# Questions tagged [philosophy-of-mathematics]

Philosophy of mathematics asks questions about mathematical theories and practices. It can include questions about the nature or reality of numbers, the ground and limits of formal systems and the nature of the different mathematical disciplines.

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### Is there a paradox of third-order arithmetic?

Calculus, sometimes analysis or second-order arithmetic, seems more intuitive when formulated in infinitesimal terms than in terms of real-valued limits. However, the meta-theory of analysis, i.e. its ...
14 votes
6 answers
4k views

### Is it possible for everything that exists to have a definition?

Is it possible for everything that exists to have a definition? I actually started out asking this in the linguistics - semantics stack and was directed here. By definition I mean at least in the ...
6 votes
10 answers
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### What is a philosophical interpretation of Bayes’s theorem when one of the probabilities is zero?

Bayes' Theorem P(H) = probability of a hypothesis P(E) = probability of evidence P(E|H) = probability of evidence given the hypothesis P(H|E) = probability of hypothesis given the evidence P(H|E) = P(...
22 votes
16 answers
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### Are numbers real?

I am confused as to what numbers are. Numbers are defined to be what they are, so numbers aren't real? But numbers are found in nature, right? So if we invented them, how can they be found in nature? ...
2 votes
2 answers
132 views

### Does paradoxes in a theory mean that the theory is incorrect and should be discarded?

I have two questions which relate to two different subjects, Science and Mathematics have different meanings of "theory", first is based on ever-growing scientific evidence while other is ...
2 votes
4 answers
138 views

### Can mathematical models be indistinguishable from the phenomena they model?

Mathematical models of the phenomena of the world, such as the weather, are used to make predictions about the outcome of the phenomena from an initial state. These models are applied as computer ...
1 vote
1 answer
63 views

### Questions about mathematical models of the real world

I'm just starting to learn about mathematical modelling but i'm getting stuck understanding how real world processes and objects are modelled by maths. The way i'm thinking about at the moment it is ...
5 votes
2 answers
263 views

### Are questions truth-apt; what is the use of assigning questions a truth-value?

Is John black (or white)? Yes he is black. No he is not (black). I don’t see how can the question be truth-apt and what use is there in assigning (or even being able to assign) a truth-value to the ...
0 votes
0 answers
69 views

### Why do variables in mathematics always represent properties of objects?

When I read about mathematical modelling, a variable is always used to represent a property of an object, for example the mass of an object. "let m be a real number that represents the mass of an ...
2 votes
5 answers
703 views

### Interpretations of quote - "Mathematics attracts because..." [closed]

I am reading "Probability and Stochastics" by Cinlar. Here is the following quote in the preface of the book. As Martin Barlow put it once, mathematics attracts us because the need to ...
11 votes
11 answers
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### Is the fact that ZFC implies that 1+1=2 an absolute truth?

This question is somehow of a follow up to to this other one, and it's something that has bugged me for a while. I understand the notion that there's no "absolute truth" in math, in the ...
3 votes
1 answer
317 views

### What is the meaning of Principle C'' in Hartry Field's 'Science Without Numbers'?

For Field, the following is 'perfectly obvious', but I would like confirmation that I understand it completely. Let A be a nominalistically statable assertion. Let A* be the assertion that results by ...
222 votes
26 answers
48k views

### Was mathematics invented or discovered?

What would it mean to say that mathematics was invented and how would this be different from saying mathematics was discovered? Is this even a serious philosophical question, or just a meaningless, ...
4 votes
4 answers
277 views

### What is intuitionistic mathematics?

What is intuitionistic mathematics? What are its claims, and what are their justifications? 1a. Intuitionism as a philosophy. L. E. J. Brouwer is credited as the originator of intuitionistic ...
3 votes
4 answers
303 views

### How is the concept of a topos in mathematics relevant to philosophy?

https://en.wikipedia.org/wiki/Topos Topoi behave much like the category of sets and possess a notion of localization; they are a direct generalization of point-set topology. My understanding is that ...
1 vote
2 answers
132 views

### Language, Meaning and Cardinality?

So I have been pondering about language. By language L I just mean a series of symbols. The upper limit of this series of symbols is Aleph-zero. Yet somehow using these symbols the human is able to ...
3 votes
3 answers
440 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 ...
20 votes
7 answers
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### To what extent can the invention of zero in India as a number be tied to Buddhist philosophy, if at all?

