It's probably fair to say that the import of foundations for the mathematics for the sciences is mostly theoretical as opposed to practical. But some of the central concerns of foundations involve natural questions, such as: Is there an algorithm to determine whether or not any given mathematical statement is a theorem? Is there an algorithm to determine whether or not any given mathematical statement is true in a given model? Is there an algorithm for determining whether any given program and input halts? Is the algorithm that determines whether any given Diophantine equation has solutions? Those are questions of clear intellectual interest even if nothing else.

Moreover, whether any given result in foundations has immediate practical consequences, the concepts and methods of foundations provide the context for the study of the branches of mathematics that do have great practical consequences. The field of computability is built from symbolic logic and mathematical logic. The invention of modern computing itself was born in this context.

Moreover, one may wish to evaluate for themself whether certain results in undecidability bear upon certain matters in combinatorics and cryptography. And, without doubt, one of the most pressing practical questions in mathematics is the P vs NP question, of which a resolution would have enormous practical import. There is currently a million dollar prize for a solution, which might be the largest offered prize now in mathematics. A solution is sought not only to satisfy theoretical curiosity but rather for the great economic impact of the question. Also, for example, Turing's theorem on the unsolvability of the halting problem is often said to entail that there can be no universal program debugger, so researchers don't have to waste efforts looking for one, but instead various other concepts and approaches are devised.

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It was posted: "[...] people talk about paradoxes as if they undermine the validity of mathematics."

Certain paradoxes, if allowed as formal contradictions, vitiate the systems in which the contradictions are derivable. The common response in foundations is to provide systems in which the contradictions are not derivable (or at least are not shown to be derivable).

It was posted: "Godel's proof of his theorem has always seemed goofy to me. I don't understand how the claim that one odd, trivial contradiction proves that math is incoherent in any meaningful way makes sense."

I don't know what is meant there by "goofy". And Godel's proofs are not proofs of contradictions. And I don't know any informed writer on mathematics who has claimed that Godel's work makes mathematics incoherent.

It was posted: "[...] allowing contradictions in math is equivalent to dropping the law of the excluded middle"

No it is not.

should math allow contradictions? I.e. should we get rid of the law of excluded middle in math — Olivier5

Retracting excluded middle wouldn't allow contradictions. The logic is monotonic: we don't get additional theorems from subsets of a consistent axiom set. To have contradictions, we have to add axioms that prove contradictions. (To prevent contradictions from entailing all statements, normally a paraconsistent system is used that retracts the law of explosion.)

This whole discussion started from the question of whether the liars paradox has any implications for the design of bridges, i.e. if the paradox undermines the basic aspects of using math to solve problems. Thoughts?

I don't know all the mathematics for engineering, but I don't imagine that reliably building bridges or other common practical endeavors depend on settling the liar paradox. But that in itself does not entail that foundations and seeking consistent axiomatizations do not have bearing on mathematics for the sciences.

“I can as a philosopher talk about mathematics because I will only deal with puzzles which arise from the words of our ordinary everyday language, such as “proof”, “ number”, “series”, “order”, etc”

And

“But I will talk about the word “ foundation” in the phase “ the foundation of mathematics”. This is a most important word and will be one of the chief words we will deal with.” From Wittgenstein’s Lectures on the Foundations of Mathematics(LFM)

Keep this mind that this is what Wittgenstein is trying to show - that philosophers of mathematics are creating bewilderment because these words are being pulled from their typical surroundings.

W “By “seeing the contradiction” do you mean “ seeing that the two ways of multiplying lead to different results”?”
T: “Yes”
W: “The trouble with this example is that there is no contraction in it at all. If you have two different ways of multiplying, why call them both multiplying? Why not call one multiplying and the other dividing, or multiplying A and the other multiplying B, or any damn thing? It is simply that you have two different kinds of calculation and you have not noticed that they give different results” LW (LFM)

This whole discussion started from the question of whether the liars paradox has any implications for the design of bridges, i.e. if the paradox undermines the basic aspects of using math to solve problems. Thoughts? — T Clark

I'm not sure but Kurt Gödel's incompleteness theorems (utilizes a variation of the liar paradox) imply that we don't know, can't prove math is consistent and while Ralph Waldo Emerson claimed "a foolish consistency is the hobgoblin of little minds...", I'm certain that a mathematical inconsistency could cause more than just bridges to collapse.

A line is not usually defined as a distance, if it is defined at all: in some systems it is a primitive element, which is not defined, but merely constrained by the axioms of that system. — SophistiCat

Funny you say this. I won't preface a statement about math objects as "usually". They're just are. Also, interesting that you mentioned constrained by the axioms of the system. Don't you want to direct that statement towards Banno's question regarding chess?

