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Questions tagged [foundations-of-mathematics]

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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” ...
Julius Hamilton's user avatar
4 votes
0 answers
80 views

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 ...
ac15's user avatar
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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 ...
Julius Hamilton's user avatar
1 vote
0 answers
32 views

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 ...
Sayaman's user avatar
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3 votes
1 answer
124 views

Does the anticlass principle solve the Burali-Forti problem?

Justification of the foundations-of-mathematics tag: I was reading through a long text on category theory, Abstract and Concrete Categories: The Joy of Cats, and they make much of the class/set ...
Kristian Berry's user avatar
3 votes
3 answers
453 views

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 ...
J Kusin's user avatar
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17 votes
21 answers
3k views

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....
Fraser Pye's user avatar
0 votes
0 answers
132 views

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 ...
DDS's user avatar
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1 answer
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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 ...
Sayetsu's user avatar
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7 votes
5 answers
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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 ...
user21312's user avatar
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2 votes
4 answers
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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 ...
Steven Harder's user avatar
3 votes
3 answers
204 views

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 ...
Loai Ghoraba's user avatar
3 votes
2 answers
135 views

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 ...
David Gudeman's user avatar
1 vote
2 answers
165 views

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, ...
Dennis Kozevnikoff's user avatar
-1 votes
3 answers
122 views

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 ...
Dennis Kozevnikoff's user avatar
2 votes
1 answer
60 views

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 ...
David Gudeman's user avatar
2 votes
1 answer
203 views

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. ...
Kristian Berry's user avatar
1 vote
0 answers
47 views

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 ...
Sayaman's user avatar
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3 votes
3 answers
97 views

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?
PDT's user avatar
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1 answer
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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 ...
Kristian Berry's user avatar
0 votes
0 answers
62 views

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 ...
Kristian Berry's user avatar
3 votes
1 answer
174 views

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 ...
user avatar
6 votes
7 answers
3k views

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 ...
Sayaman's user avatar
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2 votes
1 answer
166 views

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 ...
user107952's user avatar
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0 votes
1 answer
98 views

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 ...
Sayaman's user avatar
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-2 votes
1 answer
127 views

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 ...
Sayaman's user avatar
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2 votes
2 answers
186 views

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,...
Johannes Bauer's user avatar
-2 votes
6 answers
370 views

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 ...
François's user avatar
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2 votes
3 answers
185 views

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 ...
François's user avatar
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15 votes
7 answers
9k views

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 ...
Tetragrammaton's user avatar
2 votes
2 answers
166 views

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 ...
Zuhair's user avatar
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1 vote
1 answer
199 views

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 ...
J Kusin's user avatar
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0 votes
0 answers
49 views

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 ...
J Kusin's user avatar
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0 votes
0 answers
193 views

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 ...
Ajax's user avatar
  • 1,139
1 vote
0 answers
153 views

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 ...
J Kusin's user avatar
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1 vote
0 answers
74 views

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 ...
J Kusin's user avatar
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12 votes
7 answers
7k views

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 ...
Babu's user avatar
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5 votes
8 answers
1k views

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 ...
YetAnotherUsr's user avatar
3 votes
2 answers
564 views

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 ...
user avatar
0 votes
3 answers
229 views

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 ...
J Kusin's user avatar
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0 votes
1 answer
206 views

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 ...
user avatar
0 votes
0 answers
48 views

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 ...
Kristian Berry's user avatar
11 votes
3 answers
6k views

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 ...
Kristian Berry's user avatar
1 vote
0 answers
115 views

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 ...
Ajax's user avatar
  • 1,139
4 votes
2 answers
803 views

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 ...
Probably's user avatar
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0 votes
3 answers
176 views

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. ...
Ajax's user avatar
  • 1,139
0 votes
5 answers
240 views

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 ...
J Kusin's user avatar
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0 votes
1 answer
124 views

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 ...
RIYASUDHEEN T. K's user avatar
3 votes
3 answers
356 views

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 ...
Ajax's user avatar
  • 1,139
7 votes
4 answers
965 views

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 ...
nir's user avatar
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