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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Necessity of arithmetic truths into Godel sentences

My layman but hopeful to understand self is slowly trying to understand some of Godel and the philosophical implications of his work (uh oh). Currently my understanding is that on some level: Godel ...
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Do concepts transcend reality?

I often come to wonder about a specific, quite abstract question. Since I am not used to writing about such thing, it is very difficult for me to explain, but I will try to present my reasoning by ...
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Is there a notion of "because" in mathematics?

Sometimes, in math classes, we are asked to give justification for our mathematical assertions. We say that mathematical statement X is true because Y is true. However, I don't know if "because&...
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Unique-ness in the languages of Math and Physics?

Background So here's something I was pondering about: A teacher asks a student: "what is 3+1?" The student replies "3+1" It's not that the student's reply is wrong. But it's not an ...
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Which philosophy topics are necessary for philosophy of mathematics?

I'm currently a math major, but I'm very interested in philosophy of mathematics. I wonder if there are any prerequisites for learning philosophy of mathematics (such as studying metaphysics or ...
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is the theorem of Pythagoras right? [closed]

We don't know if the theorem of Pythagoras is right or not because we have to find a way to size things correctly, and to prove it or disprove it, and the angle too, we have to draw an right triangle ...
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Proper Philosophical Texts on the Philosophy of Science

I am a college freshman majoring in Philosophy and Physics. I am interested in the Philosophy of Physics, but before that, I would like to get an idea of general philosophical issues in the sciences. ...
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Is C.S Peirce really an independent co-discoverer of first-order logic?

According to this article copies of Frege's Begriffsschrift were both present during the early 1880s(before Peirce published his works on first-order logic) at the John Hopkins University where Peirce ...
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Why can't numbers be 'used up'?

I was speaking with a young student who has been learning about addition and subtraction (essentially functions, but he doesn't know that yet) with the idea of a 'number machine' and he could not ...
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11 votes
6 answers
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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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2 votes
5 answers
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Are mathematical proofs subject to the problem of induction?

When I consider a proof, such as Euclid's proof of the infinitude of primes, it can give a sense that something necessarily true has been obtained. I cannot remember where I got the idea, but a few ...
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Does logic give us a single definitive and universal answer for comparing the odds of unlikely events?

As an amateur who has interest in logic and mathematics I've been reading about the concept of different probability perceptions. I'd like to have your opinions over the subject below. When it comes ...
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Is there "empirical" distance without "mathematical" distance?

Mathematicians since antiquity have been thinking about length and angle, including doing things with straight-edges, rulers, compasses, and protractors. Fast-forward to modern physics, and you'll see ...
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How could second-order logic satisfy (neo) Fregean's epistemic goal?

Recently I've been reading Shapiro's Higher Order Logic in The Oxford Handbook of Philosophy of Mathematics and Logic, Chapter 25. There are some paragraphs confusing me:  One traditional goal of ...
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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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Constructing natural numbers from nothing

I found that many of us (mathematicians) try to construct natural numbers defined from the intuitive concept 'size of the set'. They take ϕ, the empty set, as the starting point, then define and ...
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Did Godel think certain math could only be understood if platonism is correct? (and correspondence and nominalism)

I’m reading Shapiro’s Thinking About Mathematics, and there’s a quote by Godel which I would like to fully understand, both his intended meaning and how it’s viewed in the wider context of mathematics,...
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Can structures in mathematical structuralism be models also?

This is a question about structures as defined by structuralism in philosophy of mathematics. From this article, it uses the term "system" which is usually (not by structuralists) denoted as ...
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Are mathematical objects a type according to type-theory?

I've been thinking about mathematical objects as a metaphysical trope, and the idea of them existing as a type has a few issues for me. Mainly the response to this question is similar to what I've ...
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2 votes
2 answers
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Kant's Prolegomena Note I - Geometry being an objective representation of nature

I'm trying to understand this part of Kant's Prolegomena to Any Future Metaphysics, Note I to "How is pure mathematics possible?": It would be completely different if the senses had to ...
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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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Kant's Prolegomena §13 - triangle example argument

In Prolegomena to Any Future Metaphysics, Kant argues that space (and time) are not qualities of objects, but a priori intuitions that allow the concepts of objects in our minds. To argue in favor of ...
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When does 'number' become 'quantity'?

Numbers themselves are simply conceptual objects, but when does number become a quantity? Is the 'cardinality' of a set a 'quantity'? it is a count but we represent it with just a number that we ...
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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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"Can be interpreted as" vs "Is"

Consider the following pairs of statements: "I see what I interpret as a chair" vs. "I see a chair." "This chair can be interpreted as a set of atoms" vs. "This ...
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Why are there no Computer Algebra Systems designed to import known mathematical identities/theorems?

