Questions tagged [philosophy]
Questions involving philosophy of mathematics. Please consider if Philosophy Stack Exchange is a better site to post your question.
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Do complex numbers really exist?
Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...
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How do I convince someone that $1+1=2$ may not necessarily be true?
Me and my friend were arguing over this "fact" that we all know and hold dear. However, I do know that $1+1=2$ is an axiom. That is why I beg to differ. Neither of us have the required ...
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How do we know what natural numbers are?
Do I get this right? Gödel's incompleteness theorem applies to first order logic as it applies to second order and any higher order logic. So there is essentially no way pinning down the natural ...
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Why is mathematical induction a valid proof technique? [duplicate]
Context: I'm studying for my discrete mathematics exam and I keep running into this question that I've failed to solve. The question is as follows.
Problem: The main form for normal induction over ...
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What is the correct reading of $\bot$?
I have some doubts about the "natural" interpretation of $\bot$ in Natural Deduction and sequent calculus.
In Prawitz (1965) $\bot$ (falsehood or absurdity) is called a sentential constant [page 14]
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How to interpret material conditional and explain it to freshmen?
After studying mathematics for some time, I am still confused.
The material conditional “$\rightarrow$” is a logical connective in classical logic. In mathematical texts one often encounters the ...
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Is formal truth in mathematical logic a generalization of everyday, intuitive truth?
I'm trying to wrap my head around the relationship between truth in formal logic, as the value a formal expression can take on, as opposed to commonplace notions of truth.
Personal background: When I ...
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Avoiding proof by induction
Proofs that proceed by induction are almost always unsatisfying to me. They do not seem to deepen understanding, I would describe something that is true by induction as being "true by a technicality". ...
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Set theoretic concepts in first order logic
I have been reading introductory texts on first order logic (for example, Leary&Kristiansen). All of them used concepts that I have heard in set theory courses - ordered pairs, functions, ...
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How far can one get in analysis without leaving $\mathbb{Q}$?
Suppose you're trying to teach analysis to a stubborn algebraist who refuses to acknowledge the existence of any characteristic $0$ field other than $\mathbb{Q}$. How ugly are things going to get for ...
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What is "ultrafinitism" and why do people believe it?
I know there's something called "ultrafinitism" which is a very radical form of constructivism that I've heard said means people don't believe that really large integers actually exist. Could someone ...
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Meaning of the word "axiom"
One usually describes an axiom to be a proposition regarded as self-evidently true without proof.
Thus, axioms are propositions we assume to be true and we use them in an axiomatic theory as premises ...
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How to think about theories that prove their own inconsistency?
There are consistent first-order theories that prove their own inconsistency. For example, construct one like this:
Assuming there is a consistent and sufficiently expressive first-order theory at ...
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Why do we not have to prove definitions?
I am a beginning level math student and I read recently (in a book written by a Ph. D in Mathematical Education) that mathematical definitions do not get "proven." As in they can't be proven. Why not? ...
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What is a number?
A dictionary I consulted said a 'number' is a 'quantity', so I looked up what quantity means and the same dictionary said it is an amount or number of some material or thing. Since quantity and ...
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