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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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 Gödel 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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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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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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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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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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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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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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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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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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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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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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Are implementations to semiotics what proofs are to syntax, and models to semantics?
I see this term implementation and its family used here and there in writings on set theory. There are implementations of natural numbers, of ordered pairs, of functions, of "mathematics in ...
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Does psychology/cognition come prior to mathematics?
I am not a hundred percent sure this belongs in Philosophy SE, but I couldn't think of a better SE to ask it, so I am asking it here. I got into an argument with a friend who claimed that the science ...
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Why was Russell discontent with Wittgenstein's view on "logic as tautologies"?
While reading Logicomix, I came across a scene that I don't quite understand.
Russell: ...Logicians are creating elaborate ways to "say the same things in different words"...this "...
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What is the idea behind "p or not p" being a tautology?
Most (all?) logic books consider "p or not p" to be a tautology, hence always true, and this is usually stated without any further discussion. (I never gave it a second thought.)
In common ...
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Is there an alternative to infinity?
We can say that a discrete set with 1 and 2 allows us to count just from 1 to 2 but a sequential set with 1 and 2 allows us to count from 1 to 2 in an infinite way (1.1, 1.2, 1.3 ...) but no man can ...
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Are there any resources that discuss the relevance of mathematical fields/problems to philosophy?
I've been enjoying reading Scott Aaronson's paper Why Philosophers Should Care About Computational Complexity. The paper discusses how the field of computational complexity is of major relevance to ...
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{Stipulativism vs. ostensivism} vs. {Formalism/if-then-ism vs. ante rem realism}
The SEP article on definitions includes the following two passages:
See Frege 1914 for a defense of the austere view that, in mathematics at least, only stipulative definitions should be countenanced....
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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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Why should universal generalization work for abstract objects?
I am reading a logic book in my free time and usually the inference rule of universal generalization is motivated by real-life examples: Imagine having the statement that all people with brown hair ...
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Is the Quine-Duhem thesis valid also for mathematics?
I have been studying the lack of consensus regarding e.g. the first Hilbert problem (whether a problem is solved or unsolved). It is clear for several of the problems that a fair amount of ...
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Forcing in other axiomatic systems [closed]
What axioms do you need before you can use forcing? Does forcing use the axiom of infinity? Is there forcing in geometry as opposed to set theory?
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Why mathematicians\logicians try to establish totally mechanical frameworks, in the first place?
As much as I know, at least for the past 200 years mathematicians\logicians are doing their best in order to reduce their works into syntax (into some totally mechanical frameworks) accoding to some ...
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How can one believe that mathematical assertions are objectively true while denying the existence of mathematical objects?
I had started reading a book called "A Historical Introduction to The Philosophy of Mathematics" where it began by outlying some common beliefs within the philosophy of Mathematics. One such ...
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How did Descartes made a logical skeptic argument against logic, without falling into a paradox, in his Metaphysical Meditations? Is it actually valid
René Descartes seems to have made some arguments against logic and mathematics in his Metaphysical Meditations, however it seems that these arguments are still logical, and the problem is whether that ...
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Definition of 'Identity' [duplicate]
This may seem like a very specific or stupid question, but I'm new to this, I'm interested in the idea of 'identity' and 'identical. I've heard some description of the idea different 'copies' or ...
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