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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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 [🍏 ]
...
user68071
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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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Is Ultrafinitism Proven by Neuronal Dynamics?
In his book Studies in No-Self Physicalism, Feng Ye suggests a radical form of physicalism to explore the meaning of mathematical concepts. From my experience with neuronal dynamics, the Hodgkin-...
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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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Can a totally ordered set with a last element but no first element exist, or is this contradictory?
Can a totally ordered set with a last element but no first element exist, or is this contradictory? An example of such a set would be a set that is ordered from largest to smallest, with there being ...
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How is the completeness of first order logic reconciled with the incompleteness of set theory?
First Order Logic (FOL) is complete in the sense that:
there is a proof procedure for FOL such that just the statements(/wffs) of FOL that are true and remain true under any re-interpretation of their ...
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How many variations on graph-theoretic/related parameters indicate alternatives to foundationalism/coherentism/infinitism?
The set theory I'm trying to work in right now is geared towards applying an "axiom of multifoundation" whose local maximum representation is:
The interpretation of the elementhood glyphs ...
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If it is trivial that 𝘟, is it trivial that it is trivial that 𝘟?
Differentiate between empty, trivial, and nontrivial solutions to problems. From a category-theoretic point of view (or maybe just mathematics/logic historically), one has that empty solutions are ...
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Entry points from philosophy into mathematics at higher levels?
Everytime I look up of the link between philosophy and mathematics, I see the topics only of the most foundational levels discussed. As in logic, and stuff. When I study higher mathematics theories, ...
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Is Neil Barton's algebraic/ontological distinction equivalent to the actualist/possibilist distinction?
In, "Multiversism and Concepts of Set: How much
relativism is acceptable?" Neil Barton distinguishes between an ontological interpretation of set-theoretic multiverses as referents and an ...
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Logic and math as a study of possibilities and not so much about human reasoning
Most of what I've come across about the "hierarchy of disciplines" seem to say that logic/math is more fundamental than physics, physics more fundamental than chemistry ... biology more ...
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How many instances of 1 are there in the expression "1+1"?
Is it just two marks/numerals representing a singular number 1, or are they actually two instances of 1?
And what about in a set with repetition such as {1, 1, 2, 3}?
And if these are actually ...
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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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Continuum and Choice sequence
I am reading a paper on Brouwer's intuitionism. It mentions that according to Brouwer, the concept of continuum is perceived as a whole by intuition. However, it also mentions setting up choice ...
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Is modern mathematics scholasticism?
I have thought a lot a about mathematics and it's foundations. There have been several attempts to give it a solid foundation, and they all failed. Frege / Russell logical atomist approach failed, ...
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Is Benacerraf's argument circular?
I'm reading Benacerraf's What numbers could not be, where he provides the following argument against platonistic account of numbers.
The only criteria we can ask for in searching the correct account ...
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What is computation?
The most common definitions of computation I have seen are in terms of "what Turing Machines, Lambda Calculus, etc. do," which is unsatisfying. The definition of computable functions does ...
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Quine-Putnam indispensability argument
If Quine-Putnam's argument is (following the SEP):
(P1) We ought to have ontological commitment to all and only the
entities that are indispensable to our best scientific theories.
(P2) Mathematical ...
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Why do some proof method feel more satisfying than others?
Let's say we are asked to show that 1+2+3.. =n(n+1)/2, then a very simple way to prove this is to use induction. The proof is simple, consider P(1) and show P(n+1) from P(n). However, it feels quite ...
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What are some sociological considerations to understands mathematics culture?
One could argue that, fundamentally, mathematics is a sociological process, as the backbone of mathematics is that of the mathematic proof, and the mathematic proof of a statement, at least as used in ...
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How does understanding of fragments differ from understanding of the whole?
Consider a person reading a mathematical proof, then each syllogism from it's antecedent maybe understood by that person, yet they may find it difficult to understand the whole proof. At times however,...
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Natural Language and Implication
I understand that relevant logic deals with a natural-language interpretation of implication, but it seems too restrictive. It does seem a bit of a reach to say that there is a conceptual link between ...
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Turing's bridging argument of conflating mathematical logic and the philosophy of mind?
So I read this paper and I'll quote the relevant parts:
'Turing's machines are humans who calculate
On Computable Numbers' thus took on the
aspect of a hybrid paper: an attempt to integrate what ...
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What is a good approach to the question "the real number 2 is the same as the complex number 2?"
If Platonism holds, mathematical objects exist independently of our minds, and so the number two exists independently of our minds, but can there be multiple number 2's independent of our minds (e.g. ...
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Kant's view on higher-dimensional geometry
According to Kant, geometry is possible because of our intuition of space. But, this intuition is presumably 3-dimensional, as we experience the world 3-dimensionally. So, how would higher-dimensional ...
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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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