-theory would be inconsistent per Tarski's theorem).
TonesInDeepFreezeJune 05, 2021 at 03:43#5467320 likes
I might be able to offer an opinion as to whether Godel has proved that there are true propositions within mathematics that cannot be mathematically proven.
It depends on the definition of 'mathematically proven'.
Is there any proof that such a "formal system that incorporates methods going beyond ZFC." will or even could be found?
It is easy to see that there are theories that are proper extensions of ZFC ['proper' meaning having theorems that ZFC does not have]. But that doesn't settle the question of whether those theories are within what we consider to be justifiable mathematics, or even the question of what it means to be justifiable mathematics.
It seems to me that for sentences of arithmetic, especially ones for which a computation exists to determine whether it holds or not, we are on quite firm ground "epistemologically" to say, without quibbling about formality, that the sentence is true when we can compute that it does hold.
What gets to me, and maybe you can clarify, is how it could be that we can “compute that it does hold” and yet not have, at some level or another, thereby “proven” it. If we are taking for granted (as proven, in some sense or another, or else just assumed) that arithmetic works the way we usually use it, and an arithmetical operation yields a certain output, have we not consequently proven (or assumed) that output as part and parcel of having proven (or assumed) that arithmetic works in such a way as that?
What gets to me, and maybe you can clarify, is how it could be that we can “compute that it does hold” and yet not have, at some level or another, thereby “proven”
Doesn’t it mean that there must always be some assumptions? Something that is assumed to be so, but can’t be proven to be so? Isn’t that what the issue is about?
Reply to Wayfarer I'm pretty sure that's not what Godel is on about at all.
But on that unrelated topic, I am vociferously opposed to justificationism, the usual kind of rationalism (contra critical rationalism), which says that you should reject everything that can't be proven conclusively "from the ground up", because per Agrippa's / Munchausen's trilemma that is inherently impossible. Instead, as a critical rationalist, I think it's fine (and necessary) to run with whatever assumptions you're inclined to, until they can be disproven.
Godel's about whether there are things that are true but aren't provable. And I don't see how we can ever do better than "Maybe? I suppose it's always technically possible, but we can never be sure whether or not there are". Because to sure, we would have to be sure that something was unprovable, and also be sure that it was true -- and I don't see how we could "be sure that it was true" without, in doing so, proving it, and so showing it to be not-unprovable.
As I understand it, Godel only shows something about the relationship between a formal system and statements in it: that some systems can't prove some things they're capable of talking about either way, even though we can know, through in a proof made in a higher-level system, that those things are true.
Reply to Wayfarer You said things that are assumed but can't be proven. By the nature of assumptions, we don't know whether or not they are true. Whether or not it's okay to believe things that are only assumptions and not proven is a different question from whether or not there definitely are things that are true but not provable.
I seem to recall reading somewhere that Gödel was a mathematical Platonist. Are you suggesting that Gödel's incompleteness theorem would be trivially true on a formalist understanding of mathematics because to be true in a language just is to be proven in that language?
I wasn't specifically referring to Gödel's theorem, but using that example, a strictly formal reading of the first incompleteness result would be like this quote in the Wiki article:
Raatikainen 2015:Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.
Here the statement is not said to be either true or false; if pressed, an anti-realist* might say that (a) the question of truth is meaningless outside the context of a particular formal system, and (b) in the context of system F the Gödel statement is neither true nor false, since it can be neither proved nor disproved in that system.
On the other hand, you have, no doubt, heard paraphrases to the effect that the Gödel statement is "true but unprovable." Such readings lean on a realist/Platonist understanding of mathematical truth. They would appeal to the structure of the Gödel statement, which states an arithmetical truth.
This is a very crude and clipped summary. Like I said, the question of truth in mathematics and its relation to provability has been investigated and debated at great length. Just searching for works with the words "truth" and "provability" or "proof" in the title will net you several pages of results on Google Scholar.
* I will withdraw the label "formalism" and use instead the more vague "anti-realism" or "anti-Platonism."
There are different versions of formalism, and it is not the case that in general formalism regards truth to be just provability.
Metaphysician UndercoverJune 05, 2021 at 12:02#5467870 likes
Sextus Empiricus against the dogmatist's criteria of truth:
[quote=Stanford Encyclopedia of Philosophy]At the end of Sextus’ discussion in PH II, he clearly signals, as one would expect, that he suspends judgment on whether there are criteria of truth:
You must realize that it is not our intention to assert that standards of truth are unreal (that would be dogmatic); rather, since the Dogmatists seem plausibly to have established that there is a standard of truth, we have set up plausible-seeming arguments in opposition to them, affirming neither that they are true nor that they are more plausible than those on the contrary side, but concluding to suspension of judgement because of the apparently equal plausibility of these arguments and those produced by the Dogmatists. (PH II 79; cf. M VII 444)[/quote]
[quote=Internet Encyclopedia of Philosophy]According to Chisholm, there are only three responses to the Problem of the Criterion: particularism, methodism, and skepticism. The particularist assumes an answer to (1) and then uses that to answer (2), whereas the methodist assumes an answer to (2) and then uses that to answer (1). The skeptic claims that you cannot answer (1) without first having an answer to (2) and you cannot answer (2) without first having an answer to (1), and so you cannot answer either. Chisholm claims that, unfortunately, regardless of which of these responses to the Problem of the Criterion we adopt we are forced to beg the question. It will be worth examining each of the responses to the Problem of the Criterion that Chisholm considers and how each begs the question against the others. [/quote]
TonesInDeepFreezeJune 05, 2021 at 14:11#5468120 likes
Something that is assumed to be so, but can’t be proven to be so? Isn’t that what the issue is about?
I don't know what issue you mean when you ask what the issue is about. But for incompleteness, it's not just a matter of having to assume things to prove things.
TonesInDeepFreezeJune 05, 2021 at 14:42#5468290 likes
Godel's about whether there are things that are true but aren't provable.
That is exactly the most salient oversimplification that causes misunderstanding.
You know the following, but it bears emphasizing:
There is no mathematical statement that isn't provable. That is, for any mathematical statement (even a self-contradiction) there are systems that prove the statement.
Godel's theorem is that for any given system S of a certain kind there are statements F in the language for S that such that S proves neither F nor ~F.
It's a matter of quantifier order:
Godel: For any system S of a certain kind, there exist statements undecided by S.
False: There exist statements F such that for any system S of a certain kind, F is undecided
Godel only shows something about the relationship between a formal system and statements in it: that some systems can't prove some things they're capable of talking about either way, even though we can know, through in a proof made in a higher-level system, that those things are true.
That seems to me to be a reasonable summary.
TonesInDeepFreezeJune 05, 2021 at 14:45#5468300 likes
Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.
— Raatikainen 2015
Here the statement is not said to be either true or false [...] in the context of system F the Gödel statement is neither true nor false, since it can be neither proved nor disproved in that system.
Yes, the theorem itself, as you quoted it, does not mention truth. But from the theorem, we do go on to remark that the undecided sentence is true.
And the statement is neither true or false in the system on an even more fundamental basis than that it is undecided by the system:
There is an even more fundamental reason that the object-theory does not yield a determination of truth. That is that the object-language does not have a truth predicate. There's a subtle difference: A theorem of the object-theory is true in any model of the theory, so in that sense one would say that the object-theory does determine the truth of certain sentences. But the object-theory does not itself have a theorem that the sentence is true in models of the theory (or else, the object-theory would be inconsistent per Tarski's theorem).
TonesInDeepFreezeJune 05, 2021 at 14:52#5468330 likes
When we're talking about plain arithmetical truths, I don't know why we would have to go down the road of wondering about realism. I mean, non-realists still recognize the truth of arithmetical statements.
TonesInDeepFreezeJune 05, 2021 at 16:00#5468500 likes
It is unprovable in the system being discussed. It is provably true in the mathematics used to discuss that system.
I should qualify that remark and others I made along the same lines.
We prove (though not in the object system) that the Godel-sentence is true on the assumption that the object-system is consistent.. That qualification might be regarded as implicit in my remarks, but it is best for me to make it explicit.
we are on quite firm ground "epistemologically" to say, without quibbling about formality, that the sentence is true when we can compute that it does hold.
I should put that remark on hold. I need to figure out whether saying that we have a "computation" is correct.
There is an even more fundamental reason that the object-theory does not yield a determination of truth. That is that the object-language does not have a truth predicate.
But if we adhere to strict "truth=provability" principle, then the sentence is not true even in the metatheory, if it assigns truth to sentences subject to their provability in the object theory.
When we're talking about plain arithmetical truths, I don't know why we would have to go down the road of wondering about realism. I mean, non-realists still recognize the truth of arithmetical statements.
Anti-realists recognize arithmetical statements as true relative to particular mathematical theories, which are as fictitious as any other such theories. Realists view some mathematical systems, such as arithmetic, as representing an objective, mind-independent reality; for them the mathematical study of such systems can be likened to scientific research.
Again, I want to disclaim that this is a simplistic caricature, but here are some statements in the same spirit by mathematician G.H. Hardy:
Mathematical theorems are true or false; their truth or falsity is absolute and independent of our knowledge of them.
G.H. Hardy:
Pure mathematics... seems to me a rock on which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way.
(Quoted from another work of Torkel Franzen: "Provability and Truth" (1987))
TonesInDeepFreezeJune 05, 2021 at 18:45#5468930 likes
But if we adhere to strict "truth=provability" principle, then the sentence is not true even in the metatheory, if it assigns truth to sentences subject to their provability in the object theory.
What "truth=provability" principle do you have in mind? What is its mathematical formulation? Meanwhile, the incompleteness theorem proves that the set of provable sentences does not equal the set of true sentences. [Often now, I'll leave tacit the usual qualifiers such as "provable in system S" and "true in the standard model".]
Reply to TonesInDeepFreeze Perhaps you could elaborate on computational proof. When I conjecture a theorem in complex analysis I usually turn to the many programs I've written for examples that will either suggest the conjecture is true or abruptly halt the process - if only temporarily - by demonstrating it is false in a particular case.
I don't think this is what you are discussing, however. I'm trying to see the link between actual mathematics and foundational mathematics in this regard.
what you [Pfhorrest] said about there always being a meta-level wherein the unprovable truths within a system can be proven seems questionable. — Janus
It is the case that there is an infinite escalation of theories, each proving arithmetical truthts not provable in the lower theories.
Would this fact render all such proofs non-exhaustive and/ or trivial — Janus
The theories are not exhaustive, indeed. But I don't see why that would make the proofs trivial.
Perhaps, then, it depends on whether we have in mind mathematical or philosophical triviality.
G.H. Hardy:Pure mathematics... seems to me a rock on which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way.
I don’t see how this can be so. The fact that 317 is a prime number is indeed not dependent on your or my assent, but it’s regardless a fact which only a rational mind can grasp, and in that sense is what can be called an intelligible object or object of rational thought. I don’t see how this undermines idealism but rather reinforces it, in my understanding of that term.
I'm talking about sentences in the language of arithmetic. I don't know whether these matters bear upon your areas of mathematics.
I am pretty rusty on this stuff, so take this modulo a grain of salt:
The Godel sentence G "says":
"For every n, it is not the case that n is the Godel-number of a proof of the sentence with Godel-number g."
G is a sentence purely in the language of arithmetic. The "it says" about proofs and Godel-numbers is seen and proven (in the meta-theory) with regard to the construction of G per the arithmetization of syntax.
And, G has Godel-number g.
The part "it is not the case that n is the Godel-number of a proof of the sentence with Godel-number g" is a computable property. Let's call it 'C'. So G is of the form:
For all n, Cn.
Now, for concision, let's say we're looking at some particular system S.
Godel-Rosser proves "If S is consistent, then both G and ~G are not theorems of S."
And let's say that by 'true' we mean true in the standard model for the language of arithmetic. Godel did not himself have formal model theory to reference, but in context we may say that his context might as well be tantamount to it. Moreover, we could dispense the formality of models by just agreeing that 'true' means what it ordinarily means to mathematicians who don't care about mathematical logic. For example, '0+0=0' is simply true and '0=1' is simply false.
