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.

'Philosophy of science is as useful to scientists as ornithology is to birds', saith Richard Feynman.

However, both ornithology and philosophy of science are still of interest!

Philosophy of science is as useful to scientists as ornithology is to birds', saith Richard Feynman. — Wayfarer

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.

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. — Pfhorrest

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.

The Game of Life simulated on the Game of life - very cool.

I didn't note anything new, although this is a particularly clear pop-explanation. Nice.

The philosophical implications of Godel's theorems are usually very overblown. — Pfhorrest

See Godel, God, and knowledge by way of an example.

I didn't note anything new, although this is a particularly clear pop-explanation. Nice. — Banno

Thank you. I thought it rather a stylish presentation. I like that guy’s channel.

Thank you. I thought it rather a stylish presentation. I like that guy’s channel. — Wayfarer

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. — Janus

I'm not saying that unprovable statements are definitely false, so this is a non-sequitur.

Figures. He seems Californian.

this is a particularly clear pop-explanation — Banno

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.

Almost no pop explanations ever mention Godel numbers in their presentation of the results. — StreetlightX

Yep.

Think of it as lies for children.

:up:

visual comments and props are the requirements of the media. — Wayfarer

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.

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). — Pfhorrest

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). — 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.

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.

Kudos! Thanks again. — TheMadFool

Aw shucks.... :yikes:

Did you notice what went wrong? — 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.

Aw shucks.... :yikes: — Wayfarer

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. — 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'm not saying that unprovable statements are definitely false, so this is a non-sequitur. — Pfhorrest

I didn't say you did say that ( although I guess it might follow from what I thought you said). You said:

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. — 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.

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. — Pfhorrest

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. — 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.

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 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)?

the version in the video — TheMadFool

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. — 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.)

Godel uses the liar paradox to wit, the sentence L = This sentence is false — 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.

I defer to your better judgement. @Wayfarer :wink:

Sarvam mithyā bravīmi — Bhartrihari

If you can correctly extract from the video that Godel's argument is circular, then the video is wrong. — TonesInDeepFreeze

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.

I defer to your better judgement — TheMadFool

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.

is it any good? — fishfry

It is terrible. I mentioned why earlier in this thread.

can you tell me if it's giving people false ideas? — fishfry

Maybe I need to be double-checked, but my reasoning tells me that undecidability follows right from incompleteness.