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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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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Differentiating between mathematical objects and their representations
The first time I came across this distinction was when I asked this question. It was highlighted to me that there is a difference between the matrix and how we represent it. There is a difference ...
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Meaning of the word Abstract
Recently I came across with a word called abstraction in mathematics which means the process of removing unimportant details from a system so that we can focus on the ones that really matter.
I found ...
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What does 'denote' mean in language and mathematics?
My initial idea of 'denote' is that is a verb that describes how an element of well-defined language such as 'cat' or '2' relates to the object/concept they refer to, however in mathematics text books,...
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Are the limit notation in mathematics an example of the use-mention distinction?
I'm trying to explain the limits of a function in calculus and I'm not sure whether to mention that the notation that to state that 'the limit of f(x) as x approaches zero' is a slight use-mention ...
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Is logical possibility the same as mathematical possibility? [closed]
Is everything that is logically possible also mathematically possible, and vice versa? Note, I am not suggesting that logic and mathematics are identical. I am merely asking whether logical ...
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Are the truths of arithmetic logically necessary? [duplicate]
Are true statements of arithmetic logically necessary? That is, is "2+3=5", the commutativity of addition of natural numbers, and the infinitude of primes, among other statements, logically ...
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Can there be a solution to these three problems?
I have read many times that some problems or logical propositions do not have solutions or are outright impossible. These are three examples of such problems:
[The Russell's paradox] which is deemed ...
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How to solve "impossible" problems?
In mathematics and philosophy there are some unsolvable problems like Russell's paradox or the liar's paradox that are usually said to be undecidable... There are also other "impossibilities"...
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Does mathematics fall within the domain of analytical philosophy? [duplicate]
Summary of the problem: I’m unsure if it is correct to think of mathematics as analytic philosophy in the domain of counting and am posting to gather perspectives.
Details and any research: make what ...
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Is it correct to say: 'Numbers are quantities'
I'm interested in whether the common view of numbers as 'quantities' is mathematically/philosophically incorrect. If you search the definition of number you get 'quantity'.
Bertrand russell's ...
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Does every mathematical question have an unambiguous answer?
Does every mathematical question have an unambiguous answer?
For example, suppose I were to assert "In the decimal expansion of pi, does there occur in at least one location a billion 1's in a ...
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The treatment of real numbers as 'objects'
In school we learn about numbers through physical amounts and we take two things and put them with two other things and call it four things in total.
Is this view of numbers as amounts slightly 'old ...
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Can we model something up to an approximation which fundamentally does not have a mathematical description?
Usually physicists assume there exists a mathematical description of reality and their models are mere approximations. So here's something that I wasn't sure about:
Let's say I have a phenomena which ...
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Visual representation of numbers?
When we learn mathematics we are given these visual representations of numbers, as things become more advanced we learn the idea of a 'number line', my question is how valid is the idea of a number ...
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Why do we equate a mathematical object with what denotes it?
Consider a matrix,
We denote matrices by an uppercase letter of the English alphabet like A, B, C, etc. Let this matrix above be denoted by A.
I can write,
I have often seen an equal to sign ...
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Peano Axioms have models other than the natural numbers, why is this ok?
Why do we say that the Peano Axioms define the natural numbers when there are more models other than natural numbers for those axioms? Maybe it is a confusion relative to the word "define", ...
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What is an instance?
I see the definition of 'instances' and it makes sense when dealing with types that define physical objects, so any cat is an instance of the type 'cat', however, why do we discuss 'instances' of ...
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Identity of mathematical objects
Leibniz law's states that if A and B have the same properties then A and B are one and the same, however we can consider mathematical objects that are isomoprhic but not identitical, they have the ...
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Is 'a level of quantity' a poor definition of 'real number'?
I was thinking about how we define numbers with respect to their uses, and came up with the definition of 'a level of quantity' which can have a different physical consequence for each quantity ...
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Is mathematics free of emotion? [closed]
I was planning to ask this on Math SE, but I decided Philosophy SE is more suited to this type of question. Is mathematics free of emotion? Certainly, solving mathematical problems and proving ...
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Was mathematics created by some being? [duplicate]
Was mathematics created by some being, like maybe God? I believe that mathematics is timeless and uncreated, but have any academic philosophers proposed that math was created by some being? By "...
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Could our natural numbers be non-standard?
This question is related. It asks: "Can truths about the natural numbers vary across possible worlds?". One comment says: "Well, no, if they use same definitions and axioms about ...
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Relationship between real quantities and numbers [closed]
Is there a definition of the relationship between real quantities and the numbers we relate to them, generally we use 'numbers' as mathematical objects with a 'proper' nouns, but we associate them ...
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Is the powerset question really an (indefinitely expansive) series of questions?
At the "end of the day," it has turned out that:
If we deny the powerset axiom, then the expression "the powerset of the zeroth aleph" refers to nothing, in which case there is an ...
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How does methodological naturalism deal with appeals to abstract objects like logical truths, mathematical truths, or natural kinds?
A core component of the modern scientific worldview and the beliefs of people and governments in western liberal democracies is that methodological naturalism is true. It is essential to scientific ...
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Is '=' a relationship between the objects or their expressions?
The Wikipedia definiton of equality gives it as a 'relationship between two expressions'
This confuses me as when we define mathematical expressions like 2+2=4 it makes no sense to say that '=' or '...
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Can the laws of mathematics and even logic change over time?
Can the laws of mathematics and even logic change over time? Like, maybe at one time there were finitely many prime numbers and now there are infinitely many? Or maybe at one time the laws of ...
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