So, either G is true or ~G is true. So, on that basis alone, we know that there is a true sentence that S does not prove. But that is not constructive - it uses excluded middle and doesn't tell us specifically which one of the two is the true one.
But we can constructively (I think?) show "If S is consistent then G is true" anyway.
Now, let's look at a certain kind of arithmetical sentence. These are sentences in the language of arithmetic all of whose quantifiers are bounded. For example:
"For all n<20, if n is prime then n has a twin prime."
For such sentences, there is an algorithm to decide their truth. Moreover, it is said that from the sentence itself, we can "read off" the algorithm (please don't ask me the technical definition of "read off" - I have not yet pursued how to formalize it).
Now, where I tripped myself up earlier in this thread is that I might have conflated the fact in the above paragraph with the fact that we do easily prove "if S is consistent then G is true", as I am not clear whether that proof is one that also is "read off" from the sentence itself, in context of the construction of the sentence vis-a-vis the arithmetization of syntax.
Metaphysician UndercoverJune 06, 2021 at 01:48#5469700 likes
Aside from whatever SophistiCat might say, it is not the case that formalism regards the incompleteness theorem in that way.
(1) Sentences are not true in a language. They are true or false in a model for a language.
(2) Sentences are not proven just in a language, but rather in a system of axioms and rules of inference.
(3) There are different versions of formalism, and it is not the case that in general formalism regards truth to be just provability.
(4) Godel's theorem in its bare form is about provability and does not need to mention the relationship between truth and proof (though a corollary does pertain to truth). The theorem is about provability, which is syntactical. Even if we had no particular notion of truth in mind, Godels' theorem goes right ahead to show that for systems of a certain kind, there are sentences in the language for the system such that neither the sentence not its negation are provable in the system.
TonesInDeepFreezeJune 06, 2021 at 02:02#5469740 likes
You claim that quote is dogmatic. What is your non-dogmatic basis for that claim?
The quote is not dogmatic, I say non-dogmatically. The quote describes the way mathematical logic uses certain terminology and certain other plain facts about mathematical logic. It is apropos to mention those terminological conventions and basics of mathematical logic, since the context of the discussion is Godel's theorem, which is a subject in mathematical logic. Meanwhile, I allow that anyone is welcome to stipulate their own terminology and even to propose an entirely different framework for consideration of mathematics. However such a proposal would be subject to the same scrutiny for coherence and rigor to which mathematical logic is subject. That is it the antithesis of dogmatism.
"It is not our business to set up prohibitions, but rather to arrive at conventions." - Rudolph Carnap
[quote=Another Carnap quote]Metaphysicians are musicians without musical ability.[/quote]
I hate all those Vienna Circle types. That’s one of the reasons I’m going to keep out of these discussions.
TonesInDeepFreezeJune 06, 2021 at 02:35#5469800 likes
The crank claims that mathematics is wrong. Not just that he proposes different mathematical conventions and definitions, but that the more ordinary conventions and definitions are blatantly wrong onto themselves. The crank cannot understand the stipulative nature of definitions. And the crank claims to find contradiction in mathematics when the crank has only found things that are not actual contradictions but instead are things that happen to be counterintuitive to him. The crank uses sophistry, evasion, raw repetition and ignorance for his position that only his own conception is right and that mathematics is wrong. The crank fancies that he eviscerates mathematics though he does not know even the least of its basics and horribly misconstrues the few bits that he has happened to come across. Thus, the preponderance of the crank's attack is the strawman. The crank is not interested in learning about the subject on which he so tendentiously opines. Instead, he is only interested in announcing his personal truths from the soapbox. The crank never (or virtually never) admits a mistake. All of that is dogmatism.
The logician says that from certain conventions, axioms, rules, and definitions, certain things follow and certain things do not follow. And the logician allows that people may set up different conventions, axioms, rules, and definitions. And the logician might even allow that proposed frameworks may have value even though they have not yet been axiomatized. The logician admits that definitions are stipulative so that definitions themselves are not inherently true, and that we may regard enquiries that proceed with different definitions. The logician seeks scrutiny of his work and is always eager to correct any errors found in his formulations. The logician admits that certain questions are not answered and that there is much still unknown. All of that is the antithesis of dogmatism.
TonesInDeepFreezeJune 06, 2021 at 02:39#5469810 likes
I hate all those Vienna Circle types. That’s one of the reasons I’m going to keep out of these discussions.
Whatever the merits or demerits of Carnap's views on metaphysics, the quote I mentioned does have wisdom.
And one may have one's own reasons for eschewing a conversation, but having a dislike of certain philosophers is not much of a rational basis for rejecting a conversation about mathematical logic.
TonesInDeepFreezeJune 06, 2021 at 03:01#5469840 likes
Now, let's look at a certain kind of arithmetical sentence. These are sentences in the language of arithmetic all of whose quantifiers are bounded.
Thanks. Certainly in the study of combinatorics there are conjectures in which all possible cases are finite in number and a computer program can do the job. Like the four color problem.
[quote=Wittgenstein, Tolstoy and the Folly of the Logical Positivists; https://philosophynow.org/issues/103/WittgensteinTolstoy_and_the_Folly_of_Logical_Positivism ] The declared aim of the Vienna Circle was to make philosophy either subservient to or somehow akin to the natural sciences. As Ray Monk says in his superb biography Ludwig Wittgenstein: The Duty of Genius (1990), “the anti-metaphysical stance that united them [was] the basis for a kind of manifesto which was published under the title The Scientific View of the World: The Vienna Circle.” Yet as Wittgenstein himself protested again and again in the Tractatus, the propositions of natural science “have nothing to do with philosophy” (6.53); “Philosophy is not one of the natural sciences” (4.111); “It is not problems of natural science which have to be solved” (6.4312); “even if all possible scientific questions be answered, the problems of life have still not been touched at all” (6.52); “There is indeed the inexpressible. This shows itself; it is the mystical” (6.522). None of these sayings could possibly be interpreted as the views of a man who had renounced metaphysics. The Logical Positivists of the Vienna Circle had got Wittgenstein wrong, and in so doing had discredited themselves.[/quote]
Meanwhile, I allow that anyone is welcome to stipulate their own terminology and even to propose an entirely different framework for consideration of mathematics.
When you reject such, and insist on the other, it's dogmaticism.
TonesInDeepFreezeJune 06, 2021 at 14:57#5470520 likes
When you reject such, and insist on the other, it's dogmaticism.
To what do 'such' and 'other' refer?
Metaphysician UndercoverJune 07, 2021 at 00:59#5472400 likes
Reply to TonesInDeepFreeze Let me explain it clearly then, since you seem to be having trouble understanding. When someone accepts, believes in, and adheres to principles which have not been empirically proven and argues a philosophy which gives the highest esteem to such principles, that is dogmatism. Many principles employed in modern mathematics, axioms, have not been empirically proven. So to believe strongly in, and argue from such principles, even labeling those who doubt these unproven principles as cranks, is dogmatism.
TonesInDeepFreezeJune 07, 2021 at 01:40#5472520 likes
Many principles employed in modern mathematics, axioms, have not been empirically proven.
It's dogmatic of you to preclude that interest in abstract mathematics must be dogmatism.
And I have not claimed that abstract mathematics has the kind of direct empirical correspondence that you dogmatically require. However, I do observe that it is used for, and has been a crucible for, the sciences and for the very technology you are using to be a condescending boor.
Moreover, whatever one's regard for mathematics, it is not dogmatism to point out what its actual formulations are as opposed to dogmatic attacking ignorance and misconstrual, such as yours, of the formulations.
labeling those who doubt these unproven principles as cranks
I have never faulted anyone for doubts about axioms or abstract mathematics. Indeed, the literature of debate regarding doubts and criticisms of various mathematical approaches fascinates and excites me and has my admiration. What I have done though is point out when people blindly attack mathematics from ignorance, confusion, stubbornness and dogmatism. There is a Grand Canyon of difference between, on one hand, doubts and reasoned critique and, on the other hand, attacks from willful ignorance, frothing confusion, and sophomoric dogmatism.
There is a Grand Canyon of difference between, on one hand, doubts and reasoned critique and, on the other hand, attacks from willful ignorance, frothing confusion, and sophomoric dogmatism.
This is true. That is why in such matters, circumspection might often be called for.
Metaphysician UndercoverJune 07, 2021 at 10:30#5473800 likes
In a conversational way, that's an okay summary. But it actually describes Tarski's result pursuant to incompleteness.
To me, it's odd that Church's theorem (undecidability of the set of theorems of the pure predicate calculus) and Tarski's theorem (the inexpressiblity of a truth predicate in a consistent theory) came in 1936, six years after incompleteness, but those two results are pretty easy corollaries. Why did it take six years to publish the proofs?
Hmm, maybe they depend on Rosser's 1936 improvement of Godel's result? I don't know.
One of these days I need to refresh my knowledge of the proof details for Church and Tarski results.
(1) That translation is different from the one in the van Heijenoort book, which, if I recall correctly is the only one approved by Godel. I don't mention that to discredit your quote or the translation it came from. Rather, just to say that in general and in principle, it may be better to refer to the approved translation.
(2) Indeed, Godel mentioned that his proof deploys the liar paradox but with 'provable' instead of 'true'. But that is not itself the observation that if we substituted 'true' for 'provable' then the system is inconsistent.
(3) Godel may have made that observation (I don't recall), and it would seem obvious anyway, but it was Tarski who put the formal cherry on top with Tarski's theorem.
Did he recognize that truth is a metamathematical notion, not part of the mathematics itself?
I'm not sure to how summarize Godel's view on that at the time of the proof. But that's not the reason for using provability rather than truth. The reason for using provability is that it works.
It's interesting that Godel landed on the idea of incompleteness from his failure to proof the consistency of analysis. Before incompleteness was even a twinkle in his eye, he was unsuccessfully trying to prove the consistency of analysis, and he saw an opportunity in that failure that would possibly prove incompleteness (I don't know the details about that though).
Note that subsequent to incompleteness, Tarski did provide a framework for handling 'truth' as a formal mathematical notion. It is metamathematical, but metamathematics is also mathematics. Mathematical logic and model theory are mathematics.
Nowhere in the paper so far as I can understand it does he make clear either that or why it doesn't work with true.
If that is the case (I don't recall all of that paper now), then it supports my point.
Anyway, his task was to prove the theorem, which he did. Explaining why it wouldn't work with 'true' is extra.
The proof works because 'provable' is arithmetizable while 'true' is not for a consistent theory. If 'true' were arithmetizable, then the theory would be inconsistent (Tarski).
No. 'true' is formalized, though not in 1930. But the important thing for incompleteness is that 'true' is not arithmetizable in a consistent theory.
More exactly: a truth predicate cannot be defined in the language of a consistent theory. In other words, a predicate T such that Tn evaluates to true in the standard model if and only if n is the Godel-number of a sentence that evaluates to true in the standard model is not definable from the language of the theory. (I think I have that right.)
TonesInDeepFreezeJune 07, 2021 at 19:53#5475600 likes
Is the sense of this reproducible here, in a conversational way. in a non-onerous number of sentences?
I can't do it some justice without some technicalities, but I will have to skip some defintions and to fudge some technicalities that would be handled better in a textbook. And to be cogent in a short space, I'll put some things in my own terms.
As is famous, Tarski proposed a correspondence notion of truth. For example:
'1+1 = 2' is true if and only if one plus one is two. [Using numerals and '+' and '=' on the left of the biconditional but words on the right of the biconditional, only to emphasize a certain difference explained in the next paragraph.]
That is not circular, since the '1+1 =2' is purely syntactical. and "'1+1 =2' is true" is a statement about the syntactical object '1+1=2', while the right side expresses a state of affairs.
Now, how do we formalize the notion of a 'state of affairs'?
Answer: With formal models.
A model is a certain kind of function from the signature (and also the universal quantifier in Enderton's book) of the formal object language:
The universal quantifier maps to a non-empty set called 'the universe'
n-ary predicate symbols (including n=0) map to n-ary relations on the universe.
n-ary function symbols (including n=0) map to n-ary functions on the universe.
