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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Let A be a finite non-empty set and S a finite symbol set. Show that there are only finitely many S-structures with A as the domain [closed]
Let A be a finite non-empty set and S a finite symbol set. Show that there are only finitely many S-structures with A as the domain
Let k be the number of elements in A,
for all constant symbols c ...
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What are the ramifications of the limitations of ZFC set theory?
In the Wikipedia article on Zermelo-Fraenkel set theory says that the theory sets out to formalize a notion of sets such that "all entities in the universe of discourse are such sets." It goes on to ...
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How does Lowenheim-Skolem theorem prove the relativity of mathematical models?
Stuart Shapiro mentioned in his book Thinking about Mathematics that Lowenheim-Skolem theorem showed the "relativity" of a model in mathematics. What does it actually mean? What does the phrase "to ...
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Mathematics Essential
I started reading History of Philosophy and readily noticed that the origins of our actual natural sciences were due to the proper use of inductive logic. Our Physics/Chemistry and Biology all are ...
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Is model theory (for logic) a kind of type theory?
Model theory is applied to axiomatic systems to give them an interpretation. Now consider a logic in axiomatic form. When we consider a model of this logic and some sentence in the logic, each ...
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Cognitive science/brain sciences and their impact on philosophy of mathematics
How does/did cognitive science influence philosophy of mathematics? I saw somewhere (Wikipedia, "Cognitive science") that it helped to create new perspective on philosophy of mathematics, but it did ...
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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 ...
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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 ...
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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 ...
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Logic and intuitions as outcomes of different languages?
So the way I see it there seem to be three different kind of languages we humans are capable of. The first is speaking language which include phrases such as: "we do not convey words, we convey ...
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Relativism and common sense in ZFC
ZFC is the most well known set theory which is considered by many as the foundation of mathematics but I am confused to understand it intuitively. Most of us have a clear understating of empty set and ...
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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 logical possibility the same as mathematical possibility? [closed]
Is everything that is logically possible also mathematically possible, and vice versa? Note, I am not suggesting that logic and mathematics are identical. I am merely asking whether logical ...
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The treatment of real numbers as 'objects'
In school we learn about numbers through physical amounts and we take two things and put them with two other things and call it four things in total.
Is this view of numbers as amounts slightly 'old ...
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Can we model something up to an approximation which fundamentally does not have a mathematical description?
Usually physicists assume there exists a mathematical description of reality and their models are mere approximations. So here's something that I wasn't sure about:
Let's say I have a phenomena which ...
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Is '=' a relationship between the objects or their expressions?
The Wikipedia definiton of equality gives it as a 'relationship between two expressions'
This confuses me as when we define mathematical expressions like 2+2=4 it makes no sense to say that '=' or '...
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Sets vs 'association'
I previously asked whether abstract objects can be split into categories, groups or sets of their component parts, and was told, definitely, and later another question occurred to me, take an ...
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Is there stance that every logical and mathematical derivation exists/is contructable but we only care about a proper subset?
I'm thinking every logical derivation as something like all the derivations in the Principle of Explosion - really everything.
It could just be a helpful interpretation, not trying to get super deep ...
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Abstract objects and changing properties
I like to use This website to explain some of the simple ways of mathematical thinking, but in the linked article by Wells, he gives his ideas on how mathematical objects are inert, but in this he ...
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What do we explictly refer to in mathematical expressios
My friend has a theory about 'instantiation' of numbers, they believe that every time we think of a number we create an 'instance' of it in our own heads, it's the same idea, but each time we think, ...
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Mathematical nominalists and realists on "there are at least as many possible intentional states as mathematical objects"?
I think this can be meaningfully asked. Intentional states
**Ideally I'm asking this about: mathematical nominalists, constructivists, intuitionists, and realists (and ideally I'd ask this about ...
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If there are theories of everything for science, are there equations of everything for math?
Some considerations:
A related question is whether math is finite, and has been asked here before. Unlike science, math does not seem to be finite. As a dear friend of mine once retorted to me, "...
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Multiple interpretations of the same syntax in mathematics?
Mathematics are doing a very odd usage of syntax and semantics. Let's take a wikipedia page as an example : https://en.wikipedia.org/wiki/Intuitionistic_logic
Here we have a syntax which is given, and ...
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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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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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Mathematical objects existing as different instances
I have a slightly complex conceptual question about the idea of 'multiple' instances of mathematical objects. In particular Real Numbers, and generally the idea of having multiple instances of ...
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What is mathematics? What are some of the most predominant philosophical definitions of mathematics?
Philosophers have given the nature of mathematics a lot of thought. As a beginner exploring philosophy, one of the questions which presents itself is 'what is X', and in this case, X is mathematics.
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How do finite processes and endless mathematical objects relate to reality?
There seem to be several philosophers who believe science (plus human norms for Sellars) can in principle leave no unanswered questions about reality. I would call this finite or exhaustible.
