Questions tagged [foundations-of-mathematics]
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98 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) ...
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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....
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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 ...
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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 &...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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Is mathematics analytic or synthetic?
This question is related to another question I posted but I think it requires its own treatment since of the already wide scope of the other question i.e. Is the classical theory of concepts ...
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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 ...
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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 ...
user37389
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Why do we have problem of concept of set?
I am reading George Boolos's "The Iterative Concept of Set," and in the first chapter of this paper, he criticizes Cantor's definition of a set as a whole or totality of objects, pointing ...
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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 ...
- 1,857
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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 ...
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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 ...
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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 ...
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Does the anticlass principle solve the Burali-Forti problem?
The text Abstract and Concrete Categories: The Joy of Cats makes much of the class/set distinction, including as a foundational matter, and situates this distinction relative to another such notion, ...
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What is the relation of mathematical propositions to natural language?
Treating natural language as a language game, what role does it play in our understanding of mathematics? Does natural language provide meaning to mathematics?
Does a proof of a conjecture, say ...
- 1,168
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Why do we need geometry for pure math?
Karl Weierstrass had a very interesting critique of Riemann's work. Supporters of Riemann, claim that a pure logician would never have been able to see the things that the "geometric imagination" of ...
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How do skeptics explain axioms not being arbitrary?
I get infinite regress but surely the axioms of ZFC or arithmetic were not so much chosen as discovered and intuited and thought about. They certainly didn't just grab whatever was around them and say ...
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Can N-valued logic (for N = 3 or more) be the basis of mathematics?
I have some questions related to multivalued logic. I am new to this forum, (I study mathematics) so I would be grateful to any useful advice. I am doubtful on even posting this question on Philosophy ...
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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 ...
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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 ...
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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 ...
user37389
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Why is it argued that an argument has one and only one conclusion?
Why can't an argument have more than just one conclusion? If we assume some premises and we assume them to be true, then by some inference rules we are sometimes able to deduce more than just one true ...
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How do we arrive at stronger theories in mathematics/logic?
A reasonable aim of formal mathematics/logic is to build systems which can "interpret" many things. As an example, ZFC can interpret a number of things. Incompleteness Theorems provide us ...
- 1,168
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Why can anything be discovered in mathematics at all?
Imagine a Perfect Mathematician that has superhuman abilities -- if you give him or her a formal foundational system for mathematics like ZFC with all the underlying logical machinery, he or she is ...
user42828
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Set theory with full comprehension
A few years ago I was reading an entry in the Stanford Encyclopedia of Philosophy related to set theory and I stumbled across a statement along these lines:
There is a set theory where full ...
user42828
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Does a Cycle Based Alternative to Set Theory Exist?
My (limited) understanding of Mathematics in general and of Set Theory (being a widely accepted foundational system of Mathematics) in particular, is that the mathematical objects described therein ...
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NBG Class Theory - Can FOL be defined in terms of "mental notions"?
I am currently exploring defining mathematical theories in terms of mental concepts rather than as being formalized in yet other mathematical theories.
In the single-sorted presentation of NBG class ...
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3
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Is the answer to whether math is discovered or invented related to theism?
I'm not asking whether mathematics is discovered or invented, rather whether being theist implies/strengthens/related to the view that it is discovered, and vice versa.
For example I came across an ...
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Has anyone discussed the analytic vs synthetic in algebra?
Let's go back to the original meanings of addition and multiplication back in ancient Sumer when arithmetic was primarily used as a tool in the trade of sheep and beer. Addition meant something like ...
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Descartes' foundationalism [closed]
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?
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Do picture proofs of the Pythagorean theorem make it empirical?
As I understand it, the Pythagorean Theorem, which defines the metric for Euclidean space, is said to be strictly mathematical in the sense that it is derived from a set of purely theoretical axioms (...
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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.
...
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1
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Why don't formalized proofs make formalism true?
All mathematicians are familiar with the (extremely plausible) fact that any ordinary mathematical proof can be formalized inside some foundational theory, e.g., ZFC.
Why doesn't this imply that ...
user42828
3
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1
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Is Mathematical Platonism meaningful to even speak about? [closed]
Certain propositions can be meaningless. How do we know if "Are there abstract mathematical entities (platonism)?" a meaningful question, and not an abuse of language?
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2
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Why can Goedel's Incompleteness Theorem be proven?
Preliminaries:
The way I understand it, Gödel took Russel's and Whitehead's Principia Mathematica (PM) and mapped strings of symbols from PM onto the integers, their Gödel numbers.
He then constructed,...
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Is mathematics based on formal logic, or vice versa?
Math is obviously based on logic in a heirarchical sense, but what about the historical sense? Is there any historical evidence of a "transition" from first order logic to mathematics? All ...
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Is it possible to create an axiomatic system where 1+1 doesn't equal 2? What would be the consequences of such a system? [closed]
1+1=2 is a result (perhaps arguably more of a definition than a theorem?) of Peano Arithmetic, as well as other systems such as ZFC. I understand that 1+1 doesn't necessarily have to equal 2 if we ...
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Inductive argument for Con(ZFC)
If you ask a mathematician, particularly a set theorist, about whether ZFC is consistent, they will answer that we can't know for sure because of Gödel's theorems. If you ask what evidence at all is ...
user42828
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3
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Is it a problem for arithmetic or our representation (or both) that there is incompleteness?
Is this a settled (as much as it can be) philosophical area? I feel like I understand that there will always be incompleteness for a finite set of axioms trying to capture all of arithmetic. But I ...
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Does set-theoretic pluralism, about axiom systems, inevitably become an invitation to non-axiomatic systems of set theory?
Per Hamkins[[11][12]] (see also his [22]), if no individual axiom is too sacred to be denied in some possible world,Q and so if no collection of such axioms is so sacred either, yet then:
The ...
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Need help understanding how certain mathemetical statements across the landscape can seemingly contradict (e.g. Cantor-Hume vs Euclid)
Here are the main components to my understanding on this issue:
Almost all of math can be given in a foundation of set theory
Different math can seemingly contradict, e.g. in Euclidean geometry ...
- 3,541
2
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1
answer
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Multigraphs, hypergraphs, and the epistemic regress
Some definitions (from what I can tell):
A multigraph is a graph where a node can connect via multiple edges.
A hypergraph is a graph where a single edge can connect more than two nodes. ...
- 22.5k
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1
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Does every mathematical question have an unambiguous answer?
Does every mathematical question have an unambiguous answer?
For example, suppose I were to assert "In the decimal expansion of pi, does there occur in at least one location a billion 1's in a ...
- 8,628
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I am looking for some ternary logic with values True/False/Absurd
I am looking for some alternative logic where a sentence p could not only be true or false but also absurd, which is different of false in such a logic.
I see that there is some content about three ...
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