For example, with the language of arithmetic, the standard model is:
the universal quantifier maps to the set of natural numbers
'=' maps to the identity relation on the set of natural numbers (that is "hardwired" since we are in a context of first order logic with identity)
'S' maps to the successor function on the set of natural numbers
'+' maps to the addition function on the set of natural numbers
'*' maps to the multiplication function on the set of natural numbers
Then the 'truth value' for sentences is inductively (mathematical induction, not empirical induction) defined (too many details for me to mention here). First are clauses for the denotations of the terms (atomic terms, then inductively, compound terms), then satisfaction for atomic formulas, then the connectives, then the quantifier, then a move from satisfaction of formulas to truth of sentences.
We are concerned not just with the object-language and a meta-language, but the object-theory and a meta-theory.
With a meta-theory, there a models of the object-theory. Per those models, sentences of the object-language have truth values. So the Godel-sentence is not provable in the object-language but it in a meta-theory, we prove that the Godel-sentence is true in the standard model for the language of arithmetic. Also, as you touched on, in the meta-theory, we prove the embedding of the Godel-sentence into the language of the meta-theory (which is tantamount to proving that the sentence is true in the standard model). That is a formal account of the matter. And in a more modern context than Godel's own context, if we want to be formal, then that is the account we most likely would adopt.
Godel himself did not refer to models. Godel's account is that the Godel-sentence is true per arithmetic, without having to specify a formal notion of 'truth'. And we should find this instructive. It seems to me that for sentences of arithmetic, especially ones for which a computation exists to determine whether it holds or not, we are on quite firm ground "epistemologically" to say, without quibbling about formality, that the sentence is true when we can compute that it does hold.
Mathematics is true a priori and so empirical validation isn't relevant.
As Wigner once suggested, there's a deep connection between mathematics and science. Per Aristotle, mathematics is the abstraction of the sensible - taking that which does not exist in separation and considering it separately:
For if attributes do not exist apart from the substances (e.g. a 'mobile' or a pale'), pale is prior to the pale man in definition, but not in substantiality. For it cannot exist separately, but is always along with the concrete thing; and by the concrete thing I mean the pale man. Therefore it is plain that neither is the result of abstraction prior nor that which is produced by adding determinants posterior; for it is by adding a determinant to pale that we speak of the pale man.
It has, then, been sufficiently pointed out that the objects of mathematics are not substances in a higher degree than bodies are, and that they are not prior to sensibles in being, but only in definition, and that they cannot exist somewhere apart. But since it was not possible for them to exist in sensibles either, it is plain that they either do not exist at all or exist in a special sense and therefore do not 'exist' without qualification. For 'exist' has many senses.
...
The same account may be given of harmonics and optics; for neither considers its objects qua sight or qua voice, but qua lines and numbers; but the latter are attributes proper to the former. And mechanics too proceeds in the same way. Therefore if we suppose attributes separated from their fellow attributes and make any inquiry concerning them as such, we shall not for this reason be in error, any more than when one draws a line on the ground and calls it a foot long when it is not; for the error is not included in the premisses.
Each question will be best investigated in this way - by setting up by an act of separation what is not separate, as the arithmetician and the geometer do.
t since it was not possible for them [mathematical objects] to exist in sensibles either, it is plain that they either do not exist at all or exist in a special sense and therefore do not 'exist' without qualification. For 'exist' has many senses..
So, does a number, say the number 7, exist? You will say - of course, you just wrote it. But that's a symbol, which denotes a quantity, a numerical value. Different symbols can refer to the same number, but the quantity or count is what the number is, and that is something that only can be grasped by a mind capable of counting; hence, it's an 'intelligible object'.
Here is a Platonic rejoinder, consisting of a passage about Augustine's view of intelligible objects.
Intelligible objects must be independent of particular minds, because they are common to all who think. In coming to grasp them, the individual mind does not alter them in any way, it cannot convert them into its exclusive possessions or transform them into parts of itself. Furthermore the mind discovers them rather than forming them or constructing them, and its grasp of them can be more or less adequate...
...certain intelligible objects - for example, the indivisible mathematical unit [i.e. prime numbers] - clearly cannot be found in the corporeal world (since all bodies are extended, and hence divisible.) These intelligible objects cannot therefore be perceived by means of the senses; they must be incorporeal and perceptible by reason alone.
...We refer to mathematical objects and truths to judge whether or not, and to what extent, our minds understand mathematics. We consult the rules of wisdom to judge whether or not, and to what extent, a person is wise. In light of these standards, we can judge whether our minds are as they should be. It makes no sense, however, to ask whether these normative intelligible objects as they should be; they simply are, and are normative for other things.
In virtue of their normative relation to reason, Augustine argues that these intelligible objects must be higher than it, as a judge is higher than what it judges. Moreover, he believes that apart from the special sort of relation they bear to reason, the intrinsic nature of these objects shows them to be higher than it. These sorts of intelligible objects are eternal and immutable; by contrast, the human mind is clearly mutable. Augustine holds that since it is evident to all who consider it that the immutable is clearly superior to the mutable (it is among the rules of wisdom he identifies), it follows that these objects are higher than reason.
TonesInDeepFreezeJune 08, 2021 at 03:20#5477100 likes
That entire passage is merely a report of notions and terminology of mathematical logic. It's nowhere even close to a philosophical statement. Except the last sentence, which does have a philosophical aspect. And philosophically it is very little - certainly not a philosophical stance and certainly not "dogmatic".
In that last sentence, I merely say that it seems to me that when we make correct computations in arithmetic, we can take the results as true. To say this more fully: Such computations may be reduced to primitive manipulation of such things as, say, plain tally marks - the most simple, most direct mathematical reasoning that I personally can imagine. Put another way, this is merely clerical attention to mechanical procedures. Now, if someone wants to express extreme doubts of computational arithmetic, then I would say, "If you think we are not justified in accepting truth from even the most simple results of manipulation of tally marks, then what mathematical knowledge do you think is justified?" I don't even claim that the person would not have a satisfactory answer. I only say that I personally don't know of one.
That is quite on the exact opposite end of the spectrum from dogmatism.
The key statement in Gödel's argument is: This sentence is unprovable.
The "argument" A (Adele) proceeds as follows,
1. If this sentence is provable then this sentence is unprovable [Gödel's key premise]
2. This sentence is provable [assume for reductio ad absurdum]
3. This sentence is unprovable [1, 2 MP]
4. This sentence is provable and this sentence is unprovable [2, 3 Conj]
5. This sentence is unprovable [2 - 4 reductio ad absurdum]
However...
Gödel's key premise is problematic,
2. If this sentence is provable then this sentence is unprovable [Gödel's key premise]
7. This sentence is unprovable or this sentence is unprovable [2 Imp]
8. This sentence is unprovable [7 Taut]
I've used only equivalence rules of natural deduction which means that Gödel's key premise, 2. If this sentence is provable then this sentence is unprovable is logically equivalent to 8. This sentence is unprovable.
If so, "argument" A becomes,
1. Thus sentence is unprovable [Gödel's key premise via substitution of "if this sentence is provable then this sentence is unprovable" = " this sentence is unprovable"]
2. This sentence is provable [assume for reductio ad absurdum]
3. This sentence is provable and this sentence is unprovable [1, 2 Conj]
4. This sentence is unprovable [2 - 3 reductio ad absurdum]
Notice, the conclusion, line 5 appears in the premises, line 1 [Gödel's key premise]. In other words, Gödel's argument begs the question, is circular and therefore, fallacious.
TonesInDeepFreezeJune 08, 2021 at 05:56#5477330 likes
Reply to TheMadFool I did start this thread, and I do think Tones asks a reasonable question. You’re continually entering these long sequences of symbolic code as if they mean something. So he’s saying, based on what? You’re claiming this is something Godel says, so, like, provide the citation.
I did start this thread, and I do think Tones asks a reasonable question. You’re continually entering these long sequences of symbolic code as if they mean something. So he’s saying, based on what? You’re claiming this is something Godel says, so, like, provide the citation.
Well, good advice Wayfarer. Truth be told, the contents of my post is drawn in full from the video in your OP. It's all in the video.
Reply to Wayfarer Please do but only if you feel like it though. TonesInDeepFreeze seems intelligent and well-read but his playing style is more brute force like the Deep Blue supercomputer which, I have to admit, defeated the world chess champion, Garry Kasparov despite...everything I guess.
t since it was not possible for them [mathematical objects] to exist in sensibles either, it is plain that they either do not exist at all or exist in a special sense and therefore do not 'exist' without qualification. For 'exist' has many senses..
— Aristotle's Metaphysics 13.1077b-1078a [Book XIII, Part 2 - Part 3]
So, does a number, say the number 7, exist? You will say - of course, you just wrote it.
But that's a symbol, which denotes a quantity, a numerical value. Different symbols can refer to the same number, but the quantity or count is what the number is, and that is something that only can be grasped by a mind capable of counting; hence, it's an 'intelligible object'.
Here is a Platonic rejoinder, consisting of a passage about Augustine's view of intelligible objects.
Augustine saw the divine mind as the ground for universals. Whereas for Aristotle, it's the concrete situations themselves, such as seven apples in a bowl, that ground the use of those terms.
In that last sentence, I merely say that it seems to me that when we make correct computations in arithmetic, we can take the results as true.
Strictly speaking, when we make correct computations in arithmetic, the results are logically valid. Following correct procedure results in a valid conclusion. Do you recognize the commonly held distinction between true and valid? A valid conclusion is not necessarily true, because it requires also that the premises are true. If we hold that axioms (as premises) are neither true nor false, or that truth and falsity is not relevant to axioms, then we cannot claim that correct computations provide us with truth.
OK. But I just wanted to point out that there are other views, such as Aristotle's, where mathematics and logic aren't considered to be a priori or exempt from empirical validation. Instead, for Aristotle, mathematics and logic were sciences of quantity and reasoning respectively. The qualitative difference to other empirical investigations is just the degree of abstraction and generality employed.
[quote=An old adage]A picture is worth a thousand words.[/quote]
[quote=Wikipedia][...]complex and sometimes multiple ideas can be conveyed by a single still image, which conveys its meaning or essence more effectively than a mere verbal description.[/quote]
I came across a nice proof of Godel's Incompleteness Theorem that utilizes computability and Cantor's diagonal argument (UC Davis lecture: Part 1 and Part 2). My summary below.
Suppose we have a list of all possible computable functions (programs) in some language (say, Python) that each accept a positive integer as input and produce either 0 or 1 as output. A computable function is a finite string of symbols that, when executed, produces an output in a finite amount of time.
Now consider a table that lists all those computable functions vertically (ordered by string length and symbol index) and the function outputs for each positive integer horizontally.
The above table shows the first three programs and the fifteenth program, with positive integer inputs 1, 2, 3 and 15. As an example, the outputs might be:
Now we define a function f-diag (called f-bar in the lecture) as:
f-diag(i) = 1 - f_i(i)
i is a positive integer that appears in three places in that definition - as the input to function f-diag, as the index to a function in the computable functions table (i.e., the i-th function), and as the input to that indexed function. Per the above table, the outputs for f-diag (calculated by inverting the diagonal elements in the table) would be:
Note that f-diag differs from every computable function by at least one input/output pair. So f-diag is not on the list of computable functions and therefore cannot itself be a computable function.
Now consider some statements about those functions.
S1: "f2(2) = 0"
S1 states that the output of function f2 with input 2 is 0. Per the table above, S1 is true. Furthermore, we can prove the statement by executing function f2 with input 2 and it will output 0 in a finite time. What "prove" means here is that there is a mechanical procedure for obtaining the output in a finite time which, in this case, is provided by function f2.
S2: "f1(3) = 0"
Per the table, S2 is false. We can prove the negation of S2 by executing function f1 with input 3 and it will output 1 in a finite time.
S3: "f-diag(2) = 1"
Per the f-diag table, S3 is true. But we lack a mechanical procedure for proving it since, as shown earlier, f-diag is not a computable function. Furthermore, any computable function is going to produce a different output to f-diag for at least one input (due to the diagonalization). So the proof system would either fail to derive a true statement about f-diag for such an input (and therefore would be incomplete) or else would derive a false statement about f-diag for such an input (and therefore would be inconsistent).
Which just is Gödel's First Incompleteness Theorem: In any rich-enough [*] formal proof system that proves only true statements there are true statements that can't be proved.
