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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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?
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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What could be a formal analysis of the logic of the proof by contradiction that the square root of 2 is not a rational number?
What could be a formal analysis of the logic of the proof by contradiction that the square root of 2 is not a rational number?
Is there a recent publication on the same subject?
EDIT
What was the ...
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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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Is topology used outside of cosmology in philosophy?
It seems like topology is used to model spacetime, but outside of cosmology, it seems like topology has absolutely no use in philosophy. Is topology used to create models that relate to abstract and ...
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Is the multiverse standpoint in set theory "ideologically committed" to plural quantification over universes/axioms?
One of the ways in which Hamkins expresses the multiverse standpoint is as the assertion that there is no "absolute background concept of sets or even ordinals." He spells out examples of ...
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Resources on the distinction between epistemology in pure and applied mathematics
I'm looking for recommendations for works that present roughly (or at least parts of) the following perspective on epistemology in mathematics. I hope having access to similar perspectives will allow ...
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How and where do I learn philosophy?
I am very new to philosophy, in what ways can I gain deeper knowledge about the subject itself and explore all the branches and truly understand what philosophy exactly is?
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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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Why don't formalized proofs make formalism true?
All mathematicians are familiar with the (extremely plausible) fact that any ordinary mathematical proof can be formalized inside some foundational theory, e.g., ZFC.
Why doesn't this imply that ...
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Bayes' Theorem and Science
More than one hypothesis may fit the data (hypotheses generation is the stock-in-trade of science)
Choosing a scientific hypothesis is not about truth. People have gone on record that inter alia it's ...
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Why are physical and logical probabilities considered separate?
It is argued that there is a difference between these probabilities. When a dice lands on 6, it is argued that because it could have landed on 1-5 by the nature of physical laws, the probability is 1/...
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If platonism was correct, would everything be real despite everything being formal?
In one of his recent essays (https://writings.stephenwolfram.com/2021/04/why-does-the-universe-exist-some-perspectives-from-our-physics-project/) the scientist Stephen Wolfram says (at the end of it, ...
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Is there a correct way to 'read' mathematical expressions?
Most written languages have a direction to be written, for example, most european languages are read from left to right, and arabic based languages are generally written right-to-left.
In mathematics, ...
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Is mathematical creativity the same as artistic creativity?
Do philosophers distinguish between mathematical creativity, and the broader artistic creativity? If so, what are the differences between these two?
A lot of people seem to treat IQ as something ...
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What does it mean to say, "Philosophy really started in 1884"?
I am looking into my old notes from a lecture I attended on Frege's Grundlagen, where the professor at some point jokingly said that philosophy started in 1884, with the publication of Grundlagen. I ...
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Mathematics and the observer in Wolfram Fundamental Physics Project...?
In this video (https://www.youtube.com/watch?v=TrnteM9E2tI&t=6633s) about mathematics in the Wolfram Physics Project, Stephen Wolfram says at minute 1:49:37 something that seems contradictory:
He ...
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Can humans symbolically manipulate X that describes themselves?
Consider the following premise:
Any statement regarding the physical world can be proven within the system of X (assume X to be something like Quantum Field Theory) by humans. One may argue that this ...
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