All Questions
Tagged with philosophy meta-math
31
questions
3
votes
0
answers
97
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Can ZFC be proven from weak systems using consistency of those systems?
Tl;dr Can we take a weak system $A_0$ then show
$$A_0 + Con(A_0)\implies Con(A_1)), \space A_1 + Con(A_1)\implies Con(A_2)), \space A_2 + Con(A_2) \implies \dots$$
terminating in ZFC?
My understanding ...
- 5,623
1
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0
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122
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What formal logic has the smallest metatheory?
I'm currently studying A Concise Introduction to Mathematical Logic (Third Edition) by Wolfgang Rautenberg. What sparked my interest in logic is my interest in foundations in general, so I was ...
- 2,059
1
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1
answer
161
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how universal is Conway's game of life? is it reasonable to expect that a technological alien civilization would recognize, say, a glider?
This is a philosophical one, so apologies if it's not appropriate. I can think of several reasons that Conway's Game of Life would be rediscovered by any mathematically inclined biological life forms. ...
- 143
4
votes
1
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141
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Is there a category (or rather a mathematical theory) for which we know a lot about, but not whether its object class is empty or not?
this is a bit of a vague question so let me describe a bit what motivates it: Yesterday I was reading the Wikipedia article about perfect numbers, where I find the section https://en.wikipedia.org/...
- 117
1
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0
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271
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How to prove the consistency of a collection of axioms?
Is there a way to prove the consistency of some chosen axioms? In the two senses following:
In each mathematical logic book, there is a special kind of deduction system, which include some logical ...
- 407
0
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0
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81
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Is it possible to list all hidden lemmas of a proof?
I'm studying Imre Lakatos' Proofs and Refutations for my master's thesis. Currently I address the concept of hidden lemmas, which I understand to mean unstated assumptions of a mathematical proof, ...
- 9
6
votes
3
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538
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Questions about foundations of mathematics
It seems to me that in trying to create a foundation of mathematics, mathematicians are trying to create a formal system that models the language used in what I view as common-sense, real mathematics. ...
user832339
4
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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
345
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Are Mathematicians Pluralists About Math?
This has been rangling around my head for awhile. With the death of Hilbert's program via Gödel's Incompleteness Theorems (and the prior damage done to Logicism via Russell's Paradox), have ...
2
votes
1
answer
111
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Metaphysical/psychological aspects of describing a formal language (mentioned in Bourbaki)
In the introduction to Bourbaki vol. 1, we read:
"It goes without saying that the description of the formalized language
is made in ordinary language, just as the rules of chess are. We do not
...
- 356
88
votes
10
answers
15k
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Is mathematics just a bunch of nested empty sets?
When in high school I used to see mathematical objects as ideal objects whose existence is independent of us. But when I learned set theory, I discovered that all mathematical objects I was studying ...
- 1,237
3
votes
3
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359
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Should axioms be seen as "building blocks of definitions"?
This question is about the difference between a definition and an axiom.
However, it does not address the following point:
Whenever we define something, this is often written as a series of axioms.
...
- 13.4k
1
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3
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359
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Good resources on the intersection of probability theory and logic from a foundations/philosophical perspective?
What are some good books, courses, or online resources for probability theory that highlights differences between classical, frequentist, Bayesian, epistemic etc.? I majored in philosophy and am now ...
1
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1
answer
209
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'The Computer and the Brain' - The mathematical language of the brain
I couldn't decide whether this question is more appropriate to post here or at the philosophy SE, but I thought I'd give people with a mathematical perspective the opportunity to help me decide. I'm ...
- 311
1
vote
1
answer
88
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What is "structure" and is it equivalent to its encoding?
I often come across a description of sets, as objects of "zero structure". I always intuitively understood Set Theory as a theory of size, meaning that the only information we get on it's objects of ...
- 820