--
[*] A system is rich-enough if it can express f-diag statements such as S3 above. f-diag is a well-defined function and S3 is a statement about positive integers just as S1 and S2 are.
Comments (148)
Remember, math is not like constructing a skyscraper, putting in a firm foundation before building the edifice. In math the edifice was largely in place, and the foundations were added afterwards.
However, both ornithology and philosophy of science are still of interest!
To be fair, Feynman liked to play a wiseass like that; but in fact he was quite a thoughtful philosopher of science.
It oversimplifies to the point of being terribly misleading. One glaring mistake is not recognizing that undecidability follows immediately from incompleteness.
And the visual gimmicks and props are not helpful.
//one of the comments: "There is something genuinely reassuring in knowing that nearly 5 million persons have watched this video in just over 1 week..."//
It basically just boils down to how any language capable of formulating e.g. a proof of arithmetic is also capable of formulating self-referential sentences to which there cannot be assigned only one or the other boolean truth value: they must be assigned by the language either neither truth value (so the language is incomplete) or else both truth values (so the language is inconsistent).
It's only in a meta-language, being used to discuss that language as an object itself, that we can say that some such statements (in the object language, about the object language) are true; but in that meta-language we can also prove that those statements are true. The meta-language will itself also be able to formulate statements about itself to which it cannot consistently assign any single truth value, but those statements in turn can only be called "true" in a meta-meta-language, which will also be able to prove those statements (in the first meta-language, about the first meta-language).
There's never a statement in any given language that is both definitely true according to the rules of that language and also not provable in that language, because to be definitely true according to the rules of a language just is to be provable in that language.
If we were to take away anything of philosophical import from Godel, it would be that we should be using either a paraconsistent logic (where statements can be both true and false without explosion) or an intuitionist logic (where statements can be neither true nor false).
What you say just seems wrong for the simple reason that the truth of statements that are not provable cannot be ruled out; we don't know if they are true or not. In other words there can be truths which we cannot determine to be such, or at least it cannot be ruled out that there are.
I didn't note anything new, although this is a particularly clear pop-explanation. Nice.
See Godel, God, and knowledge by way of an example.
Thank you. I thought it rather a stylish presentation. I like that guy’s channel.
It is a good channel. Also I think he lives somewhere near me these days because I keep seeing familiar places in the backgrounds of his videos... like in this one, he appears to be hiking near Camino Cielo above Santa Barbara, and in another recent one about soft robots he was at UCSB.
Quoting Janus
I'm not saying that unprovable statements are definitely false, so this is a non-sequitur.
Yeah, I particularly liked his use of Godel flash cards, as it were, to get the point across. Almost no pop explanations ever mention Godel numbers in their presentation of the results.
Yep.
Think of it as lies for children.
Visual gimmicks and props are not required. One can give a talk orally and with supporting text and/or non-gimmicky visuals.
And I don't even object to visuals and props, except my point is that the ones in that video are stupid. The video is a collection of baubles.
And the video is a shallow attempt at entertainment while being not very informative, not clear even as a simplification, and egregiously misleading at certain points.
An example of the stupidity is spending time on Godel's inanition. It has nothing to do with the subject. And the video includes several seconds holding on a cartoon representation of a plate of food, suggesting that is what Godel passed up in refusing to eat. As if we need to be shown what a plate of food looks like. How childishly stupid.
That is not at all a reasonable summary of Godel's theorem. Just to start: languages are not what are complete or incomplete, but rather theories are complete or incomplete. Also, it is crucial to understand that Godel's theorem has a purely syntactic part that does not require semantic notions of truth and falsehood.
Quoting Pfhorrest
Paraconsistency is a way out of incompleteness, but not on account of considerations of truth and falsehood but because contradictions are allowed. Again, Godel's theorem has a purely syntactical aspect as well as its semantical implications too.
Intuitionist formal logic is a proper subset of classical formal logic. Intuitionist logic is not a way out of incompleteness.
"I'm going to give you an extremely simplified version of some very complicated mathematics. These simplifications gloss over crucial technical details; thus the simplifications may be misleading if one does not at some point go on to understand the actual mathematics. So, we must be extremely careful not to extrapolate philosophical conclusions from our very cursory treatment of this technical subject."
First off, thank you for the video. It's uncanny, you know, how you seem to be able to find good quality videos on the www and by quality I'm not referring to the video resolution. You've made what is essentially chance into an art. It must take both intelligence and loads of luck to boot to turn what is essentially a roll of a die into a skill. Kudos! Thanks again.
Last I checked, Godel's incompleteness employs a variation of the liar sentence which, as you know, is "this sentence is false." According to the video, Godel's version of it is, K (for Kürt) = "the sentence with Godel number g is unprovable", the sentence with Godel number g being K itself. Thus, if K's provable, then it's unprovable [inconsistent because of the contradiction] and if K's unprovable then some mathematical truths are unprovable [incomplete].
As you might've already guessed, at the heart of Godel's therems lies the liar paradox. Before I go any further I need to draw your attention to the rather odd fact that Godel and anyone else who uses different versions of the liar sentence for whatever purposes is, all said and done, resorting to a L-I-A-R. Would you or anyone put to service a liar to prove something, anything? Perhaps I'm being too dramatic and perhaps I'm barking up the wrong tree; after all, the word "liar" may have been used just to grab our attention - only for effect, nothing else.
That out of the way, let's revisit K = the sentence with Godel number g is unprovable and the argument presented in the video which hopefully is a variation, salva veritate, of Godel's own.
Argument A [Adele, Godel's wife]
1. K is provable [assume for reductio ad absurdum]
2. If K is provable then K is unprovable [K = the sentence with Godel number g (K itself) is unprovable]
3. K is unprovable [1, 2 Modus Ponens]
4. K is provable and K is unprovable [contradiction] [..Math is inconsistent]
Ergo,
5. K is unprovable [1 - 4 reductio ad absurdum][..Math is incomplete]
A few points that seem worth mentioning.
a) Look at N (Nimbursky, middle name of Godel's wife) = premise 2. If K is provable then K is unprovable [K = the sentence with Godel number g (K itself) is unprovable]. The assumption that has to be made for argument A to do its job of breaking math as it were is that N makes sense, in logical terms, makes sense implies that a truth value can be assigned to it.
The first clue that something's off is that N is a derivative of the liar sentence and we know that the liar sentence doesn't make sense. One could say that the liar sentence is a poisoned well so to speak and every bucket of water, N being one, drawn from it will be lethal or, in this case, highly dubious. Common sense! No?
b) Consider now the fact that argument A is a reductio ad absurdum which, as you know, derives a conclusion and uses that to reject/negate one or more of the assumptions made in the preceding lines of an argument. If you're not familiar, a reductio argument looks like this:
1. p
2. q & ~q [inferred from p]
Ergo,
3. ~p
Now in the argument A, the following assumptions/premises occur
1. K is provable
2. If K is provable then K is unprovable [K = the sentence with Godel number g (K itself) is unprovable]
The two assumptions lead to the contradiction below,
4. K is provable and K is unprovable
We are now justified in rejecting "one" of the premises but it doesn't necessarily have to be the one Godel has rejected which is 1. K is provable. After all, a reductio absurdum doesn't actually identify which premise is false. A reductio ad absurdum is like a detective in faer earlier stages of a murder investigation - fae knows only that someone is the murderer but doesn't know who the murderer is. Thus, I could reject N = 2. If K is provable then K is unprovable [K = the sentence with Godel number g (K itself) is unprovable and if I do that Godel's argument falls apart.
Given premise 2. If K is provable then K is unprovable [K = the sentence with Godel number g (K itself) is unprovable] has highly ignoble origins (the liar sentence), shouldn't we reject it rather than reject 1. K is provable, a perfectly reasonable proposition?
c) There's another issue with statements like N = 2. If K is provable then K is unprovable [K = the sentence with Godel number g (K itself) is unprovable].
Given a proposition P,
1. If ~P then P
2. ~~P or P [from 1 implication]
3. P or P [2 double negation]
4. P [3 tautology]
The statement, If ~P then P can be thought of as P itself, it can be reduced to P. In other words, the conditional if ~P then P is an illusion of sorts because it actually means P
Let's look at the version of the liar sentence that Godel uses which is, if K is provable then K is unprovable.
1. If K is provable then K is unprovable
2. ~K is provable or K is unprovable [from 1 implication]
3. K is unprovable or K is unprovable [from 2, ~K is provable = K is unprovable]
4. K is unprovable [3 tautology]
In essence, 1. K is provable then K is unprovable is logically equivalent to (I've used only equivalence rules of natural deduction), is nothing but, the statement 4. K is unprovable wearing heavy disguise.
What this means is that Godel's argument as presented in the video becomes,
1. K is provable [assume for reductio ad absurdum]
2. K is unprovable [If K is provable then K is unprovable = K is unprovable]
3. K is provable and K is unprovable [1, 2 Conjunction]
Ergo,
4. K is unprovable [1 - 3 reductio ad absurdum]
Did you notice what went wrong? The conclusion, 4. K is unprovable is also a premise 2. K is unprovable. A petitio principii.
Aw shucks.... :yikes:
Quoting TheMadFool
I must confess that I didn't. I've not studied symbolic logic, and my mathematics is rudimentary, but I'm interested in why Godel's Theorems 'are unquestionably among the most philosophically important logico-mathematical discoveries ever made' (says this article.) I feel as though I intuitively understand why that is, but when I read up on it, I find it very hard to follow. Which is why I found that video was helpful although as we see, opinions are divided.
Seriously, luck's on your side and/or you know exactly which words to type into the search box if every search you do takes you to high quality material.
Quoting Wayfarer
Get your hands on an introductory course on logic. It'll take about a month to get a good understanding of basic logic. Some call it, derogatorily I suspect, baby logic but, if you ask me, that's a misnomer. I guarantee that you won't regret it.
Returning to the main point in re Godel's argument, the version in the video, it proceeds as follows:
a) K = the sentence with the Godel number g is unprovable
b). The sentence with the Godel number g is K itself.
Suppose there's a proof for K. It would prove K is unprovable. That's a contradiction: the unprovable is provable.
1. If K is provable then K is unprovable (Godel's key premise)
2. K is provable (assume for reductio ad absurdum)
3. K is unprovable (from 1, 2)
4. K is provable and K is unprovable (2, 3 taken together)
Ergo,
5. K is unprovable (1 to 4 reductio ad absurdum)
The problem is premise 1. If K is provable then K is unprovable is logically equivalent to the statement, K is unprovable. See vide infra,
If K is provable then K is unprovable = K is unprovable or K is unprovable = K is unprovable
In other words, I can substitute "K is unprovable" for "K is provable then K is unprovable" and then Godel's argument becomes,
1. K is unprovable [because, if K is provable then K is unprovable = K is unprovable]
2. K is provable
4. K is provable and K is unprovable (2, 3 taken together)
Ergo,
5. K is unprovable (1 to 4 reductio ad absurdum)
Notice statement 1 (Godel's key premise) = statement 5 (the conclusion). This is, as you already know, a circulus in probando (circular argument).
I didn't say you did say that ( although I guess it might follow from what I thought you said). You said:
Quoting Pfhorrest
There may be statements which are definitely true, which really just means true, (since, logically, anything true must definitely be true) even if they cannot be proven, unless by "definitely true' you mean known to be true. I didn't think you meant that, because that seems a silly thing to say, but if you did mean that then what you say is trivially true, has no bearing on Gödel's theorems, and is thus irrelevant.
Quoting Janus
What is mathematical truth is an open question in the philosophy of mathematics that has been much debated over the last 100 years, since Tarsky resurfaced it. This exchange illustrates the traditional axis of controversy between Platonists and formalists, realists and anti-realists.
