It's another example of people mistaking words for reality, the map for the territory. — T Clark

Except that Wittgenstein rejected the idea that words represent reality and maps represent territories.

It seems strange to say that we made up numbers like e or π. We don't know what the 10000000000000 trillion digit of e is, yet if we invented e shouldn't we know that? — Amalac

e and like are less numbers than they are recursive processes. We made up the process. To be more precise, all our mathematics is parasitic on our notion of the object, which is why modern mathematics emerged in tandem with the modern scientific notion of the empirical object. Empirical objects (not just perfect circles ) are subjective constructions, abstractions, idealizations. Such idealizations made mathematical calculation possible.

That's because you take the whole question of 'can the liar's paradox break bridges' a bit too literally. The real question hidden behind this tag line is: should math allow contradictions? I.e. should we get rid of the law of excluded middle in math, or would that lead to poorly designed bridges? — Olivier5

I think what Wittgenstein was saying is that the trivial inconsistencies associated with the paradoxes don't matter. Are meaningless. There's a good chance I'm wrong about that, but that's how I read the article @Banno linked to.

I don't see what this has to do with the law of the excluded middle.

Except that Wittgenstein rejected the idea that words represent reality and maps represent territories. — Joshs

Yes. Wittgenstein and I agree. Wittgenstein and I both think that mathematical inconsistencies are meaningless. I think. Maybe. I think that's what the article said.

Things in engineering are usually defined as sometimes overdetermined or positively redundant, in practice.

Epistemic closure of mathematics or its inability being used in practice doesn't prohibit a computer from modelling a bridge — Shawn

If that's what LW was saying, i.e. that we can in practice tolerate a few inconsistencies here or there in math as long as we know how to deal with them in practice, without otherwise departing from 'either p or non p', then I can agree. That looks like a reasonable position to me. But it seems he was arguing for treating contradictions not as a problem but as some sort of creative source of inspiration. That's often a good idea in real life, in literature, in proverbs and in art, all contexts where the law of the excluded middle is at best an ideal, at worse a distraction. But I'm not sure it's a good idea in mathematics.

I think what Wittgenstein was saying is that the trivial inconsistencies associated with the paradoxes don't matter. Are meaningless. There's a good chance I'm wrong about that, but that's how I read the article Banno linked to. — T Clark

I don't read this in the only (?) direct quote provided in that article, which reads as follows:

“Why are people afraid of contradictions? It is easy to understand why they should be afraid of contradictions in orders, descriptions, etc. outside mathematics. The question is: Why should they be afraid of contradictions inside mathematics?”

(emphasis mine)

Also from the article (though not a direct quote):

In relevance to this essay, Alan Turing (1912–1954) strongly disagreed withLudwig Wittgenstein’s argument that mathematicians and philosophers should happily allow contradictions to exist within mathematical systems.

The liar's paradox, like all logical paradoxes, has a simple non paradoxical solution. It's only an apparent paradox. So of course it can't break bridges or lead to poorly conceived ones. But happily or even casually allowing contradictions in math is equivalent to dropping the law of the excluded middle from mathematical logic, with far reaching consequences.

Luckily it will never happen. Math is too serious a matter to be left to philosophers.

Maths is made up. — Banno

Isn't it? It's just the best way for humanity to create proofs.Best way to "verify" its reality. Well not to say the only one which can be so accurate.
But rather than that I see no reason at all Maths to play any significance role to the universe itself.

Aliens might have created their own way ("Maths") which fits their senses better and can describe their reality better.
Maths is just an excellent necessary human invention.

Wittgenstein and I both think that mathematical inconsistencies are meaningless. I think. Maybe. I think that's what the article said — T Clark

The liar's paradox, like all logical paradoxes, has a simple non paradoxical solution. It's only an apparent paradox. So of course it can't break bridges or lead to poorly conceived ones. — Olivier5

I don’t think the solution you have in mind has anything to do with what Wittgenstein was trying to illustrate here.

happily or even casually allowing contradictions in math is equivalent to dropping the law of the excluded middle from mathematical logic, with far reaching consequences. — Olivier5

We don’t have to drop the law of the excluded middle, it deconstructs itself.

