All Questions
Tagged with philosophy proof-theory
27
questions
0
votes
1
answer
170
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What are the merits of having a "good" proof system?
Background: My understanding is that model-theoretic semantics (MTS) and proof-theoretic semantics (PTS) differ in the following ways. In MTS, you first define the notion of truth in models and then ...
user1048887
8
votes
3
answers
230
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If I can prove $Y$ from "$X$ is true" and from "$X$ is false", can I prove $Y$ without using $X$ at all?
Suppose I have a statement $X$, for which I do not know whether it is true or false. And suppose further that I want to prove a statement $Y$:
I first assume that $X$ is true, and I construct an ...
- 30.1k
7
votes
2
answers
220
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Can a mathematical proof always be objectively determined as correct or incorrect?
Fields medalist Michael Atiyah claimed a simple proof of the Riemann hypothesis, but many mathematicians rejected his proof. Am I right in saying that Atiyah's proof is either objectively correct (...
- 5,451
4
votes
1
answer
331
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Arithmetic systems without Induction
It's often said that AC is a controversial axiom and so often in my math classes when it is used a brief comment is made to the effect of "we can prove this without Zorn's Lemma but it's more work". ...
- 6,776
2
votes
2
answers
2k
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Why and how is logic related to set theory?
I am learning set theory on my own at the moment, and I realised I can't avoid not to learn logics. There is a strong connection between these two. such that, proofs for sets are based on logics.
I ...
- 45
1
vote
1
answer
93
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Assumptions necessary to justify the method of proof by contradiction.
It seems to me, these assumptions are necessary to demonstrate proof by contradiction:
i) Every proposition must belong to $T$ or $F$.
ii) No proposition belongs to both $T$ and $F$
iii) If having $...
- 2,266
0
votes
0
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74
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Can we be sure proofs have no errors?
My current understanding is that work submitted to journals has mathematicians look over it for errors. Mathematics is deductive, yet with this being the burden of proof, how can we know for sure ...
- 626
2
votes
1
answer
56
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About proofs that we cannot verify every step by hand
For something I am planning to write, I need to clarify few issues with respect to computer-aided proofs that we cannot verify every step by hand. For example, the proof for the four-color map theorem....
- 1,115
27
votes
10
answers
6k
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What exactly is circular reasoning?
The way I used to be getting it was that circular reasoning occurs when a proof contains its thesis within its assumptions. Then, everything such a proof "proves" is that this particular statement ...
- 903
5
votes
2
answers
976
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Self-verifying theory! But what can we learn from it?
There is this seemingly surprising result that no consistent and sufficiently expressive theory can prove its own consistency. But what would be the actual benefit from knowing that a theory $T$ can ...
- 30.1k
11
votes
3
answers
2k
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Consistency of ZFC and proof by contradiction
I will start off by saying that I am an elementary student of mathematics and do not possess the deep and rigorous knowledge of most members of this site. Nonetheless, whilst learning how to do a ...
- 4,322
8
votes
3
answers
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How do we prove a set axioms never lead to a contradiction?
How can we be sure that a set of axioms will never lead to a contradiction? If there's a contradiction, we will find it first or later. But if there's no one, how can we be sure we choosen reasonably ...
- 983
6
votes
1
answer
962
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Why do the axioms of equality suffice?
In this answer, Henning Makholm axiomatizes the notion of equality as follows:
Reflexive axiom, Symmetry axiom and Transitive axiom:
The properties we need are the pure equality axioms:
$x=...
- 69
2
votes
2
answers
155
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Are theorems like subroutines for math? [closed]
I've been developing more appetite for math just lately, as I study electromagnetics to deepen my understanding of electric circuits and devices.
I'm finding that doing derivations as exercises helps ...
- 191
3
votes
3
answers
221
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On "why" questions in mathematics
In response to the question How would one be able to prove mathematically that $1+1 = 2$?, Asaf Karagila explains:
In a more general setting, one needs to remember that $0,1,2,3,…$ are just symbols. ...
- 1,354