As I mentioned earlier, Godel uses the liar paradox to wit, the sentence L = This sentence is false. Such sentences are referred to as self-referential but that's an incomplete description. There are two characteristics that L has,
1. Self-reference. This sentence is false (not true).
2. Negation that causes, how shall I put it?, tension between what's being negated and what's part and parcel of the self that's being referred to. This sentence is false (not true)
A few things that come to mind:
a) Descartes' cogito argument. A variation of it would be: I do not exist. When one uses the "I", it appears that existence is baked into it. Then comes the negation "do not exist" which denies what the "I" incorporates viz. existence.
b) An interesting but probably nonsubstantive quality of L is that it refers to itself, yes, but, if my English is correct, in the third person ("this") and not "I" (first person). It kinda creeps me out - there's another possibly but not necessarily dangerous agency - the true but hidden liar - who our poorly evolved "spider sense" has detected and that's why we feel more comfortable using "this" and not "I". Warning! I'm prone to flights of fancy but then there's the Cartesian deus deceptor problem we haven't yet solved.
c) What about the Buddhist notion of anatta (non-self)?
If you can correctly extract from the video that Godel's argument is circular, then the video is wrong. We should learn Godel's argument from a carefully written exposition, not from a merely breezy cartoon version.
Quoting TheMadFool
What is the introductory course you have taken?
I see you use some sentential logic, but a good understanding of Godel's theorem requires also predicate logic, some set theory, and a first course in mathematical logic.
(By the way, in a conversational context, spelling out and numbering, as you do, such basic sentential logic as applications of modus ponens and conjunction is gratuitous pedanticism that only clutters up whatever it is you mean to say. People don't need to have such basic reasoning annotated for them.)
Quoting TheMadFool
That is flat out incorrect. Godel uses an argument analogous to the liar paradox, but not the liar paradox and nothing like "this sentence is false". Rather, it mathematically renders "this sentence is not provable in system P".
You are spreading disinformation, Porky.
[quote=Bhartrihari]Sarvam mithy? brav?mi[/quote]
I haven't watched the vid, is it any good? Veritasium is usually pretty good but not always. When I saw that he'd done one on this subject my first reaction was, "Not this sh*t again." Then I remembered that every day, there are people hearing about incompleteness for the first time. So it's fine that people are doing new videos on it. On the other hand, it's labeled, "The hole in mathematics," or "The fatal flaw in mathematics," or some such nonsense, and clearly that's giving a lot of people some wrong ideas. After all, computer science doesn't have a "hole" or a "fatal flaw" just because the halting problem is unsolvable, and that amounts to the same thing.
If you watched this vid, can you tell me if it's giving people false ideas? Or is the video accurate and people are getting the false ideas by themselves?
To me, incompleteness is not a hole or a flaw. It's deeply liberating. It shows that mathematics can never be reduced to a mechanical calculation. Mathematical truth will always transcend mere rules.
You mentioned the benefit of a course in logic. In another thread, I have listed what I consider to be the best textbooks leading to the incompleteness thereom. If you like, I can link to that post. And, for a more casual, everyperson read, I highly recommend:
Godel's Theorem: An Incomplete Guide To Its Use And Abuse - Torkel Franzen
It's readable for people with just a modest knowledge of logic and math, authoritative, pays attention to crucial technicalities but not bogged down with them, very nicely written, entertaining and witty too.
It is terrible. I mentioned why earlier in this thread.
Maybe I need to be double-checked, but my reasoning tells me that undecidability follows right from incompleteness.
Sorry to hear that, I really enjoy his physics videos. He has one on why nobody has ever measured the speed of light that's most ... illuminating.
That article gives an answer. It's a great article. (By the way, Panu Raattkainen is a top notch source on the subject.)
All of the various self-reference paradoxes have always seemed trivial to me, e.g. "This sentence is false." Who cares? Russell's paradox seems just the same, just dolled up in mathematical/logical language. Ditto with Godel's incompleteness theorem. Do these "paradoxes" really have a significant, real-time, practical impact on the effective use of mathematics and computer science in the real world? Or is it only guys who are too smart playing around with trivia as if it mattered?
I seem to recall reading somewhere that Gödel was a mathematical Platonist. Are you suggesting that Gödel's incompleteness theorem would be trivially true on a formalist understanding of mathematics because to be true in a language just is to be proven in that language?
My disagreement with @Pfhorrest seemed to perhaps hinge on his use of the term "definitely true". He hasn't responded to say whether he would claim that "There's never a statement in any given language that is both true according to the rules of that language and also not provable in that language, because to be true according to the rules of a language just is to be provable in that language." (The statement he made leaving out the word "definite").
Would a formalist allow that there could be mathematical truths that cannot be proven? If so do formalists accept that Gödel has proven that there are such truths? If not should they reject his theorem altogether?
[quote=Godel and the nature of mathematical truth, Rebecca Goldstein; https://www.edge.org/conversation/rebecca_newberger_goldstein-godel-and-the-nature-of-mathematical-truth ] Gödel was a mathematical realist, a Platonist. He believed that what makes mathematics true is that it's descriptive—not of empirical reality, of course, but of an abstract reality. Mathematical intuition is something analogous to a kind of sense perception. In his essay "What Is Cantor's Continuum Hypothesis?", Gödel wrote that we're not seeing things that just happen to be true, we're seeing things that must be true. The world of abstract entities is a necessary world—that's why we can deduce our descriptions of it through pure reason.[/quote]
I skimmed it. I will go back and read it again. I found the book you mention, it seems eminently readable from the preview, I will add it to my list. Thank you.
I suspect there's something you're not seeing here. When Bertrand Russell told Gottlieb Frege about the 'barber paradox' it had a momentous impact on Frege's whole life work. I think, in layman's terms, what is at stake is elucidating a set of mathematical and logical principles which are both consistent and complete 'all the way down', so to speak. As the Verisatum video mentions, David Hilbert had said 'we can know, we must know', referring to the 'formalist program', the aim of which was to produce such a complete and consistent set of principles. So, I think paradoxes of self-reference, and later, Godel's theorem, are seen to undermine forever this possibility. 'We don't know, we can't know'. So it has bearing on the limitations of knowledge, as far as I can discern.
From the intro to Gödel's Theorem: An Incomplete Guide to Its Use and Abuse.
You might find the answers in there.
If I may interrupt for a second, Bertrand Russell did not approve of re-stating his paradox as the barber paradox:
[quote=Russell]You can modify its form; some forms of modification are valid and some are not. I once had a form suggested to me which was not valid, namely the question whether the barber shaves himself or not. You can define the barber as "one who shaves all those, and those only, who do not shave themselves". The question is, does the barber shave himself? In this form the contradiction is not very difficult to solve. But in our previous form I think it is clear that you can only get around it by observing that the whole question whether a class is or is not a member of itself is nonsense, i.e. that no class either is or is not a member of itself, and that it is not even true to say that, because the whole form of words is just noise without meaning.[/quote]
Whilst the equation is true according to the rules of the math. The rules of the math cannot prove the equation true. To prove the equation true we need to look outside the rules of the math.
"self-reference" used pejoratively in reference to Godel's theorem is a red herring. The self-reference is seen by looking outside the object language. The theorem can be proven in finitistic combinatorial arithmetic. The proof methods are no more suspect than those of proof in finitistic combinatorial arithmetic.
Quoting T Clark
The proof of the incompleteness theorem does not rely on paradox. Anyway, it's pretty rare for the various non-foundational branches of mathematics, especially applied mathematics to be concerned with the incompleteness theorem. But there are important mathematical questions that are elucidated by the incompleteness theorem, including "There is no general method for deciding whether or not a given Diophantine equation has a solution." That settled a question that even a student of high school algebra might wonder about. Basic mathematical curiosity alone leads to the question whether there is a mechanical procedure to determine whether any given Diophantine equation has a solution. And there are other answers in mathematics that incompleteness elucidates. And the methods and context of the incompleteness theorem led to the earliest developments in computability and recursion theory, as those even became branches of mathematics in light of the techniques and context of the incompleteness proof. And, for philosophy of mathematics, Godel's theorem is a central concern. Perhaps most saliently is that (put roughly) incompleteness settles that Hilbert's hope for axioms that would settle all mathematical questions cannot be achieved.
To be clear, Franzen is taking exception to the theorem being incorrectly co-opted in many of those context.
Aside from whatever SophistiCat might say, it is not the case that formalism regards the incompleteness theorem in that way.
(1) Sentences are not true in a language. They are true or false in a model for a language.
(2) Sentences are not proven just in a language, but rather in a system of axioms and rules of inference.
(3) There are different versions of formalism, and it is not the case that in general formalism regards truth to be just provability.
(4) Godel's theorem in its bare form is about provability and does not need to mention the relationship between truth and proof (though a corollary does pertain to truth). The theorem is about provability, which is syntactical. Even if we had no particular notion of truth in mind, Godels' theorem goes right ahead to show that for systems of a certain kind, there are sentences in the language for the system such that neither the sentence not its negation are provable in the system.
Quoting Janus
Any formal statement can be proven in some system or another. The incompleteness theorem, where it bears on truth, gives us that for a given system S of a certain kind, there are true arithmetical statements that are not provable in S. But those statements are provable in other systems. Even if a statement is arithmetically false, there are systems that prove the statement. Even if a statement if logically false, there are system that prove that statement (though, of course, those systems are inconsistent). So, if one is self-admittedly speaking only quite loosely to say "there are truths that cannot be proven" then we must regard that as standing for the more careful, "for a given system S of a certain kind, there are arithmetic truths that are nor provable in S".
Yes, your quote from the book is well taken. And it is clear in the context of the book, but some people might not realize that context, so I just wanted to underline it.
No.
(1) The Godel sentence is not an equation.
(2) "rules of math" is unclear.
(3) We don't look outside the "rules of math" even given a reasonable understanding of what 'the rules of math" might mean.
(4) A correct way to say it this:
For a given system S of a certain kind, there are statements that are arithmetically true but that are not provable in S.
"of a certain kind" can be rendered as "that is recursively axiomatizable, consistent, and and extension of Robinsion arithmetic"
"arithmetically true" can be rendered as "true in the standard model of the language of first order Peano arithmetic):
So the statement is not provable in the system. But when we look at the standard model, we see that the statement is true. Godel didn't himself refer to models, but it's the way we would formalize it now. And if we don't want to be so pedantic to formalize with models, we can say that we see that the statement is true by "outside the system" just looking at the way the statement was formulated and how it relates to ordinary arithmetic.
That's pretty close without splitting hairs technically, as we would split those hairs in a more formal treatment.
But then, on the other hand, we have many popular intellectuals and scientists proselytising science as a world-view, telling us that science understands the world better than we do ourselves. I think that’s why Godel’s theorem has been seized on - rightly or wrongly - as a foil against the proselytising scientists - the Hawkings and De Grasse Tysons of the world who seem to claim ‘secret knowledge’ that none of us can access without years of study, but who also evince little insight into classical humanism and philosophy proper.
Anyway, I suppose to answer my own objection, that is the rationale behind books such as Franzen’s, which at least enable the educated layman to better consider the issue on its own terms.
I don't take exception to the Goldstein quote. But her book about incompleteness needs to be read critically. As I recall (though I can't cite specifics from memory) she gets too casual sometimes to the point of misstating certain key points.
The Franzen book is the one to read.
Thank You Wayfarer
[quote=Wikipedia]Relationship with the liar paradox [of Godel's Incompleteness Theorems]
Gödel specifically cites Richard's paradox and the liar paradox as semantical analogues to his syntactical incompleteness result in the introductory section of "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I". The liar paradox is the sentence "This sentence is false." An analysis of the liar sentence shows that it cannot be true (for then, as it asserts, it is false), nor can it be false (for then, it is true). A Gödel sentence G for a system F makes a similar assertion to the liar sentence, but with truth replaced by provability: G says "G is not provable in the system F." The analysis of the truth and provability of G is a formalized version of the analysis of the truth of the liar sentence.[/quote]
Please carry out your own investigations into the issue if you're interested of course. Good luck.
Yes, analogues.
I think you're exactly right. That was the point of my post. It's not that I think they're wrong and I'm right. I just don't get it. I'm hoping someone will answer my questions - why do these seemingly trivial paradoxes and inconsistencies matter so much? Where do they meet the world?
Quoting Wayfarer
But why, in any practical sense, does that matter? Does it make mathematics less useful or effective in any significant way? It seems Platonic. Forms existing without relation to physical reality.
Quoting TonesInDeepFreeze
Basic mathematical curiosity is a pretty good reason to study something. I don't have any problem with that.