Computer Algebra Systems (CAS) are philosophically interesting in that they are an aspect of the long history of treating mind as mechanism. In this respect, mathematics may be thought of as ...
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3 answers
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How is 'Pure Intuition' possible according to Kant?

One of the key passages is from Prolegomena to Any Future Metaphysics (1783). According to Kant pure intution is the means to obtain mathematical theorems as synthetic a priori propositions. This ...
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Do the limitations of science prevent the creation of only one true version of science?

I was recently intrigued by the following comment made by Ricky Gervais in this discussion with Stephen Colbert (Timestamp: 3:50). If we took every science book, right, and every fact and destroyed ...
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Can set theory be non-extensional?

Here is Juliet Floyd stating "Wittgenstein's non-extensionalism, like Russell's in Principia, precluded development of an extensional theory of the infinite (set theory). https://youtu.be/...
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Does Gödel’s findings boil down to part of classical mathematics (as opposed to computation) is flawed?

According to artificial intelligence researcher Joscha Bach, only classical mathematics is affected by Gödel’s incompleteness theorem however not computation where calculations are performed in a step-...
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Have the indispensability arguments been examined for probability theory?

Indispensability arguments are widely known in the philosophy of mathematics, the idea being (roughly) that we should commit to the existence of those mathematical entities that are indispensable for ...
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Are numbers, given just as mathematical objects, quantities in themselves?

If we are talking just about '5', without it being with respect to any 'amount', does the idea of the number itself as a point on a line imply that it is itself some kind of abstract 'quantity'. It ...
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1 vote
2 answers
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What role does philosophical position play in foundational mathematical research?

Does philosophical position (platonist, formalist, etc) play a role in thinking about mathematics, and the subsequent research? That is, for example, did "platonist" position, or "...
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Gödel’s Incompleteness Theorem: How can truth go deeper than proof?

My current understanding: Math starts with a set of basic (purportedly self-evident) statements that are taken as a given without the need to prove them true, like e.g., a + b = b + a etc. Such ...
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Can a backwards infinite regress account for its own existence?

Suppose we have a domain of discourse D with an infinite collection of elements, and suppose that it is the case that the existence of each element x is dependent upon another element y (or collection ...
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A fair critique of Wittgenstein's insight?

So I'm part of this math meme group and this was posted I'm not an expert in "modal homotopy type theory" but are both claims true? And is this a fair critique of Wittgenstein's insight?
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Do breakthroughs in mathematics lead to breakthroughs in other scientific disciplines, vice versa, both, or is there no relationship?

I asked this question in the Mathematics StackExchange, but I was told it might be better posted here in the Philosophy StackExchange. I heard a professor say once that Einstein's mathematics led him ...
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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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Intuition behind existence of "function sets"

The usual axioms ensure the existence of certain sets that serve as functions. For example (which is chosen arbitrarily) the function f which maps real values of x to x^2+2 can be represented by the ...
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References on Metaphysical Dependence on Set Theory

I'm new here. I was having a hard time reading and comprehending John Wigglesworth's thesis Metaphysical Dependence and Set Theory. I was hoping if there is anything out there talking about the matter;...
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3 votes
3 answers
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In category theory, why do we meet more left adjoints than right adjoints

In this answer, the author states that "many of the naturally occurring functors we meet tend to have left adjoint but often they lack right adjoints". Is there any philosophical explanation ...
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Could everything exist? What would this even mean?

I saw something that said the reason the universe exists is that everything exists, in an infinite multiverse. This then answers why the laws of physics of our universe are the way they are, which is ...
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3 answers
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What created the bias in humans that make us intuitively claim the refutation of the Continuum Hypothesis?

So I was watching this youtube video. We've got a well ordering of the real numbers but just between zero and one that'll do okay now comes a little statistical argument. You and I are gonna throw ...
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2 votes
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What in mathematics has the property of beauty?

I might or might not be at an impasse in my writing... I have around 200 pages of notes, and I finally sat down and tried to compile some of the material, but I feel like the presentation is off ...
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2 votes
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Is Frege and Russell's definition of number reducible to Hilbert's?

This is my first post on this site and I hope that its length/format is not inconsistent with any moderation guidelines. My question relates to the distinction that can be drawn between the logicist ...
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1 vote
1 answer
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References for the notion of grounding, applied to mathematical truths

I am interested in papers that discuss the notion of grounding and applies it to mathematical statements. For example, the facts that 1+1=2 and 2+2=4 collectively ground their conjunction 1+1=2 AND 2+...
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2 votes
4 answers
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Is Mathematics a form of experience?

When someone experiences the mental clarity of 2 + 2 = 4, is this a form of experience similar to let's say, seeing red, or the sour taste of a pickle. On the one hand it seems like it is a form of ...
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5 votes
8 answers
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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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