I'm certain that a mathematical inconsistency could cause more than just bridges to collapse. — TheMadFool

As far as I have seen, which, admittedly isn't far, the inconsistencies in math are analogous to "This sentence is not true." The proof of Godel's first incompleteness theorem uses similar slight of hand to show that, as Wikipedia says:

...no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers.

From what I've read, the foofaraw about these ideas comes from the fact that they crush logician's and mathematician's dreams of a perfect formal logical system, not from any impact to any mathematical system that could have an impact on the real world.

Am I sure about this? No way, but it seems like that's what Wittgenstein was saying in the linked article that @Banno provided. Is it possible I have misunderstood? You betcha.

As far as I have seen, which, admittedly isn't far, the inconsistencies in math are analogous to "This sentence is not true." The proof of Godel's first incompleteness theorem uses similar slight of hand to show that, as Wikipedia says:

...no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers. — T Clark

Well, let's use the sour grapes technique on math - if we can't discover every truth in math because of Gödel's incompleteness theorems, maybe we can prove that all such mathematical truths are merely trivial, truisms like 1 = 1, not worth knowing at all or, more accurately, obviously true if there can be such a thing in math. I dunno.

A digression no doubt but something worth looking into, no?

From what I've read, the foofaraw about these ideas comes from the fact that they crush logician's and mathematician's dreams of a perfect formal logical system, not from any impact to any mathematical system that could have an impact on the real world.

Am I sure about this? No way, but it seems like that's what Wittgenstein was saying in the linked article that Banno provided. Is it possible I have misunderstood? You betcha. — T Clark

Is the liar statement (this sentence is false) more about language than about logic?

Is the liar statement (this sentence is false) more about language than about logic? — TheMadFool

The Russell paradox is basically the same thing using sets instead of sentences.

:ok:

Godel's 2nd incompleteness theorem is not that certain systems can't be proven consistent, but rather that if they are consistent then they can't be proved consistent by certain means. For example if PA is consistent, then PA cannot be proved consistent by PA itself, though it can be proven consistent with, for example, Gentzen's proof. Moreover, the 2nd incompleteness theorem is about formal provability of consistency and does not itself say anything directly about knowledge, which is a philosophical issue, not covered by the mere mathematics of Godel's proof..

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The true arithmetical statements that are unprovable include ones that are not trivial, and especially are not of the form of logical truths (since, of course, all logical truths are provable). One can easily look up the subject of substantive mathematical statements (ones worth knowing whether they are theorems) that are undecidable in the pertinent systems.

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It was said, "[...] inconsistencies in math are analogous to "This sentence is not true."

A system is inconsistent if and only if it has a theorem of the form P & ~P. Whatever is meant by "analogous", we know that there are inconsistent systems having nothing to do with the liar paradox.

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Russell's paradox is couched in terms of sets when discussing set theory, but the basic paradox does not require any notion of sets whatsoever. That is illustrated by using "shaves" rather than "member of". The result is that for any 2-place relation whatsoever, call it 'R', it is not the case that there is an x such that for all y, we have y bears R to x if and only if y does not bear R to y. Symbolically:

~ExAy(Ryx <-> ~Ryy)

There is no mention of the notion of 'set' there.

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Among the salient uses of the liar paradox for mathematical logic is Tarski's theorem. That theorem is that systems of a certain kind that also can form their own truth predicate can thereby form the liar paradox so that such systems are inconsistent.

What's up with the liar sentence?

point still holds. You can look at a reflection of your eyes but you don't see yourself seeing. You only see — Wayfarer

[from the thread: An Analysis Of The Shadows]

What's the deal with self-reference (self-reflection)?

Moreover, the 2nd incompleteness theorem is about formal provability of consistency and does not itself say anything directly about knowledge, which is a philosophical issue, not covered by the mere mathematics of Godel's proof.. — TonesInDeepFreeze

Fitch's Paradox Of Knowability

Assumption: Everything is knowable.
Conclusion: Everything is known.

:chin:

The law of excluded middle is thus a kind of useful fiction. — Joshs

Why then, i guess your answer to Wittgenstein ought to be: don't dispose of the LEM, it's useful...

It feels good that Wittgenstein agrees with me, even if Turing and you do not. — T Clark

Wonderful. On my side it feels very good to agree with Turing -- enigma buster & war hero, inventor of the computer -- and to disagree with Wittgenstein, whom I consider a fake philosopher.

Inconsistent mathematical systems are a thing. — Banno

Fair enough. As I said, if anyone wants to build a parallel form of mathematics where the LEM does not apply, I see no objection whatsoever. As long as they don't build any bridge with it...