I just want to make it clear - I don't doubt the results of these brilliant mathematicians work. I'm not like one of those relativity deniers who think that I can see something that mathematicians and scientists have worked on for centuries.
See sections 4.4 and 4.5 here:
https://plato.stanford.edu/entries/goedel-incompleteness/
For philosophical concerns about mathemtics see:
https://www.cairn.info/revue-internationale-de-philosophie-2005-4-page-513.htm
and
section 6 here again:
https://plato.stanford.edu/entries/goedel-incompleteness/
I'll spend some time with your referenced text. Then I'll come back later and demonstrate more of my mathematical ignorance.
Thanks.
Isn't it all to do with the foundations of mathematics and logic? 'Foundations' suggests to me something real, a system of thought that can be anchored against a, or the, absolute. Both Frege and Russell were attempting that in different ways. There's an SEP article on Russell's paradox here. So it may not matter in the practical sense of you and I carrying on with our lives, but it is a philosophical issue of great significance.
Even as regards to 'Platonic forms' - if they're real, not in the sense of being only 'in someone's mind', then that's significant. Because it suggests that 'what is real' extends well beyond what is, well, materially existent. If forms, and numbers, are real, then they're real in a different sense to the objects of physics, no matter how subtle. That's what fascinates me about platonism.
Quoting T Clark
Hey we're all in the same boat! In fact I bet my ignorance is bigger than yours!
I resist philosophical labels, but I've come to the conclusion that I probably am a pragmatist. I have a strong resistance to philosophical issues that don't have practical consequences. As I said, let me spend some time reading.
Quoting Wayfarer
Oh, yeah! We'll see about that.
I too will plead, as at this time, especially as I am rusty in the subject, I wouldn't be able to marshal enough knowledge to explicate the details of the mathematics mentioned there.
Here's a more direct reference to Godel's incompleteness theorem vis-à-vis the Liar paradox.
[quote=Wikipedia]Gödel's incompleteness theorems are two fundamental theorems of mathematical logic which state inherent limitations of sufficiently powerful axiomatic systems for mathematics. The theorems were proven by Kurt Gödel in 1931, and are important in the philosophy of mathematics. Roughly speaking, in proving the first incompleteness theorem, Gödel used a modified version of the liar paradox, replacing "this sentence is false" with "this sentence is not provable", called the "Gödel sentence G"[/quote]
Yes, there is an analogy and similarity, but also the modification used makes a great difference too.
Correct but it's not just self-reference, it's also negation of some kind. The self-referential sentence, "I exist" doesn't create problems like the self-referential negation, "I don't exist."
Let's look at the liar sentence L = this sentence (L itself) is false.
According to most books, the logic is as below,
Option 1
1. If L is true then L is false (the liar sentence)
2. L is true (assume)
3. L is false (1, 2 modus ponens)
4. L is true and L is false (2, 3 together, contradiction)
Ergo,
5. L is false (2 - 4 reductio ad absurdum)
Option 2
6. If L is false then L is true (the liar sentence)
7. L is false (assume)
8. L is true (6, 7 modus ponens)
9. L is true and L is false (7, 8 together, contradiction)
Ergo,
10. L is true (7 - 9 reductio ad absurdum)
Now, what's interesting is,
1. If L is true then L is false = L is false or L is false = L is false
That means the argument in option 1 becomes,
11. L is false = If L is true then L is false
12. L is true (assume)
13. L is true and L is false (11, 12 together, contradiction)
Ergo,
14. L is false (12 - 13 reductio ad absurdum)
However, notice line 11 (premise) = line 14 (conclusion). In other words, what was to be proved was assumed beforehand among the premises. The argument is circular which simply means 14. L is false is unwarranted.
Similarly, revisiting option 2, the statement 6. If L is false then L is true = L is true or L is true = L is true
The argument for option 2 then becomes,
15. L is true = If L is false then L is true
16. L is false (assume)
17. L is true and L is false (15, 16 together, contradiction)
Ergo,
18. L is true (16 - 17 reductio ad absurdum)
Notice here again that line 15 (premise) = line 18 (conclusion). Put simply, the conclusion has been assumed in the premises. Circular argument, which means we're not justified in concluding 18. L is true.
What does this all mean? We can arbitrarily assign a truth value to the liar sentence (true/false) but that's where it all stops - all logic beyond that is going to be circular and useless. Since the alleged contradiction of the liar paradox can only occur after an inference which begins with an assumption of a truth value for the liar sentence, and since, as explained above, all such arguments are circular, we're no longer justified to infer anything at all (that includes any further truth value for the liar sentence) from the initial assigned truth value for the liar paradox. Ergo, there being no inferrable truth value, there can be no contradiction. In short, the liar paradox doesn't entail a contradiction at all.
Read more there for more explanation.
On the other hand, see page 44 of Franzen's'Godel's Theorem: An Incomplete Guide To Its Use And Abuse' for a different take on the matter.
In any case, in whatever sense we may reasonably say the proof uses "self-reference", the proof is not vitiated by it, as the proof can be carried out in finitistic combinatorial arithmetic.
The equal sign there and the ones following it are not syntactical. Perhaps you mean the biconditional. If so, rewrite to see whether your argument still holds up,
Leaving out the "definitely"s completely changes the meaning, so no, I wouldn't claim that modified sentence.
We cannot know for sure ("definitely", or "certainly") that some proposition is true, without in the process having proven it, so we cannot know for sure that any given proposition is true-but-unprovable, because to be sure of the first part we would have to violate that second part.
It might in some principled way remain the case that something or another could be true but not provable, but we could never say for sure that we had an example of that, because in somehow saying for sure that something was true, we just would be proving it.
It is unprovable in the system being discussed. It is provably true in the mathematics used to discuss that system.
Quoting Pfhorrest
We have a sure example. It's as sure as finitistic combinatorial arithmetic. It's a quite complicated calculation, but it is still a finite calculation.
I'm back and I've read the texts you referenced. They were very interesting but not very helpful in answering my questions. I don't need to go any further with this except to say I was surprised to see that some of the philosophical claims associated with Godel's theorum are similar to the mystical/philosophical claims associated with quantum mechanics. These in particular struck me:
proves that Mechanism is false, that is, that minds cannot be explained as machines.
They all insist that Gödel’s theorems imply that the human mind infinitely surpasses the power of any finite machine or formal system.
I'm going to leave it here. I appreciate your help. This was fun.
Right, which is what I said a few posts back in that exchange. It's only in the meta-language, not the object language, that we can assign a truth value to the proposition, and in the meta-language we are also able to prove that proposition. The meta-language will itself also contain propositions which it cannot prove, and to which it cannot assign truth values; and in yet another language being used to discuss that first meta-language as an object language, a truth-value can be assigned to the unprovable statements of the first meta-language, but in that even higher-order language those statements can also be proved. You never have both a definite truth value and unprovability on the same level. "True but unprovable" only works when you mix levels: it's true, according to the higher-level system we're using to discuss the lower-level system, and unprovable, according to that lower-level system; but that lower-level system has no idea whether or not it's true (because it's unprovable), and in the higher-level system it's provable (which is how it can be known true).
I agree that we cannot know for sure that some proposition is true without having proven it, and thus it would seem to follow that we cannot know that any given proposition is "true-but-unprovable". But the truth of any proposition is not dependent on our having proven it, but only our knowing of the truth of it is.
If I understood mathematics much better than I do I might be able to offer an opinion as to whether Godel has proved that there are true propositions within mathematics that cannot be mathematically proven.
Perhaps the answer to that last question is no] it seems: is this from the SEP is correct and if I have correctly interpreted its meaning:
"A common misunderstanding is to interpret Gödel’s first theorem as showing that there are truths that cannot be proved. This is, however, incorrect, for the incompleteness theorem does not deal with provability in any absolute sense, but only concerns derivability in some particular formal system or another. For any statement A unprovable in a particular formal system F, there are, trivially, other formal systems in which A is provable (take A as an axiom). On the other hand, there is the extremely powerful standard axiom system of Zermelo-Fraenkel set theory (denoted as ZF, or, with the axiom of choice, ZFC; see the section on the axioms of ZFC in the entry on set theory), which is more than sufficient for the derivation of all ordinary mathematics. Now there are, by Gödel’s first theorem, arithmetical truths that are not provable even in ZFC. Proving them would thus require a formal system that incorporates methods going beyond ZFC. There is thus a sense in which such truths are not provable using today’s “ordinary” mathematical methods and axioms, nor can they be proved in a way that mathematicians would today regard as unproblematic and conclusive.".
Is there any proof that such a "formal system that incorporates methods going beyond ZFC." will or even could be found? In light of all this what you said about there always being a meta-level wherein the unprovable truths within a system can be proven seems questionable.
Even if there were always a meta-level capable of (non-trivially) proving any true mathematical proposition, then there would still seem to be a kind of infinite regress of incomplete systems. Would this
fact render all such proofs non-exhaustive and/ or trivial I wonder (in my mathematical ignorance)?
The general idea you expressed is okay, but I suggest some clarifications and context (much of which you likely know already).
We are concerned not just with the object-language and a meta-language, but the object-theory and a meta-theory.
With a meta-theory, there a models of the object-theory. Per those models, sentences of the object-language have truth values. So the Godel-sentence is not provable in the object-language but it in a meta-theory, we prove that the Godel-sentence is true in the standard model for the language of arithmetic. Also, as you touched on, in the meta-theory, we prove the embedding of the Godel-sentence into the language of the meta-theory (which is tantamount to proving that the sentence is true in the standard model). That is a formal account of the matter. And in a more modern context than Godel's own context, if we want to be formal, then that is the account we most likely would adopt.
Godel himself did not refer to models. Godel's account is that the Godel-sentence is true per arithmetic, without having to specify a formal notion of 'truth'. And we should find this instructive. It seems to me that for sentences of arithmetic, especially ones for which a computation exists to determine whether it holds or not, we are on quite firm ground "epistemologically" to say, without quibbling about formality, that the sentence is true when we can compute that it does hold.
Quoting Pfhorrest
There is an even more fundamental reason that the object-theory does not yield a determination of truth. That is that the object-language does not have a truth predicate. There's a subtle difference: A theorem of the object-theory is true in any model of the theory, so in that sense one would say that the object-theory does determine the truth of certain sentences. But the object-theory does not itself have a theorem that the sentence is true in models of the theory (or else, the object-theory would be inconsistent per Tarski's theorem).
It depends on the definition of 'mathematically proven'.
Quoting Janus
It is easy to see that there are theories that are proper extensions of ZFC ['proper' meaning having theorems that ZFC does not have]. But that doesn't settle the question of whether those theories are within what we consider to be justifiable mathematics, or even the question of what it means to be justifiable mathematics.
Quoting Janus
It is the case that there is an infinite escalation of theories, each proving arithmetical truthts not provable in the lower theories.
Quoting Janus
The theories are not exhaustive, indeed. But I don't see why that would make the proofs trivial.
What gets to me, and maybe you can clarify, is how it could be that we can “compute that it does hold” and yet not have, at some level or another, thereby “proven” it. If we are taking for granted (as proven, in some sense or another, or else just assumed) that arithmetic works the way we usually use it, and an arithmetical operation yields a certain output, have we not consequently proven (or assumed) that output as part and parcel of having proven (or assumed) that arithmetic works in such a way as that?
Doesn’t it mean that there must always be some assumptions? Something that is assumed to be so, but can’t be proven to be so? Isn’t that what the issue is about?
But on that unrelated topic, I am vociferously opposed to justificationism, the usual kind of rationalism (contra critical rationalism), which says that you should reject everything that can't be proven conclusively "from the ground up", because per Agrippa's / Munchausen's trilemma that is inherently impossible. Instead, as a critical rationalist, I think it's fine (and necessary) to run with whatever assumptions you're inclined to, until they can be disproven.
Godel's about whether there are things that are true but aren't provable. And I don't see how we can ever do better than "Maybe? I suppose it's always technically possible, but we can never be sure whether or not there are". Because to sure, we would have to be sure that something was unprovable, and also be sure that it was true -- and I don't see how we could "be sure that it was true" without, in doing so, proving it, and so showing it to be not-unprovable.
