should math allow contradictions? I.e. should we get rid of the law of excluded middle in math — Olivier5
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
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?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
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. — T Clark
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? — TheMadFool
[from the thread: An Analysis Of The Shadows]point still holds. You can look at a reflection of your eyes but you don't see yourself seeing. You only see — Wayfarer
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
Funny you say this. I won't preface a statement about math objects as "usually". They're just are. — Caldwell
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
Again, retracting the law of excluded middle does not provide contradictions. — TonesInDeepFreeze
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
What was Banno's question? — SophistiCat
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
Maths is made up. — Banno
if number is real but not material, then you have something real but not material, — Wayfarer
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
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