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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Besides the Axioms---What are the First Principles of Euclidean Geometry?
Regarding 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 ...
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The mathematical concept of a limit in terms of the classical theory of concepts
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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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 "...
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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 ...
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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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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?
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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 ...
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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?
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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, ...
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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Constructivism and function definition in mathematics
In this blog post, we find the following passage:
This connects with something Thomas Forster said, when he rightly highlighted the distinctively modern conception of a function as any old pairing of ...
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Omniscience leads to necessitarianism
You have probably seen these types of arguments before on incompatibility of omniscience and free will. The question is are these arguments valid and what can be a good refutation?
Let G= x is known ...
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Why is there something instead of nothing?
A simple but fundamental question.
The "something" means the whole Universe (known and unknown), it could be represented as the reality version of the set of all sets, which is itself debated. It ...
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Are "A ∧ A" and "A ∨ A" degenerate expressions?
Although some time ago I had become somewhat familiarized with the notion of degeneracy in mathematics and physics, in my musings on the trivial/nontrivial distinction I found that both Wikipedia and ...
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Which field is more rigorous, mathematics or philosophy?
I don't know if this question is best suited for this stack exchange, but I couldn't think of a better stack exchange. I want to know, which field of study is more rigorous, mathematics or philosophy? ...
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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 ...
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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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Is life experience a necessity to generate mathematical thought?
The beginning of mathematics for the Human race was motivated by Human experience. This made me wonder, suppose if there was a concious being existing in deep space devoid of any life experience, ...
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Applicability of Mathematics
Suppose that an alien civilization exists, in a planet somehow similar to our own (oxygen-based, plants, animals), in an evolutionary stage similar to ours (large cities, advanced communications, near-...
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Can location be assigned to an entity, given a lack of length, depth, or width?
If one is to postulate an entity that has a complete or absolute lack of height, depth, and width, can such an entity be located anywhere? Or does attribution of location to an entity entail length, ...
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Can Mathematics Fully Describe the Universe?
To what extent mathematics can capture all physical phenomena? Drawing an analogy from computer science: finite automata can handle regular expressions (does "(([a-z]))" match "((h))&...
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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 there a One True Set Theory?
From the description of Category Theory in nlab:
Category theory is a structural approach to mathematics that can (through such methods as Lawvere's ETCS) provide foundations of mathematics and (...
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Which mathematical operations leave the ontology invariant? [closed]
So usually one maps a math equation to an ontology in physics. Imagine me modelling a ball rolling up an inclined plane at an arbitrary angle. Now, the moment I make the inclined angle 90 degrees to ...
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Rather than "ought to be true = is true" being impossible, might it not just be a trivial stage of moral representation?
I just finished reading Eugenia Cheng's essay on moral phraseology in mathematics, and so I want to go over something she says on pg. 20:
A recent lecturer of Part III Category Theory declared that ...
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Mathematical Platonism. Are numbers real?
Often heard this being asked: Are numbers real?
As an answer I offer my own analysis for what its worth.
The color green is considered real. As per scientists it's only distinguishing quality is that ...
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How to understand the notion of majority when comparing infinite sets?
Suppose I make the argument:
It is very unlikely that in a naturalistic universe, the constants have life sustaining values, since the majority of metaphysically possible universes do not have such ...
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Why would infinite monkeys not produce the works of Shakespeare?
Apologies if this is a very basic/obvious question. I have no training in philosophy, but have been making my way through Peter Adamson's History of Philosophy podcast.
Recently I listened to his ...
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Would "to avoid the class/set distinction" be, or not be, an ad hoc reason to propose a couniversal set?
Once upon a time, von Neumann proposed the axiom of limitation-of-size, which says that any class "too large to be a set" is then a "proper class," meaning that there is a ...
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Why is mathematics so fantastically successful at describing the universe?
Anyone who has studied physics will quickly see how fantastically successful mathematics is at describing the universe. The famous physicist Richard Feynman said in his book "the character of physical ...
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Probabilities and Certainties on the Monkey Axis: Yet more about those monkey typists
I was reading with some interest the answers and comments to this question about that familiar, weird and somewhat inhumane infinite-monkey experiment which, somehow, is still generating fresh and ...
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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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Justification of implication-introduction and modus ponens
Given only the definition of material implication through the truth table
A | B | A → B
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f | f | t
f | t | t
t | f | f
t | t | t
(where, as usual, "t" ...
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How can zero exist if zero is nothing [closed]
I understant why it has to exist, but how can zero exist, if zero is nothing, then nothing is something wich means that zero cant exist, ive seen similar questions but i still dont get it, help
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Why not move from proof numbers to theories instead of theories to proof numbers?
In mathematics, they do this thing where they figure out what are called "proof-theoretic ordinals" (see this section of the SEP article on proof theory for background details) of theories, ...
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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 ...
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Would Frege's version of the empty set contain "parafinitesimal elements," at least from the multiversal standpoint?
Frege's definition of the empty set was not a raw extensional one: he did not simply write the partial string {} and say, "That's it: that's the empty set." His account was more intensional: ...
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Axioms, meaning, and notation
According to at least one philosophy of mathematics, the axioms determine the meaning of the primitive symbols that are used in the axioms. The phrase "used in" is somewhat imprecise, so ...
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