As I understand it, Godel only shows something about the relationship between a formal system and statements in it: that some systems can't prove some things they're capable of talking about either way, even though we can know, through in a proof made in a higher-level system, that those things are true.
Isn’t that what I said?
I wasn't specifically referring to Gödel's theorem, but using that example, a strictly formal reading of the first incompleteness result would be like this quote in the Wiki article:
Here the statement is not said to be either true or false; if pressed, an anti-realist* might say that (a) the question of truth is meaningless outside the context of a particular formal system, and (b) in the context of system F the Gödel statement is neither true nor false, since it can be neither proved nor disproved in that system.
On the other hand, you have, no doubt, heard paraphrases to the effect that the Gödel statement is "true but unprovable." Such readings lean on a realist/Platonist understanding of mathematical truth. They would appeal to the structure of the Gödel statement, which states an arithmetical truth.
This is a very crude and clipped summary. Like I said, the question of truth in mathematics and its relation to provability has been investigated and debated at great length. Just searching for works with the words "truth" and "provability" or "proof" in the title will net you several pages of results on Google Scholar.
* I will withdraw the label "formalism" and use instead the more vague "anti-realism" or "anti-Platonism."
Quoting TonesInDeepFreeze
[quote=Stanford Encyclopedia of Philosophy]At the end of Sextus’ discussion in PH II, he clearly signals, as one would expect, that he suspends judgment on whether there are criteria of truth:
You must realize that it is not our intention to assert that standards of truth are unreal (that would be dogmatic); rather, since the Dogmatists seem plausibly to have established that there is a standard of truth, we have set up plausible-seeming arguments in opposition to them, affirming neither that they are true nor that they are more plausible than those on the contrary side, but concluding to suspension of judgement because of the apparently equal plausibility of these arguments and those produced by the Dogmatists. (PH II 79; cf. M VII 444)[/quote]
[quote=Internet Encyclopedia of Philosophy]According to Chisholm, there are only three responses to the Problem of the Criterion: particularism, methodism, and skepticism. The particularist assumes an answer to (1) and then uses that to answer (2), whereas the methodist assumes an answer to (2) and then uses that to answer (1). The skeptic claims that you cannot answer (1) without first having an answer to (2) and you cannot answer (2) without first having an answer to (1), and so you cannot answer either. Chisholm claims that, unfortunately, regardless of which of these responses to the Problem of the Criterion we adopt we are forced to beg the question. It will be worth examining each of the responses to the Problem of the Criterion that Chisholm considers and how each begs the question against the others. [/quote]
Because the reckoning itself is not necessarily in a formal context, so it is not formal proof, though it could be.
I don't know what issue you mean when you ask what the issue is about. But for incompleteness, it's not just a matter of having to assume things to prove things.
That is exactly the most salient oversimplification that causes misunderstanding.
You know the following, but it bears emphasizing:
There is no mathematical statement that isn't provable. That is, for any mathematical statement (even a self-contradiction) there are systems that prove the statement.
Godel's theorem is that for any given system S of a certain kind there are statements F in the language for S that such that S proves neither F nor ~F.
It's a matter of quantifier order:
Godel: For any system S of a certain kind, there exist statements undecided by S.
False: There exist statements F such that for any system S of a certain kind, F is undecided
Quoting Pfhorrest
That seems to me to be a reasonable summary.
I don't think he is. The distinction he's making is very important.
Yes, the theorem itself, as you quoted it, does not mention truth. But from the theorem, we do go on to remark that the undecided sentence is true.
And the statement is neither true or false in the system on an even more fundamental basis than that it is undecided by the system:
Quoting TonesInDeepFreeze
When we're talking about plain arithmetical truths, I don't know why we would have to go down the road of wondering about realism. I mean, non-realists still recognize the truth of arithmetical statements.
I should qualify that remark and others I made along the same lines.
We prove (though not in the object system) that the Godel-sentence is true on the assumption that the object-system is consistent.. That qualification might be regarded as implicit in my remarks, but it is best for me to make it explicit.
Quoting TonesInDeepFreeze
I should put that remark on hold. I need to figure out whether saying that we have a "computation" is correct.
Quoting TonesInDeepFreeze
But if we adhere to strict "truth=provability" principle, then the sentence is not true even in the metatheory, if it assigns truth to sentences subject to their provability in the object theory.
Quoting TonesInDeepFreeze
Anti-realists recognize arithmetical statements as true relative to particular mathematical theories, which are as fictitious as any other such theories. Realists view some mathematical systems, such as arithmetic, as representing an objective, mind-independent reality; for them the mathematical study of such systems can be likened to scientific research.
Again, I want to disclaim that this is a simplistic caricature, but here are some statements in the same spirit by mathematician G.H. Hardy:
(Quoted from another work of Torkel Franzen: "Provability and Truth" (1987))
What "truth=provability" principle do you have in mind? What is its mathematical formulation? Meanwhile, the incompleteness theorem proves that the set of provable sentences does not equal the set of true sentences. [Often now, I'll leave tacit the usual qualifiers such as "provable in system S" and "true in the standard model".]
Quoting SophistiCat
True relative to models of theories. (Of course though, if P is a theorem of a consistent theory S then P is true in any model of S.)
Quoting SophistiCat
Not necessarily for arithmetical theories or even the arithmetical part of broader theories.
I don't think this is what you are discussing, however. I'm trying to see the link between actual mathematics and foundational mathematics in this regard.
So much depends on definitions, it seems.
Quoting TonesInDeepFreeze
Perhaps, then, it depends on whether we have in mind mathematical or philosophical triviality.
Thanks. I have learned from this thread to avoid discussion of this topic in future.
I don’t see how this can be so. The fact that 317 is a prime number is indeed not dependent on your or my assent, but it’s regardless a fact which only a rational mind can grasp, and in that sense is what can be called an intelligible object or object of rational thought. I don’t see how this undermines idealism but rather reinforces it, in my understanding of that term.
This is a great way of stating it! Thank you. :-)
I want to rephrase it a little more consistently so the simple shift in quantification order is more apparent to others:
Godel:
for any system S of a certain kind,
there is at least one statement F such that
S cannot decide F.
False:
there is at least one statement F such that
for any system S of a certain kind,
S cannot decide F.
I'm talking about sentences in the language of arithmetic. I don't know whether these matters bear upon your areas of mathematics.
I am pretty rusty on this stuff, so take this modulo a grain of salt:
The Godel sentence G "says":
"For every n, it is not the case that n is the Godel-number of a proof of the sentence with Godel-number g."
G is a sentence purely in the language of arithmetic. The "it says" about proofs and Godel-numbers is seen and proven (in the meta-theory) with regard to the construction of G per the arithmetization of syntax.
And, G has Godel-number g.
The part "it is not the case that n is the Godel-number of a proof of the sentence with Godel-number g" is a computable property. Let's call it 'C'. So G is of the form:
For all n, Cn.
Now, for concision, let's say we're looking at some particular system S.
Godel-Rosser proves "If S is consistent, then both G and ~G are not theorems of S."
And let's say that by 'true' we mean true in the standard model for the language of arithmetic. Godel did not himself have formal model theory to reference, but in context we may say that his context might as well be tantamount to it. Moreover, we could dispense the formality of models by just agreeing that 'true' means what it ordinarily means to mathematicians who don't care about mathematical logic. For example, '0+0=0' is simply true and '0=1' is simply false.
So, either G is true or ~G is true. So, on that basis alone, we know that there is a true sentence that S does not prove. But that is not constructive - it uses excluded middle and doesn't tell us specifically which one of the two is the true one.
But we can constructively (I think?) show "If S is consistent then G is true" anyway.
Now, let's look at a certain kind of arithmetical sentence. These are sentences in the language of arithmetic all of whose quantifiers are bounded. For example:
"For all n<20, if n is prime then n has a twin prime."
For such sentences, there is an algorithm to decide their truth. Moreover, it is said that from the sentence itself, we can "read off" the algorithm (please don't ask me the technical definition of "read off" - I have not yet pursued how to formalize it).
Now, where I tripped myself up earlier in this thread is that I might have conflated the fact in the above paragraph with the fact that we do easily prove "if S is consistent then G is true", as I am not clear whether that proof is one that also is "read off" from the sentence itself, in context of the construction of the sentence vis-a-vis the arithmetization of syntax.
Wise decision, the dogmatic don't provide reasonable discourse.
What dogmatism do you think you have witnessed?
Quoting TonesInDeepFreeze
You claim that quote is dogmatic. What is your non-dogmatic basis for that claim?
The quote is not dogmatic, I say non-dogmatically. The quote describes the way mathematical logic uses certain terminology and certain other plain facts about mathematical logic. It is apropos to mention those terminological conventions and basics of mathematical logic, since the context of the discussion is Godel's theorem, which is a subject in mathematical logic. Meanwhile, I allow that anyone is welcome to stipulate their own terminology and even to propose an entirely different framework for consideration of mathematics. However such a proposal would be subject to the same scrutiny for coherence and rigor to which mathematical logic is subject. That is it the antithesis of dogmatism.
"It is not our business to set up prohibitions, but rather to arrive at conventions." - Rudolph Carnap
I hate all those Vienna Circle types. That’s one of the reasons I’m going to keep out of these discussions.
The logician says that from certain conventions, axioms, rules, and definitions, certain things follow and certain things do not follow. And the logician allows that people may set up different conventions, axioms, rules, and definitions. And the logician might even allow that proposed frameworks may have value even though they have not yet been axiomatized. The logician admits that definitions are stipulative so that definitions themselves are not inherently true, and that we may regard enquiries that proceed with different definitions. The logician seeks scrutiny of his work and is always eager to correct any errors found in his formulations. The logician admits that certain questions are not answered and that there is much still unknown. All of that is the antithesis of dogmatism.
Whatever the merits or demerits of Carnap's views on metaphysics, the quote I mentioned does have wisdom.
And one may have one's own reasons for eschewing a conversation, but having a dislike of certain philosophers is not much of a rational basis for rejecting a conversation about mathematical logic.
Okay, I admit, that does read like a 'Goofus and Gallant'.
Thanks. Certainly in the study of combinatorics there are conjectures in which all possible cases are finite in number and a computer program can do the job. Like the four color problem.
[quote=Wittgenstein, Tolstoy and the Folly of the Logical Positivists; https://philosophynow.org/issues/103/WittgensteinTolstoy_and_the_Folly_of_Logical_Positivism ] The declared aim of the Vienna Circle was to make philosophy either subservient to or somehow akin to the natural sciences. As Ray Monk says in his superb biography Ludwig Wittgenstein: The Duty of Genius (1990), “the anti-metaphysical stance that united them [was] the basis for a kind of manifesto which was published under the title The Scientific View of the World: The Vienna Circle.” Yet as Wittgenstein himself protested again and again in the Tractatus, the propositions of natural science “have nothing to do with philosophy” (6.53); “Philosophy is not one of the natural sciences” (4.111); “It is not problems of natural science which have to be solved” (6.4312); “even if all possible scientific questions be answered, the problems of life have still not been touched at all” (6.52); “There is indeed the inexpressible. This shows itself; it is the mystical” (6.522). None of these sayings could possibly be interpreted as the views of a man who had renounced metaphysics. The Logical Positivists of the Vienna Circle had got Wittgenstein wrong, and in so doing had discredited themselves.[/quote]
What I meant was, I will henceforth refrain from invoking Godel’s theorems to make philosophical claims.
When you reject such, and insist on the other, it's dogmaticism.
To what do 'such' and 'other' refer?
That's condescension coming from a person who can least afford it.
Quoting Metaphysician Undercover
I haven't argued a philosophy.
Quoting Metaphysician Undercover
It's dogmatic of you to preclude that interest in abstract mathematics must be dogmatism.
And I have not claimed that abstract mathematics has the kind of direct empirical correspondence that you dogmatically require. However, I do observe that it is used for, and has been a crucible for, the sciences and for the very technology you are using to be a condescending boor.
Moreover, whatever one's regard for mathematics, it is not dogmatism to point out what its actual formulations are as opposed to dogmatic attacking ignorance and misconstrual, such as yours, of the formulations.
