Questions tagged [foundations-of-mathematics]
76 questions
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Why do universities not teach constructive mathematics to CS undergraduates?
I had a conversation with a user on the Internet. And it did indeed wake my interest regarding something that I had also been asking myself long ago. Why do so many universities still teach beginners in computer science classical mathematics and not…
Tetragrammaton
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In simple terms, what is the difference between logic in mathematics and philosophy?
I want to understand the difference between mathematical and philosophical logic. I actually thought they were the same till I read this post. Concisely speaking, what is the difference between how a philosopher conceives as logic vs how a…
Reine Abstraktion
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What is the difference between logic and mathematics?
I’ve read the article in the SEP about the philosophy of mathematics. I believe I follow most of it.
However, I am a bit puzzled by something that may be due to some basic misunderstanding on my part. When it is stated that the goal of (classical)…
Martin C.
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9
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Set theory vs. type theory vs. category theory
IIRC, in the univalent-foundations program (per Voevodsky), category theory is represented as a possible sort of evolution or new wave of type theory. Maybe my memory is off, but anyway, in nlab they say:
Type theory and certain kinds of category…
Kristian Berry
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What are the main issues on which the schools of Intuitionism, Formalism, and Logicism disagree?
What is the difference between Intuitionism, Formalism, and Logicism? Namely - on which issues do they disagree? And what is the relation of those schools of thought to Platonism, Nominalism, and Fictionalism?
Jordan S
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7
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If most numbers are uncomputable, in what sense do they exist?
Since the set of computer programs is countable and the set of real numbers is uncountable, then it means most real numbers are incomputable. i.e. there does not exist an algorithm to compute their digits one by one (each digit in finite time) -…
nir
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Is mathematical creativity the same as artistic creativity?
Do philosophers distinguish between mathematical creativity, and the broader artistic creativity? If so, what are the differences between these two?
A lot of people seem to treat IQ as something unrelated to creativity when it seems to be the case…
Sayaman
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5
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Does Münchausen's trilemma apply to mathematics?
I'm a mathematician/statistician, and I've been recently reading about epistemology and philosophy of science in my particular field of study.
In statistics, there is a deep concern for the objective validity of statements in science, meaning by…
YetAnotherUsr
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5
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What separates mathematics from logic? Can "mathematical" operations be applied to logical systems?
In my 'Introduction to Logic' class, my professor told us that half of the class will be based on "mathematical" operations withing logic. After looking through the textbook, I realized that he meant things like the predicate calculus and…
mark-antoin9977
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What's so bad about giving up the Axiom of Choice?
The Axiom of Choice (AoC) in set theory famously gives rise to controversial and counterintuitive theorems. (Examples: Banach-Tarski paradox and existence of non-measurable sets.)
I'm aware of some of the theorems that depend on AoC: All vector…
WillG
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4
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Descartes' foundationalism
Is the cogito an axiom from which we can reason axioms of mathematics? Was Descartes' aim to make mathematics (and other fields of knowledge) reducible to the cogito?
Prince Deepthinker
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Do Godel's incompleteness theorems create a contradiction/paradox?
I have seen Godel's theorems presented as a paradox. However, I was only able to infer it's supposed to be one because it proves mathematics to be incapable to be consistent AND complete at the same time and I don't see incompleteness as…
Probably
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Is defining the concept of Probability still an open problem in the Philosophy of Science?
There exist several interpretations of the concept of Probability:
https://en.wikipedia.org/wiki/Probability_interpretations
Being the assumption of Repeatability an important difference between them.
I was wondering if the interpretation of the…
Thomas
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What are the limitations of the language of mathematics?
I was told that mathematics cannot express qualitatively what the elements of a set are, such that you cannot say for example that the members of a set consists of white tigers. So mathematics cannot add qualitative details to a mathematics concept…
Sayaman
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Did Descartes believe arguments for Euclid's parallel postulate were cogent?
If Descartes wanted to found philosophy on the certainty of mathematics, it seems he must have considered arguments for Euclid's parallel postulate cogent, or at least not doubted them.
Gerolamo Saccheri (1667—1733), one of the early forerunners of…
Geremia
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