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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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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Has this mathematically proven solipsism? [duplicate]
http://bc.upjp2.edu.pl/Content/5621/35_PDFsam_Ca%C5%82o%C5%9B%C4%87%20ze%20znakiem%20wodnym3.pdf
It's not so much the math as it is these things in the link:
More generally, there can be no deductive ...
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What is the scope of "philosophy of mathematics"?
Beside the foundations and logic, are there fields of mathematics (maybe following the MSC2020 Mathematics Subject Classification) which are currently of interest to philosophers of mathematics?
More ...
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Was Wittgenstein a mathematical finitist?
Wittgenstein was a notorious critic of set theory, calling it "laughable nonsense". However, he also wholeheartedly rejected intuitionist logic of Brouwer and Weyl, saying "it is ...
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Why did it take 3000 years for theories of mathematical foundations to emerge? [closed]
Humans have been doing mathematics for at least 3000 years. In ancient Egypt they did some advanced trigonometry and number theory. Mathematics is thousands of years old, but for some reason it was ...
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Does the preface paradox undermine long mathematical proofs?
Descartes, IIRC, somewhere says something about the vagaries of memory influencing our justification for believing in our memory, and thence for believing in proofs involving many steps that we have ...
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The massive problem with regarding string manipulations as the foundation of mathematics
Formalists believe that mathematics is just a game of string manipulation, not much different from other games like Ludo or chess. I think string manipulation is an extremely useful way to think about ...
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Intuitionism, the law of excluded middle and mental construct
I don't get why LEM is rejected in intuitionistic logic. The basic idea behind intuitionism is that math is a mental construct. But how does this make LEM not acceptable? I've seen some similar posts ...
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Is partial symmetry one of the most fundamental concepts or laws of reality?
Brain is partially symmetric, planets are, most of the object that look symmetric, are actually partially symmetric. Is partial symmetry in some sense a fundamental concept of our mind or fundamental ...
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How does one test their credences of belief?
Suppose I feel that event A is more plausible than event B. How can I test, verify, or falsify this?
For example, suppose I have a belief that my partner is cheating on me. Suppose I have another ...
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Does single case chance actually exist?
Does chance actually exist for a single case? Even for a coin, what does it mean to say that there is a 50% chance that the next coin toss will land on heads?
Someone might say that this means that if ...
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Is there any philosophy that specifically argues against subjective probability?
When I say subjective probability, I am referring to the notion of defining a probability in relation to a credence of belief. For example, one may say that there’s a very high probability that the ...
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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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Is this a legitimate way to reframe structuralism in the philosophy of mathematics?
As an umbrella term, "structuralism" has to cover realist and nonrealist versions, while also carrying through the theme of its name nontrivially (for there is a trivial way to make ...
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Do intuitionists and predicativists have an overly "absolute" concept of infinity?
Sifting through the historical data, I get the impression that intuitionism is not strictly a case of finitism (much less ultrafinitism), but more like "parafinitism". Predicativism, in turn,...
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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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What's the constructivist's view to the S4 modal logic?
Intuitionistic logic can be translated to S4 modal logic by parsing intuitionistic P→Q to classical □(P→Q). There is no other way round, for there is no intuitionistic equivalent to ◊P. To analyze ...
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Does a formula denote it's truth value once the variable is assigned?
In a lot of systems like boolean algebra '=' is treated as a function that takes two inputs and yields a truth value.
In first-order logic we often use an expression like 1. p(x)=(x+1=2) and to treat '...
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"Impredicative" definitions in mathematics
In this blog post, the following definition of an "impredicative definition" is offered:
A definition is said to be impredicative if it defines an object E by means of a quantification over a ...
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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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May different philosophies of math correspond to different subdisciplines in math?
Two ways of saying this:
Each philosophy of math has a direct counterpart mathematics: for intuitionism, it's a mathematics of intuitions (not as "hunches" but in the Kantian sense); for ...
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How do we use topology to model knowledge?
The topology of knowledge: In this application, topological spaces are used to model the structure of knowledge, where the open sets correspond to coherent bodies of knowledge and the closure ...
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Does "mixing up the notation" bring out a synthetic character for basic arithmetic?
There's this thing in the work of Immanuel Kant and Hannah Arendt where they'll slip into Greek and/or Latin, sometimes in the middle of a sentence (even if for just a word there), or sometimes like ...
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How would we define a topos between Classical Logic and Paraconsistent Logic, and determine what the homomorphism between them would be? [closed]
How would we define a topos between Classical Logic and Paraconsistent Logic, and determine what the homomorphism between them would be? I learned that topoi can't be used for philosophical ideas, but ...
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How can mathematical results impact the physical world?
In his 2007 book I Am a Strange Loop, Douglas Hofstadter uses an analogy based on a domino computer.
Indeed, it is possible to build logical doors made of dominoes (see e.g. here) and realize simple ...
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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 the principle of mathematical induction a purely logic statement? [closed]
Mathematical induction states that if a proposition P(0) is true, and if the implication P(n) ⇒
P(n+1) is true, then it must be the case that P(n) is true for all natural numbers n.
My main question ...
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What is the meaning of assertion?
I often see the word "assertion" in books of philosophy of language or logic. They may list a sentence like
Snow is white.
Then somewhere in the context, they may write "assertion of ...
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The demarcation problem of mathematics
The demarcation problem in the context of philosophy is usually used to mean the demarcation problem of science, the problem of separating science from non-science. However, what about the demarcation ...
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Consistency versus Provable consistency?
So influenced by perhaps Penrose and this question, I was under the impression in the Continuum hypothesis (CH) using ZFC axioms one cannot prove nor disprove the CH. From Wikipedia:
Cantor believed ...
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Is musical formalism a better version of formalism (in the philosophy of mathematics) than game-theoretic formalism?
There is a sense in which it is "true that" in the game of chess, a knight can move in an L-pattern, a queen can move in direct lines from end to end of the board, and pawns turn into queens ...
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Can the continuum hypothesis be settled in physics?
Can the continuum hypothesis be settled in physics? In a lecture mathematician Woodin considers the possibility:
Develops the mathematical physics of a mathematical understanding of
the physical ...
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What does it mean for something to be "more likely"? Whether you would bet on it? Whether history suggests it to be true? Or both?
What does it mean for A to be more likely than B? For example, suppose two people are throwing darts. The first person gets a bulls eye 6 out of 10 times. The second person misses every single time by ...
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Is the axiom of dependent choice constructive?
Page xvii of Schechter's Handbook of Analysis and its Foundation says that the Principle of Dependent Choice(DC) is constructive.
Is DC considered constructive? Different debaters may have different ...
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How is the concept of a topos in mathematics relevant to philosophy?
https://en.wikipedia.org/wiki/Topos
Topoi behave much like the category of sets and possess a notion of
localization; they are a direct generalization of point-set topology.
My understanding is that ...
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