“ In the decimal expansion of TT either the group "7777"
occurs, or it does not—there is no third possibility." That is to say:
"God sees—but we don't know." But what does that mean?—We use a picture; the picture of a visible series which one person sees the whole of and another not. The law of excluded middle says here: It must either look like this, or like that. So it really—and this is a truism—says nothing at all, but gives us a picture. And the problem ought now to be: does reality accord with the picture or not? And this picture seems to determine what we have to do, what to look for, and how—but it does not do so, just because we do not know how it is to be applied. Here saying "There is no third possibility" or "But there can't be a third possibility!"—expresses our inability to turn our eyes away from this picture: a picture which looks as if it must already contain both the problem and its solution, while all the time we feel that it is not so.”
(Philosophical Investigations 352)

How is the halting problem relevant to the number of spoons on the table?

As if that any consistent formal system within which a certain amount of elementary arithmetic can be carried out is incomplete implied that the number of spoons on the table is indeterminate.

Hence,

...what Wittgenstein was saying is that the trivial inconsistencies associated with the paradoxes don't matter. — T Clark

and almost

that we can in practice tolerate a few inconsistencies here or there in math as long as we know how to deal with them in practice, without otherwise departing from 'either p or non p', then I can agree. — Olivier5

but not

If one allows contradictions within mathematics, they will spread everywhere, — Olivier5

We don’t have to drop the law of the excluded middle, it deconstructs itself. — Joshs

Ridiculous.

I don't read this in the only (?) direct quote provided in that article, which reads as follows:

“Why are people afraid of contradictions? It is easy to understand why they should be afraid of contradictions in orders, descriptions, etc. outside mathematics. The question is: Why should they be afraid of contradictions inside mathematics?”
(emphasis mine)

Also from the article (though not a direct quote):

In relevance to this essay, Alan Turing (1912–1954) strongly disagreed with Ludwig Wittgenstein’s argument that mathematicians and philosophers should happily allow contradictions to exist within mathematical systems. — Olivier5

This from the article's author:

All the above means that if mathematics is a human invention, then any contradictions and paradoxes there are (within mathematics) must be down to… us. And if they’re down to us, then they aren’t telling us anything about the physical world (which includes Turing’s bridge — see later) or even about a platonic world of numbers — because such as thing doesn’t even exist.

And this:

Wittgenstein’s argument (at least as it can be seen) was that the Liar paradox does indeed lead to this bizarre conclusion because — in a strong sense - it was designed to do so. That is, it is part of a language-game which was specifically created to bring about a paradox. And because it’s a self-enclosed and artificial language-game, then Wittgenstein also asked “where will the harm come” from allowing such a contradiction or paradox?

And this:

Indeed many (pure) mathematicians have often noted the complete irrelevance of much of this paradoxical and foundational stuff to what they do. Thus if it’s irrelevant to many mathematicians, then surely it would be even more irrelevant to the designers who use mathematics in the design of their bridges.

Whether or not Wittgenstein means what I said he means, I think this shows that the author of the article thinks Wittgenstein means what I said he means.

This is a really interesting discussion.

We don’t have to drop the law of the excluded middle, it deconstructs itself.
— Joshs

Ridiculous — Olivier5

Did you read the Wittgenstein quote? Do you understand what he’s trying to say?

Except that Wittgenstein rejected the idea that words represent reality and maps represent territories. — Joshs

I baulked at this the first time I read it, but on re-reading I gather the emphasis is on representing; as in word do not represent reality, but are the reality. The world is what is the case.

No models, nothign between "the cat is on the mat" being true and the cat 's being on the mat.

IS that right?

Did you read the Wittgenstein quote? Do you understand what he’s trying to say? — Joshs

He is saying that the LEM is not always useful. So? We should give it up just like that?

When you find out that your cellphone cannot mow the lawn, does your cellphone deconstruct itself?

No models, nothign between "the cat is on the mat" being true and the cat 's being on the mat.

IS that right? — Banno

But then we have to wade into the messiness of ‘use’. The cat is on the mat is no less complicated than the sense of any particular word. How is it being used in a particular context? We’d have run through a potential infinity of such uses before we came upon that use in which the concept of ‘truth’ becomes relevant. But having done so , what can we conclude about the status of ‘truth’? Can we save some sense of it that doesn’t get sucked down into the relativity of use? Is ‘true’ just another thing we say in certain contexts for certain purposes?

