Questions tagged [foundations-of-mathematics]
105 questions
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Is infinity a number?
So I've been on a number of math fora, part of learning some calculus (not much of set theory, no). To my surprise I found what I would describe as strong resistance from some folks against (using) infinity in calculus. I find this baffling knowing…
Hudjefa
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What is a natural number?
It’s been on my mind lately. I do maths and work with them daily, but I’m not entirely sure of what they really are.
I understand they are symbols at a surface level, but there is obviously more to it. I’ve seen the set theoretic definition where…
Frazer
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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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Does logic "come before" mathematics?
I always thought of mathematics as being founded on logic. After all, even the most basic mathematical definition is based on logic. When we enunciate ZFC axioms, we're relying on the concepts of "axiom" and "axiomatic system", which are concepts of…
Elvis
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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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Do mathematicians care about the validity ("truth") of the axioms?
Vladimir Arnold (b. 1937) once said that David Hilbert (b. 1862) and Bourbaki (f. 1934) "proclaimed that the goal of their science was the investigation of all corollaries of arbitrary systems of axioms." (Arnold: Swimming Against the Tide (2014),…
Viktor K.
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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…
Clemens Bartholdy
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How do philosophers of mathematics understand the difference between set theory, type theory, and category theory?
In the Univalent Foundations program (per Voevodsky), category theory is presented as the evolution of, or a new wave of, type theory. In the nLab, it’s written:
Type theory and certain kinds of category theory are closely related. By a…
user40843
12
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7 answers
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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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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Philosophical implications of adopting category theory (as foundational) for traditional questions about the nature of mathematical objects?
Category theory can be seen as foundational theory of mathematics since it brings together different subdomains and gives a more abstract and general framework to "ground" those subdomains. I've only recently started to study this, and I wonder what…
mudskipper
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How do we justify the Power Set Axiom?
The more I have been deeply pondering the foundation of mathematics the more it seems like the root of all evil and ambiguity comes from the (seemingly harmless) Power Set Axiom. I'm curious as to the defenses from an intuitive or maybe even…
BENG
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I want to study philosophy but only the epistemic/ontological/phenomenological side (no ethics, politics) what should I study? (i.e.: study guide)
Good afternoon, I hope you are having a lovely day.
I am a student of mathematics and logic and plan to specialize into philosophy of mathematics. I already know quite a bit of background on formal logic (up to compactness and completeness) and I'm…
rutabulum
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Intuition for potential infinity in mathematics
Is there a kind of "consensus" towards the meaning & intuition of the concept of "potential infinity" that goes back to Aristotle and is promoted by Edward Nelson, e.g. in the paper Hilbert's mistake?
Nelson distinguishes it strictly from "completed…
user267839
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What does it mean to say that two theorems (provable statements) are 'equivalent'?
sometimes one sees/reads assertions such as "[the bounded inverse theorem] is equivalent to both the open mapping theorem and the closed graph theorem", but taken formally and literally this would amount to the trivial observation that (φ∧ψ) → (φ↔ψ)…
acb1516
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