Sellars:
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Questions about Reichenbach's Principle and causes
Is "statistical dependences need to be explained causally" an accurate depiction of Reichenbach's Principle? (Rob Spekkens https://youtu.be/n8NRSPCekmI?t=1575)
Does one need to accept this ...
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How to prove consistency of theory with metalanguage?
I am familiar with first-order model theory. I also know that Tarski's definition of truth was made precisely in order to avoid paradoxes related to metalanguage such as the Liar. My question is: how ...
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Can you cross a space from which a two dimensional plane is missing?
If I travel through 3d space, will my travel be stopped abruptly if I encounter a 2d plane without space? That is if a 2d plane of space is missing?
You can consider every type of motion, continuous, ...
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Is a complete mathematical description of reality possible?
There are definitely states of systems(like mind) which are not quantifiable. For mathematics to work in principle, we need states which are quantifiable or measurable. So, does this go to show that ...
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Is Fourier transform a human made tool or an act of nature? [duplicate]
I am a PhD students in physics, and my father is a Math researcher. One time, I asked him
"Doesn't the fact that we can use math to explain things that happen in front of us, tell us that math is ...
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Is there more than one form of logic in mathematics?
Is there more than one form of logic in mathematics? I would be inclined that mathematics only cover one type of formal logic, but I would be interested to know if there are variants thereof or ...
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Does the following argument about the ontological nature of math exhibit poor reasoning?
Argument
P1: Mathematics is the substrate upon which all natural phenomena occur and necessarily governs phenomena in the physical world.
P2: One can experience something that is not mathematically ...
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How can we differentiate between change and progress in the area of math and ethics?
I'm studying epistemology, and I want to use reason and language as tools for carrying an investigation. How do I discuss the subjectivity inherent in change and progress, and also whether change and ...
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Statements about real world
We make statements like "This table is composed from atoms". This statement must be true or false. But what if tomorrow the atomic theory is completely abandoned and we work with another ...
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What is Number again
I came here having asked on the math stack exchange site about number. There are several responses to that question or one similar that suggest that here is the best place to ask the question. On ...
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Which problems do you consider as most important open problems in philosophy of mathematics?
At the "intersection" of mathematics and philosophy, or, rather, within their "union", surely some problems are still open and no general consensus is attained when those problems are discussed.
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Are there universes where rules of mathematics do not follow?
According to Max Tegmark the ultimate reality is the Mathematical world. Mathematically possibility also refers to physical possibility. Can there be such a type of universe where mathematical ...
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Comparisons between two notions of existence
I have the following, rather naive question:
To what extent can the a priori existence of mathematical objects be reasonably compared with the seemingly a posteriori existence of objects established ...
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Kant's Notion of Synthetic A Prioiri as Logical Entailment
Is there something wrong about interpreting Kant's notion of synthetic a priori statements to be logical entailments?
I understand, I think, that Kant didn't want to say such statements (e.g math ...
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Mechanics of Perception
How is perception formed? By perception I mean 'thought' or 'idea' of the World.
What I see by itself does not contribute anything to thought. Only an acknowledgement can contribute to structuring of ...
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Are there recent coherence theory of truth for mathematical truths?
Are there any recent works (papers, books, etc) in philosophy of mathematics where it is given an account of mathematical truth in terms of a coherence theory of mathematical truth?
I am interested ...
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Examples of theories that assume the existence of an “External Reality”?
In this paper written by physicist Max Tegmark (https://arxiv.org/pdf/0704.0646.pdf) it talks about "External Reality Hypothesis". Specifically, he says:
Although many physicists subscribe to the ...
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Can math be done without syllogisms? [closed]
Question seems self explanatory. Is there anything in mathematics that can be stated to be true without using a logical syllogism?
Had a discussion with somebody about this recently.
Sorry if this is ...
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Philosophy - If Space and Time are infinite and therefore infinite copies of us would end up existing, then wouldn't we still be gone after we die?
I have been pondering a question in my head. If Space and Time are infinite, then does that mean that Nietzsche's Eternal Return theory is true in the way that my life would recur, that when 'I' ('I' ...
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doomsday argument -- microstate-vs-macrostate objection
(Note: crossposted from https://math.stackexchange.com/questions/2856241/ where some comments suggested maybe the question isn't appropriate on math.se, although I thought the "statistical-inference" ...
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Correct Way of Handling A Corollary of A Corollary?
I have a conclusion S that is moderately interesting. While the corollary of S is more interesting, the corollary of the corollary of S is extremely interesting. Should I just label them corollary 1 ...
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Are propositions in mathematics synthetic or analytic?
I'm reading Kant's Critique of Pure Reason and I understand that he thought that "Mathematical Judgements are all synthetic". I would like to know where does this debate lies or if it is of interest ...