Whether or not Wittgenstein means what I said he means, I think this shows that the author of the article thinks Wittgenstein means what I said he means. — T Clark

Wittgenstein is literally asking why should one be afraid of contradictions in mathematics. What you or the author are saying, I don't know. I would answer that mathematics as we know them are built on the LEM, so the reason why we should be afraid of contradictions in mathematics is to keep that body of work alive and well. Now if anyone wants to build a parallel form of mathematics where the LEM does not apply, be my guest.

Just don't build any bridge with it.

Is there a transcript for this discussion between Wittgenstein and Turing? It would be interesting to see the entire conversation.

When you find out that your cellphone cannot cut the grass, does your cellphone become useless to you? — Olivier5

No, but my sense of its usefulness changes. Logical propositions have to do with things being the case or not. So they presuppose that the things we are passing judgement on just sit there being what there are indepdently of our judgements about them. This way of looking at logical propositions doesn’t recognize that before something can be the case or not, there has to be agreement on the sense of what it is to be a ‘case’. That is to say, words have an infinity of potential senses , and which sense is being generated is a function of the context of use. Logic can pretend that such constant subtle shifts in sense do not exist because they don’t often amount to enough of a disagreement to become noticeable. For most intents and purposes , we can assume that we are all on the same page when inquiring whether something is the case or not. But this is because the generality of logic was designed to mask these usually subtle interpersonal( and infra-personal)
differences. The law of excluded middle is thus a kind of useful fiction.

It's in lectures 21 and 22 of Wittgenstein's Lectures on the Foundations of Mathcrnatics, Cambridge 1939. Turing is present throughout the book on and off.

I would answer that mathematics as we know them are built on the LEM — Olivier5

So one might think, but because maths is made up we can develop little intricacies by thinking otherwise. Inconsistent mathematical systems are a thing. They are based on rethinking logical explosion rather than excluded middle.

Here's some inconsistent geometry:

Wittgenstein is literally asking why should one be afraid of contradictions in mathematics. What you or the author are saying, I don't know. I would answer that mathematics as we know them are built on the LEM, so the reason why we should be afraid of contradictions in mathematics is to keep that body of work alive and well. — Olivier5

I'm ok with that. As I said before, this is not my area of expertise. It feels good that Wittgenstein agrees with me, even if Turing and you do not.

I love these illusions. But they have much less to do with geometry than they do with how we use our background knowledge of perspective to fill in shapes. Artists can figure out the ‘trick’ quicker than most of the rest of us. But then maybe that makes it a good illustration of mathematical
contradiction in Wittgenstein’s sense after all.

...they have much less to do with geometry... — Joshs

But what happens when you take the images seriously? see Chris Mortensen.

eg: Inconsistent Mathematics, Reutersvärd, And Buddhism: An Interview With Chris Mortensen
or
Review of 'Inconsistent Geometry', by Chris Mortensen

More Australians mucking stuff up.

Ah, thanks. :up:

I tried excerpting the relevant bits from a pirated pdf found online, but it needs considerable reformatting. The whole book is essential reading if you're interested in Wittgenstein.

I'm interested in seeing more in detail what Turing said, not so much Wittgenstein himself. Mathematics, much less foundations of mathematics, is beyond me.

But some paradoxes are interesting. It may be due to mathematical considerations, or linguistic ambiguity or lack of comprehension, so looking at Turing's reply might be instructive.

Turing: The sort of case which I had in mind was the case where you have a logical system, a system of calculations, which you use in order to build bridges. You give this system to your clerks and they build a bridge with it and the bridge falls down. You then find a contradiction in the system.--- Or suppose that one had two systems, one of which has always in the past been used satisfactorily for building bridges. Then the other system is used and the bridge falls down. When the two systems are then compared, it is found that the results which they give do not agree.

So the point in what you quote, as I take it, would be to avoid building bridges that fall. A system with a paradox is a kind of faulty logic system.

But there's many reasons why they could fall not limited to paradox. I mean, why do paradoxes arise at all? :chin:

The bridge example doesn't seem to originate with Turing, but comes up in an earlier lecture:

Watson: The reason why one thinks that in all such cases of agreement and disagreement there must be a right and a wrong is that in the past there have been mistakes in mathematical tables, with the result that if one used these tables when building a bridge, it would probably fall down.
Wittgenstein: The point is that these tables do not by themselves determine that one builds the bridge in this way; only the tables together with a certain scientific theory determine that.