The Wikipedia entry on zero suggests that the ancient Greeks were unsure about the ontological status of zero. They asked themselves, 'How can nothing be something?' whereas in Buddhism, Sunyata or ...
11 votes
11 answers
3k views

### Can axioms be false?

I have often wondered, can axioms be false? For example, I could take as an axiom that "Dogs don't exist", but that is false. To give a more mathematical example, I could take as an axiom ...
4 votes
1 answer
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### Demonstrate that a term cannot be well-typed?

This problem is coming from Exercise 3.3 in Bacon's A Philosophical Introduction to Higher-order Logics. I am trying to do my due-diligence here and not skip problems, but this one stuck out to me. ...
3 votes
3 answers
574 views

### Is graph theory a good model for Seven Bridges of Koenigsberg?

I've asked this question on MathSE, but apparently people over there don't like philosophy. Seven Bridges of Koenigsberg is the problem whose solution (by Euler) gave a rise to graph theory and (...
2 votes
2 answers
189 views

### What philosophy of mathematics denies the existence of uncountable sets?

Finitism denies the existence of infinite mathematical objects (e.g. quantification over infinite domains is not considered meaningful). Is the position that denies the existence of uncountable sets (...
17 votes
21 answers
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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....
3 votes
1 answer
153 views

### Is the classical theory of concepts compatible with logical positivism's view on analyticity of mathematics?

Doing some work on theory of mathematical concepts and need a good framework that suits my own views. Is the classical theory of concepts, which seems to no to suffer very much when considered in ...
30 votes
10 answers
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### Isn't the notion that everything will occur in an infinite timeline an example of the gambler's fallacy?

I've seen a few different formulations of this, but the most famous is "monkeys on a typewriter" - that if you put a team of monkeys on a typewriter, given infinite time, they will ...
3 votes
6 answers
167 views

### Do contingent propositions about the world rely on the consistency of mathematics?

Assume that a contradiction in mathematics is discovered, say '0=1'. Then, by the principle of explosion from classical logic (by the rules of which, arguably, the world adheres as well) we can derive ...
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1 answer
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### If mathematics is invented for a deterministic reason, then do we discover that prior reason through our inventiveness? [closed]

Consider that mathematics is invented in some predetermined way. Would it then still be possible to claim that it is really invented and not discovered? Is there a point, modulo determinism, where the ...
3 votes
3 answers
275 views

### How to apply the classical theory of concepts on the mathematical concept of a limit?

I am studying the limit concept from mathematics using the classical theory of concepts. According to this theory a concept is; "A structured mental representation which is characterised by a ...
10 votes
7 answers
2k views

### Is the axiom of infinity truly an axiom?

I hope I can communicate my concerns effectively, so I can reach an understanding about a topic that I've been reflecting and researching intensely on for a few days. I am thinking about actually ...
0 votes
0 answers
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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 ...
1 vote
1 answer
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### Gödel's Asymmetry

First of all, The Liar sentence, off of which Gödel constructed his argument. L = This sentence is false. As the story goes, L implies contradiction AND ~L implies contradiction. So far so bad. Then ...
8 votes
4 answers
2k views

### If Large Language Models can do Maths, is Formalism true?

A slightly flippant question, but curious to see what my platonist rivals might have to say! One of the proported reasons that Open-AI was having business politics trouble was the suggestion that ...
4 votes
2 answers
333 views

### Why do constructive mathematicians claim that mathematical truth is temporal?

(crossposted here, wasn't sure where it belongs...) It seems to me (and correct me if this is a misconception) that the traditional divide in the interpretation and practice of mathematics is between ...
3 votes
4 answers
205 views

