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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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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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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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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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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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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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 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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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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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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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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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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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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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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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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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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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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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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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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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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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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Why do numbers apply to such disparate concepts?
I understand numbers to be defined as objects defined to have certain convenient properties in relation to certain operations. It is very surprising that the exact same group objects should be ...
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Can this be an example of sophism?
Foreword:
0 is considered an even number,
but if 0 would be an even number,
then 0 apples would count an even number of apples.
Example:
3 apples [🍏🍏🍏]
2 apples [🍏🍏 ]
1 apple [🍏 ]
...
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Do some philosophers-of-mathematics give priorities to different epistemologies of math, rather than (over)committing to one epistemology?
Take Kant and Gödel, for example. Kant was neither just an intuitionist nor just a formalist, nor even absolutely a non-realist (the forms of space and time are, after all, empirically real and ...
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Does the incomputability of kolmogorov complexity imply that we will never have a final theory of everything?
The Kolmogorov Complexity is the size of the simplest program that produces a specific output. By the Curry-Howard Correspondence, "programs" are isomorphic to "axiomatic systems" ...
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Gödel's Incompleteness Theorem
I'll keep this short and sweet.
Construct Axiomatic System A in which we can do math.
Gödel Sentence G = G is unprovable in A.
Gödel's Argument (I)
If G is provable then there's proof that G has no ...
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A problem I noticed with if-then-ism in the philosophy of mathematics
In the philosophy of mathematics, if-then-ism is the view that mathematical assertions of existence, like the statement that there exist numbers which are their own squares, should, strictly speaking, ...
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Is mathematics based on formal logic, or vice versa?
Math is obviously based on logic in a heirarchical sense, but what about the historical sense? Is there any historical evidence of a "transition" from first order logic to mathematics? All ...
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Difference between how a physicist and mathematician approach science?
I am acutely aware of the fact that the marriage between mathematics
and physics, which was so enormously fruitful in past centuries, has
recently ended in divorce.
Freeman Dyson
Is there a ...
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Implicit Models and Probability - are degrees of belief/truth/existence a complete free-for-all?
Or, to put it another way, as long as you model your statements using the grammatical framework of our modern logical idioms, is it appropriate practice to assign a probability to any utterance at all,...
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Are there different forms of rigor, and if so, are some forms of rigor more rigorous than others?
Is there only one kind of rigor? Or does rigor come in different forms, like mathematical rigor, philosophical rigor, and scientific rigor? And if it does, are some forms of rigor more rigorous than ...
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Does significance testing contain a logical flaw or not?
This question was sparked from a comment Conifold had made. Link to comment here: Is probabilistic modus tollens a fallacy?
He says, and I quote, “The valid form used in significance testing is: If P ...
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To what extent can an equation be appropriately described as an emergent property of its variables?
If I think about a variable's value as the result of a measurement or counting, it makes sense to me to think that such a measurement wouldn't magically square itself. That measurement can only be ...
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Why is equality not considered the fundamental unit / principle in nature?
(I don't know how to ask this other than by laying out my worldview. Needless to say, this is here to be dissected and disemboweled. I realize the broad sweeps will irk people. I hope that by ...
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Is the answer to whether math is discovered or invented related to theism?
I'm not asking whether mathematics is discovered or invented, rather whether being theist implies/strengthens/related to the view that it is discovered, and vice versa.
For example I came across an ...
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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 criteria determines if a proposition is mathematical or empirical?
It seems that there is a distinction between mathematical vs empirical statements.
For example, consider the proposition “All even numbers greater than two are a sum of two prime numbers.” This ...
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How do modern platonists explain the objective, specific connections between the physical and abstract?
There seems to be an entirely objective, human-independent way in which specific physical objects relate/correspond to specific abstract objects.
Example, we don't think the abstract inverse cube law ...
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Does the Universe tend towards complexity/elegance?
So many of our scientific theories suggest that, from singularity or homogeneity, everything grows more and more complex. I say 'so many' here because I'm not speaking only about cosmology (and ...
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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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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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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 ...
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Two kinds of abstract objects - circles and sets
Both circles and sets are considered abstract objects. I can visualise a circle in my mind (can 'see it through my mind's eye') but can't visualise a set or a number. I have no picture of a set in my ...
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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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Did Kant believe that the a priori truths don't coincide with the necessary truths?
I just started to read about Kant's metaphysical distinction between analytic vs synthetic truths (necessary vs contingent) and his epistemological distinction between a priori vs a posteriori truths. ...
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In what way are (natural) numbers perfect?
In mathematics, one often makes the remark that what is being talked about is a perfect idealized object. "Our planet is a sphere, but it's not really a perfect mathematical sphere (that is ...
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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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Is it legitimate in science to use two contradictory axiomatic systems?
For example, in Zermelo–Fraenkel set theory (ZF), the addition of the axiom of determinacy(AD) is inconsistent with the addition of the axiom of choice(AC). Is it legitimate to adopt ZFC (ZF+AC) as ...
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