Questions tagged [foundations-of-mathematics]
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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 ...
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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 ...
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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 ...
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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 ...
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Why can't mathematics prove that Tic-tac-toe is a simple game? [closed]
Why can't mathematics prove that Tic-tac-toe is a simple game? Tell me if I am wrong, but we can prove that chess is less complex than tic-tac-toe, by counting the number of actions possible if we ...
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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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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 ...
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Why do mathematical platonists believe in the abstract when math clearly comes from FOL, a non-abstract?
To assure ourselves first order logic is as free of paradox, errors, and impermanence, mathematicians and logicians "grounded" math in a language/system everyone can agree upon. Here is a ...
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How can you analogize mathematical induction to dominoes falling, if some domino can fail to topple?
This analogy doesn't convince me, because what if some domino (after b, the base case) fails to topple? In real life, a domino can remain standing upright if it got placed too far apart from the ...
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Mathematical "forms" as a relation of varying arity
This might be more a MathSE question, but on the other hand, it would involve a peculiar reimagining of the relation between set theory and type theory, so I'll try it out here.
OK, so earlier I ...
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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 ...
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How do logicians think of strength of proof systems?
I want to understand how logicians reason about strengths of proof systems and argue relative strengths of proof systems. I want to appreciate the validity of the reasoning by which we establish ...
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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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Working of Mathematical Induction
I am aware of what proof by Mathematical Induction is. I have also used it in numerous proofs. However, I don't understand formal correctness/validity of the method down to the level of Peano Axioms. ...
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Is there any conflict with Holism and equals and plus signs of mathematics?
Edit - better phrasing/summary:
Maybe this phrasing helps "the same object expressed in different ways". That's one meaning behind 'equals'. 10 = 1+...4 --> 10 really is 1+...4. So if ...
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Can Mathematics be a tool to analyse immaterial existences
Doubt: Can we say that Mathematical thoughts (arguments) can be independent of physical (Time-space continuum or material) world since it is an abstract science. In other words can Mathematics be a ...
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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 ...
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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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Is there a weaker/general version of Incompleteness Theorem which holds for every formal axiomatic system?
Is there a general version of Godel's Incompleteness Theorem which holds for any formal axiomatic system (and not just those capable of modelling basic arithmetic)?
If no, is it absurd to ask why such ...
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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 ...
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Why is the definition of the real numbers not contradictory? [closed]
I understand that a set whose members can, in principle, be enumerated (by having a formula) can be considered as a well-defined set. Therefore, set of all even numbers, multiples of 3, and so on ...
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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 ...
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Is Constructivism (philosophy of mathematics) against classical logic?
Is Constructivism (philosophy of mathematics) against classical logic? I might be wrong, but mathematics' main branch of logic is based on classical logic, and I was wondering if Constructivism was ...
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What is the current status of Foundation-of-Mathematics programmes?
I have been reading 'A Very Short Introduction to Mathematics' by Timothy Gowers and at one point he mentions that most of the mathematical proofs can be finally resolved to a set of logical ...
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Is there some non-classical logic where the van der Waerden theorem does not apply?
The van der Waerden theorem is a theorem in the branch of mathematics called Ramsey theory which states that for any given positive integers r and k, there is some number N such that if the integers {...
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What is the connection between Lawvere and Cantor?
Lawvere wrote in a couple papers that Cantors word “menge” which is usually understood as “set” is actually a cohesive type. And the “kardinale” is the abstraction from this by getting rid of the ...
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Books on Philosophy of Mathematics [duplicate]
I want to buy a philosophy of mathematics book. I have three options in mind: Philosophy of Mathematics by Øystein Linnebo, Philosophy of Mathematics: Selected Readings by Paul Benacerraf, or
The ...
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