### What is an object's properties?

What can we consider an object's properties, for example, when can we consider an object's properties as 'changing'? For example, if I move an object from my desk to my table, has it changed? If I ...
5 votes
4 answers
566 views

### What is it that is done when we DO mathematics?

I want to understand more deeply and philosophically what exactly mathematicians do. Wikipedia lists some major subareas like analysis, geometry but ends its lead paragraph with There is no general ...
1 vote
2 answers
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### Looking for a reference on a kind of mathematical platonism

I've been doing some introductory reading on the philosophy of mathematics in an attempt to find well expressed views similar to the following. I haven't been successful. The view is that ...
1 vote
1 answer
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### How is synthetic knowledge produced in fictionalism?

With the Greek gods being fictional there is still objective knowledge - how many Greek female gods are there, etc. (Or if that's still too ambiguous, how many Greek gods are named Zeus). But "...
3 votes
5 answers
278 views

### Conceptual difference between probability vs percentages

Suppose there is a medical study which finds that having some Z gene is relate to a disease Y by a by 50%. Now, would it be correct to interpret this is as a probabilistic result? That is, there is a ...
1 vote
2 answers
82 views

### What does philosophy have to do with category theory? [closed]

Category theory seems very abstract and unrelated to philosophy. Why does it seem to be a part of philosophy? Is category theory used in philosophy and in the development of logical arguments? Isn't ...
1 vote
1 answer
254 views

### Does deflationary truth collapse into a correspondence theory?

If you ask what justifies a deflationary account of truth, doesn’t that reveal an implicit isomorphism within the justification thus collapsing the account into a traditional correspondence theory?
7 votes
5 answers
2k views

### 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 ...
4 votes
5 answers
390 views

### Does panpsychism imply mathematical entities are conscious?

Does panpsychism claim or logically imply that even mathematical entities, like numbers and functions and sets, are conscious entities? Or is it restricted to physical objects?
5 votes
6 answers
615 views

### Are laws separate “objects” or are they inextricably part of the universe?

This question came forth from a discussion I was having. Suppose that the universe is deterministic because of some laws. But those laws themselves exist for no reason. Does this mean that the laws, ...
3 votes
1 answer
147 views

### Can abstract concepts be represented by types in mathematics?

I am reading about type theory along with abstraction and am wondering how they relate. Am i right in thinking that an abstract concept (from the result of abstraction) can be represented by a type in ...
4 votes
5 answers
2k views

### Why do we have a problem about understanding the concept of the "empty set"?

The title seems quite bit more generalized, but I'm saying about the philosophers and mathematicians in the past who discussed about the concept of nothing, or the empty set. I'm ...
3 votes
2 answers
110 views

### Are Bourbaki and Deligne Mathematical Realists?

The following are two closely related questions. What was Bourbaki's position on the ontological status of mathematical objects? Were they some kind of Realist/Platonist or were they Formalist? ...
2 votes
4 answers
929 views

### Is Fermat's last theorem a logical necessity or a different kind of necessary truth?

Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than 2. The question was, is this a logically necessary ...
0 votes
1 answer
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### Inverted spatial qualia: a detectable example?

The SEP article on inverted qualia discusses this mostly as follows: One of [Frege's] theses in The Foundations of Arithmetic is that arithmetic is “objective”, which he explains as follows: What is ...
3 votes
3 answers
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### Why are empirical and theoretical knowledge connected?

There is a web of issues pertaining to the theoretical underpinnings of science that I would like to read more about, and so if anyone could take a look at the following claims and questions and ...
118 votes
22 answers
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### Why don't fair coin tosses "add up"? Or... is "gambler's fallacy" really valid?

I have always been perplexed by a seeming paradox in probability that I'm sure has some simple, well-known explanation. We say that a "fair coin" or whatever has "no memory." At each toss the odds ...