Again, retracting the law of excluded middle does not provide contradictions. Intuitionist mathematics eschews the law of excluded middle. If classical mathematics is consistent, then perforce intuitionist mathematics is consistent.

Funny you say this. I won't preface a statement about math objects as "usually". They're just are. — Caldwell

Within the context of a given mathematical system, yes. But there is more than one system, and hence more than one way to define/describe a line. For example, in analytical geometry a line is a collection of points, because that's just how analytical geometry is built up. Roughly speaking, you start with numbers, from numbers you build points and spaces, and from that you build all the geometrical objects, including lines. That is not how lines are introduced in Euclidean geometry though. Euclid himself doesn't really define a line - he just gives an intuitive picture of what he is going to talk about. The real "definition" of a line comes in the form of axioms that constrain its properties.

Also, interesting that you mentioned constrained by the axioms of the system. Don't you want to direct that statement towards Banno's question regarding chess? — Caldwell

Funny you should mention chess, because chess pieces are a good example of use-definition. A formal description of a chess game would not have a formal definition of a chess piece - it's just an abstract object to which we give a name. Its meaning is given by the use to which it is put in the game: the rules of how different pieces move, etc.

What was @Banno's question?

Again, retracting the law of excluded middle does not provide contradictions. — TonesInDeepFreeze

The point is that allowing contradictions as fine and mellow in mathematics would contradict the LEM.

This whole discussion started from the question of whether the liars paradox has any implications for the design of bridges, i.e. if the paradox undermines the basic aspects of using math to solve problems. Thoughts? — T Clark

As already explained, this is not really the question at hand, rather it is a bit of a caricature of the more general question at hand, which was: How should we treat logical contradictions in mathematics? Should we reject or minimize them, as if they were a problem, or should we rather welcome them and treat them as a source of creativity?

What was Banno's question? — SophistiCat

I think he asked: Did we invent or did we discover chess? This I read as a parallel to the question of whether math are invented or discovered.

As already explained, this is not really the question at hand, rather it is a bit of a caricature of the more general question at hand, which was: How should we treat logical contradictions in mathematics? Should we reject or minimize them, as if they were a problem, or should we rather welcome them and treat them as a source of creativity? — Olivier5

Yes, the liar's paradox statement is shorthand for the overall argument.

This question was aimed at @TonesInDeepFreeze, who seems to understand this better than the rest of us.

Yes, the liar's paradox statement is shorthand for the overall argument. — T Clark

I think it's always a good idea to ask precise questions; that was the sense of my input.

To be clear, Fitch's paradox is not a conclusion that the truth of all statements is known, but rather the conclusion is that it is not the case that the truth of all statements is knowable, since by a modal argument, if the truth of all statements is knowable then the truth of all statements is known, while it is not the case that the truth of all statements is known.

Anyway, I don't know what was intended by mentioning Fitch's paradox in connection with my remark about the second incompleteness theorem.

If a system is inconsistent, then the system contradicts every statement in the system, not just the law of excluded middle. So it is pointless to adduce the law of excluded middle in this way. There seems to be expressed a very basic misunderstanding of logical and mathematical theories: It is a crucial and basic point that needs to be understood: A consistent system cannot be made inconsistent by retracting axioms.

Maths is made up. — Banno

I asked Banno what is the payoff for this belief. The payoff is, it deflates the argument that number is real but not material. Numbers exist in minds, minds depend on brains, without brains there can be no minds and therefore no numbers, goes the reasoning. Because if number is real but not material, then you have something real but not material, meaning materialism is false. And that is a no-go in secular scientific culture. Ought not to over-complicate it.

if number is real but not material, then you have something real but not material, — Wayfarer

So you infer that there is a ghost in the machine in order that there can be five spoons on the table even when they go uncounted. Presumably the spirit counts the spoons when no one is around in the quad. I hadn't fully understood that you were so close to Berkeley.

But you have misunderstood my position. I have repeatedly gone through the process of disavowing materialism. Here's a list of thread I have created that touch on the topic:

Philosophical Plumbing — Mary Midgley

Causality, Determination and such stuff.

Nothing to do with Dennett's "Quining Qualia"

Midgley vs Dawkins, Nietzsche, Hobbes, Mackie, Rand, Singer...

Anscombe's "Modern Moral Philosophy"

Subject and object

You participated in most of them. Indeed, let's not over-complicate it, but when you are less distracted we might engage a bit of nuance.

If a system is inconsistent, then the system contradicts every statement in the system, not just the law of excluded middle. So it is pointless to adduce the law of excluded middle in this way. — TonesInDeepFreeze

I wouldn't call it pointless to point at one consequence among many. Your choice of word. But we agree on the rest.

Oh, and while going through my old threads, there is this one: "1" does not refer to anything.