Questions tagged [foundations-of-mathematics]
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98 questions
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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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Does 1 multiplied by 1 equal 2? [closed]
I have seen on online chatter that this subject is open to speculation.
I would like to attempt to resolve this issue.
Will anybody sell me their house if they agree that this equation is true?
If you ...
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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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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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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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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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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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Why is this argument valid?
I m reading Linnet's paper 'pluralities and set' where his claim said that collapse principle lead contradiction if we didn't assume 'it is possible to quantify over absolutely everything'
He uses ...
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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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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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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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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 ...
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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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The smallest possible formal definition of FOL
I find the common presentation of first order logic somewhat confusing. I feel that I often don’t understand why we need the exact terms and concepts we do.
My current recapitulation of “standard FOL” ...
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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 ...
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Is there a set theory which implies the interval [0, 1] but no more?
A deductive system (as a collection of judgments and rules of inference) can be used to describe something commonly called a “set theory”. We can imagine a priori there are certain properties we would ...
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Is there a limited number of 'pragmatic' logic rules?
What you have cited is a pragmatic limit, as you have not seen logic
systems with more than 8 or so precepts.
IF there were such a limit to precept quantity,
then YES there would be a limit to the ...
user37389
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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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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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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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What are the First Principles of Euclidean Geometry (Besides the Axioms)?
On first principles, Wikipedia says:
A first principle is an axiom that cannot be deduced from any other within that system. The classic example is that of Euclid's Elements; its hundreds of ...
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Do Gödel's incompleteness theorems and Tarski's theorem of indefinability of truth show we can never discover and prove every truth?
I thought I had a grasp on this. Do Gödel's apply to just math; logic, too; or more, and what does its applicability entail? If it applies to math, does it apply to physics? Similarly with Tarski: can ...
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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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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 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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Is modern mathematics scholasticism?
I have thought a lot a about mathematics and it's foundations. There have been several attempts to give it a solid foundation, and they all failed. Frege / Russell logical atomist approach failed, ...
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Why did it take 3000 years for theories of mathematical foundations to emerge? [closed]
Humans have been doing mathematics for at least 3000 years. In ancient Egypt they did some advanced trigonometry and number theory. Mathematics is thousands of years old, but for some reason it was ...
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Is there a historically plausible account of the real numbers?
Frege's ill-fated program to define the natural numbers in terms of abstraction was intended as a genuine account of what natural numbers are, not just a way of encoding numbers in set theory like ...
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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. ...
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How would we define a topos between Classical Logic and Paraconsistent Logic, and determine what the homomorphism between them would be? [closed]
How would we define a topos between Classical Logic and Paraconsistent Logic, and determine what the homomorphism between them would be? I learned that topoi can't be used for philosophical ideas, but ...
user37389
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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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Is the response (in the mathematics community) to Wiles' proof of Fermat's Last Theorem, evidence for social constructivism about math?
Wiles' proof initially involved reference to functional equivalents of inaccessible cardinals (here, Grothendieck universes). Rather than take this as evidence for the meaningfulness and usefulness of ...
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Is category theory an example of foundherentism?
After reading this essay about the history of type theory, I have refined my assessment of the set- vs. type-theory question in two ways. More similarly to what I was thinking before, I still ground ...
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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
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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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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 ...
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How do we determine if a statement can't be proved by mathematics alone?
How do we determine if a statement can't be proved by mathematics alone? It seems mathematics can only prove something that can be defined purely in mathematics terms, but can't prove simple ...
user37389
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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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Why do people still use classical logic? [closed]
It seems to me very crazy that mathematicians reactions to Godël incompleteness theorem have been mostly to agree that there are statements of the language which can neither be proved nor disproved ...
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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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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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Per Mathematical Structuralism, can a pure mathematical theory have semantics that is not closed on isomorphism?
This question is the philosophical side of a question that I've recently posted to MathOverflow. Here, I'm specifically asking about the output of Mathematical Structuralism on that question that I'll ...
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Do any philosophers say surreal numbers are reason to doubt platonism?
Not trying to be inflammatory at all, this is a genuine (maybe dumb) question.
Especially in regards to the genesis of the surreals, which was Conway thinking about Go endgames. They seem among the ...
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Realism as necessary for impredicative mathematics to avoid viscous circle, but not really?
Here is an quote from Godel from Shapiro’s Thinking About Mathematics:
“…the vicious circle…applies only if the entities are constructed by ourselves. In this case, there must clearly exist a ...
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What is mathematical analysis?
Hilbert's aim to reduce all mathematics to finite logical system was shown impossible by Goedel. He did mathematical analysis of logic itself (Goedel numbering). Turing defined algorithms, and ...
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Two questions about mathematical platonism
Any set, number, shape, definition, axiom, etc we write down or think about is not the ideal platonic version. But surely the mathematical platonist thinks humans are closer to that unreachable goal ...
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Is there a naturalized intuitionist mathematics? Is it Kantian?
I have in mind an interpretation of mathematics as intuitionalism, where intuitions are subjective (built from personal experience), but subjective experience is ultimately explained “objectively” a ...
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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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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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