Quoting Metaphysician Undercover
I have never faulted anyone for doubts about axioms or abstract mathematics. Indeed, the literature of debate regarding doubts and criticisms of various mathematical approaches fascinates and excites me and has my admiration. What I have done though is point out when people blindly attack mathematics from ignorance, confusion, stubbornness and dogmatism. There is a Grand Canyon of difference between, on one hand, doubts and reasoned critique and, on the other hand, attacks from willful ignorance, frothing confusion, and sophomoric dogmatism.
Oh no, @Metaphysician Undercover has plenty of that.
He doesn't have the actual superiority to spend. He's in overdraft with just that one pathetic attempt.
Mathematics is true a priori and so empirical validation isn't relevant.
Quoting TonesInDeepFreeze
This is true. That is why in such matters, circumspection might often be called for.
Take a look at my quote above, and the context from where it's taken. You are arguing a philosophy of truth.
Quoting Wayfarer
That's exactly what makes arguing for mathematics as the purveyor of truth, dogmatism.
I made no argument for a philosophy regarding truth.
Quoting Metaphysician Undercover
And I didn't argue that mathematics is a purveyor truth known a priori and that empirical concerns are not relevant.
Your claim that I am dogmatic is unsupported.
In a conversational way, that's an okay summary. But it actually describes Tarski's result pursuant to incompleteness.
To me, it's odd that Church's theorem (undecidability of the set of theorems of the pure predicate calculus) and Tarski's theorem (the inexpressiblity of a truth predicate in a consistent theory) came in 1936, six years after incompleteness, but those two results are pretty easy corollaries. Why did it take six years to publish the proofs?
Hmm, maybe they depend on Rosser's 1936 improvement of Godel's result? I don't know.
One of these days I need to refresh my knowledge of the proof details for Church and Tarski results.
(1) That translation is different from the one in the van Heijenoort book, which, if I recall correctly is the only one approved by Godel. I don't mention that to discredit your quote or the translation it came from. Rather, just to say that in general and in principle, it may be better to refer to the approved translation.
(2) Indeed, Godel mentioned that his proof deploys the liar paradox but with 'provable' instead of 'true'. But that is not itself the observation that if we substituted 'true' for 'provable' then the system is inconsistent.
(3) Godel may have made that observation (I don't recall), and it would seem obvious anyway, but it was Tarski who put the formal cherry on top with Tarski's theorem.
Quoting tim wood
That is clear from the proof. It doesn't work with 'true' but it works with 'provable'.
Quoting tim wood
I'm not sure to how summarize Godel's view on that at the time of the proof. But that's not the reason for using provability rather than truth. The reason for using provability is that it works.
It's interesting that Godel landed on the idea of incompleteness from his failure to proof the consistency of analysis. Before incompleteness was even a twinkle in his eye, he was unsuccessfully trying to prove the consistency of analysis, and he saw an opportunity in that failure that would possibly prove incompleteness (I don't know the details about that though).
Note that subsequent to incompleteness, Tarski did provide a framework for handling 'truth' as a formal mathematical notion. It is metamathematical, but metamathematics is also mathematics. Mathematical logic and model theory are mathematics.
It's merely an informal heuristic expression.
Quoting tim wood
If that is the case (I don't recall all of that paper now), then it supports my point.
Anyway, his task was to prove the theorem, which he did. Explaining why it wouldn't work with 'true' is extra.
The proof works because 'provable' is arithmetizable while 'true' is not for a consistent theory. If 'true' were arithmetizable, then the theory would be inconsistent (Tarski).
Quoting tim wood
No. 'true' is formalized, though not in 1930. But the important thing for incompleteness is that 'true' is not arithmetizable in a consistent theory.
More exactly: a truth predicate cannot be defined in the language of a consistent theory. In other words, a predicate T such that Tn evaluates to true in the standard model if and only if n is the Godel-number of a sentence that evaluates to true in the standard model is not definable from the language of the theory. (I think I have that right.)
I can't do it some justice without some technicalities, but I will have to skip some defintions and to fudge some technicalities that would be handled better in a textbook. And to be cogent in a short space, I'll put some things in my own terms.
As is famous, Tarski proposed a correspondence notion of truth. For example:
'1+1 = 2' is true if and only if one plus one is two. [Using numerals and '+' and '=' on the left of the biconditional but words on the right of the biconditional, only to emphasize a certain difference explained in the next paragraph.]
That is not circular, since the '1+1 =2' is purely syntactical. and "'1+1 =2' is true" is a statement about the syntactical object '1+1=2', while the right side expresses a state of affairs.
Now, how do we formalize the notion of a 'state of affairs'?
Answer: With formal models.
A model is a certain kind of function from the signature (and also the universal quantifier in Enderton's book) of the formal object language:
The universal quantifier maps to a non-empty set called 'the universe'
n-ary predicate symbols (including n=0) map to n-ary relations on the universe.
n-ary function symbols (including n=0) map to n-ary functions on the universe.
For example, with the language of arithmetic, the standard model is:
the universal quantifier maps to the set of natural numbers
'=' maps to the identity relation on the set of natural numbers (that is "hardwired" since we are in a context of first order logic with identity)
'S' maps to the successor function on the set of natural numbers
'+' maps to the addition function on the set of natural numbers
'*' maps to the multiplication function on the set of natural numbers
Then the 'truth value' for sentences is inductively (mathematical induction, not empirical induction) defined (too many details for me to mention here). First are clauses for the denotations of the terms (atomic terms, then inductively, compound terms), then satisfaction for atomic formulas, then the connectives, then the quantifier, then a move from satisfaction of formulas to truth of sentences.
Then what would you call the following?
Quoting TonesInDeepFreeze
As Wigner once suggested, there's a deep connection between mathematics and science. Per Aristotle, mathematics is the abstraction of the sensible - taking that which does not exist in separation and considering it separately:
Quoting Aristotle's Metaphysics 13.1077b-1078a [Book XIII, Part 2 - Part 3]
So, does a number, say the number 7, exist? You will say - of course, you just wrote it. But that's a symbol, which denotes a quantity, a numerical value. Different symbols can refer to the same number, but the quantity or count is what the number is, and that is something that only can be grasped by a mind capable of counting; hence, it's an 'intelligible object'.
Here is a Platonic rejoinder, consisting of a passage about Augustine's view of intelligible objects.
Quoting Cambridge Companion to Augustine
That entire passage is merely a report of notions and terminology of mathematical logic. It's nowhere even close to a philosophical statement. Except the last sentence, which does have a philosophical aspect. And philosophically it is very little - certainly not a philosophical stance and certainly not "dogmatic".
In that last sentence, I merely say that it seems to me that when we make correct computations in arithmetic, we can take the results as true. To say this more fully: Such computations may be reduced to primitive manipulation of such things as, say, plain tally marks - the most simple, most direct mathematical reasoning that I personally can imagine. Put another way, this is merely clerical attention to mechanical procedures. Now, if someone wants to express extreme doubts of computational arithmetic, then I would say, "If you think we are not justified in accepting truth from even the most simple results of manipulation of tally marks, then what mathematical knowledge do you think is justified?" I don't even claim that the person would not have a satisfactory answer. I only say that I personally don't know of one.
That is quite on the exact opposite end of the spectrum from dogmatism.
The "argument" A (Adele) proceeds as follows,
1. If this sentence is provable then this sentence is unprovable [Gödel's key premise]
2. This sentence is provable [assume for reductio ad absurdum]
3. This sentence is unprovable [1, 2 MP]
4. This sentence is provable and this sentence is unprovable [2, 3 Conj]
5. This sentence is unprovable [2 - 4 reductio ad absurdum]
However...
Gödel's key premise is problematic,
2. If this sentence is provable then this sentence is unprovable [Gödel's key premise]
7. This sentence is unprovable or this sentence is unprovable [2 Imp]
8. This sentence is unprovable [7 Taut]
I've used only equivalence rules of natural deduction which means that Gödel's key premise, 2. If this sentence is provable then this sentence is unprovable is logically equivalent to 8. This sentence is unprovable.
If so, "argument" A becomes,
1. Thus sentence is unprovable [Gödel's key premise via substitution of "if this sentence is provable then this sentence is unprovable" = " this sentence is unprovable"]
2. This sentence is provable [assume for reductio ad absurdum]
3. This sentence is provable and this sentence is unprovable [1, 2 Conj]
4. This sentence is unprovable [2 - 3 reductio ad absurdum]
Notice, the conclusion, line 5 appears in the premises, line 1 [Gödel's key premise]. In other words, Gödel's argument begs the question, is circular and therefore, fallacious.
That is is nothing like Godel's proof. On so many levels it is nonsensical.
What actual version of a Godel's proof have you read in a paper or book?
Well, let's you tell us how Gödel's argument works. C'mon. Out with it!
I asked you what version of Godel's proof have you read in a paper or book. That is, what writing did you base your previous post on?
Well, good advice Wayfarer. Truth be told, the contents of my post is drawn in full from the video in your OP. It's all in the video.
No, that's not Aristotle's position. See below.
Quoting Wayfarer
Augustine saw the divine mind as the ground for universals. Whereas for Aristotle, it's the concrete situations themselves, such as seven apples in a bowl, that ground the use of those terms.
Strictly speaking, when we make correct computations in arithmetic, the results are logically valid. Following correct procedure results in a valid conclusion. Do you recognize the commonly held distinction between true and valid? A valid conclusion is not necessarily true, because it requires also that the premises are true. If we hold that axioms (as premises) are neither true nor false, or that truth and falsity is not relevant to axioms, then we cannot claim that correct computations provide us with truth.
OK. But I just wanted to point out that there are other views, such as Aristotle's, where mathematics and logic aren't considered to be a priori or exempt from empirical validation. Instead, for Aristotle, mathematics and logic were sciences of quantity and reasoning respectively. The qualitative difference to other empirical investigations is just the degree of abstraction and generality employed.
Which reminds me of this:
[quote=An old adage]A picture is worth a thousand words.[/quote]
[quote=Wikipedia][...]complex and sometimes multiple ideas can be conveyed by a single still image, which conveys its meaning or essence more effectively than a mere verbal description.[/quote]
Suppose we have a list of all possible computable functions (programs) in some language (say, Python) that each accept a positive integer as input and produce either 0 or 1 as output. A computable function is a finite string of symbols that, when executed, produces an output in a finite amount of time.
Now consider a table that lists all those computable functions vertically (ordered by string length and symbol index) and the function outputs for each positive integer horizontally.
The above table shows the first three programs and the fifteenth program, with positive integer inputs 1, 2, 3 and 15. As an example, the outputs might be:
Now we define a function f-diag (called f-bar in the lecture) as:
i is a positive integer that appears in three places in that definition - as the input to function f-diag, as the index to a function in the computable functions table (i.e., the i-th function), and as the input to that indexed function. Per the above table, the outputs for f-diag (calculated by inverting the diagonal elements in the table) would be:
Note that f-diag differs from every computable function by at least one input/output pair. So f-diag is not on the list of computable functions and therefore cannot itself be a computable function.
Now consider some statements about those functions.
S1 states that the output of function f2 with input 2 is 0. Per the table above, S1 is true. Furthermore, we can prove the statement by executing function f2 with input 2 and it will output 0 in a finite time. What "prove" means here is that there is a mechanical procedure for obtaining the output in a finite time which, in this case, is provided by function f2.
Per the table, S2 is false. We can prove the negation of S2 by executing function f1 with input 3 and it will output 1 in a finite time.
Per the f-diag table, S3 is true. But we lack a mechanical procedure for proving it since, as shown earlier, f-diag is not a computable function. Furthermore, any computable function is going to produce a different output to f-diag for at least one input (due to the diagonalization). So the proof system would either fail to derive a true statement about f-diag for such an input (and therefore would be incomplete) or else would derive a false statement about f-diag for such an input (and therefore would be inconsistent).
Which just is Gödel's First Incompleteness Theorem: In any rich-enough [*] formal proof system that proves only true statements there are true statements that can't be proved.
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[*] A system is rich-enough if it can express f-diag statements such as S3 above. f-diag is a well-defined function and S3 is a statement about positive integers just as S1 and S2 are.