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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what's means scope of further modal operators?
I am reading page 315 of Parsons' Sets, Classes, and Truth.
He presents the comprehension principle in the following form, but at the same time, he argues that this does not prevent Russell's paradox....
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Is Intuition Indispensable in Mathematics?
As a mathematician, I operate as a platonist (as I'm sure most do). I have an intuitive conception of a discrete order of globs or objects starting at some point and continuing off into the distance. ...
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Can there be mathematical facts of the matter without mathematical realism?
Can statements involving mathematical expressions or comparisons be true without mathematical realism? For example, consider the following statements:
"Peter is taller than John."
"The ...
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If mathematics isn't fundamental to the universe, will we need to develop a new, non-mathematical approach to physics? [closed]
Suppose, for the sake of argument, that mathematical realism is false and math is not fundamental to the universe.
Physics, as a discipline, aims to understand reality at its most basic level, and ...
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Is it possible to do physics without mathematics?
In my last question If physics can be reduced to mathematics (and thus to logic), does this mean that (physical) causation is ultimately reducible to implication?, I entertained the notion that ...
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If physics can be reduced to mathematics (and thus to logic), does this mean that (physical) causation is ultimately reducible to implication?
I was reading What is the difference between implication and causality? and began considering the relationship between causation, physical laws, and logic. If causation in the physical universe is ...
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Why do constructivists accept the countable choice?
There are variants of constructive mathematics. Many of them (e.g. Bishop's constructive mathematics) seem to accept the axiom of countable choice.
I agree that the axiom of choice is not constructive,...
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What do all branches of Mathematics have in common to be considered "Mathematics", or parts of the same field?
At some point in my life I think I've read what all branches of Mathematics had in common were numbers. But then I remembered a branch of the many Mathematics I had when I was an university student, ...
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Is there any mathematically valid statement that defies logic?
I know that in most cases mathematical statements are also logically sound and valid, and vice versa. However, my question is, is there a mathematical statement, expression or theory that does not ...
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A priori knowledge vs. formalism
Mathematical truths are commonly used as an example of a priori knowledge in the Kantian sense. As a basic example, 2+2=4 is true regardless of anyones experience, it is true before any person writes ...
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What is the importance of philosophy in the development of scientific and abstract thinking?
Is there evidence for this? In which books or articles could I find more information?
I've been researching, and it seems that logic and epistemology are relevant for developing scientific thinking, ...
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Did Gauss criticize cardinals?
In a well-known passage, Gauss criticized the use of infinity in mathematics in the following terms:
I protest first of all against the use of an infinite quantity as a completed one, which is never ...
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Is there a simple paradox like Russell's paradox that arises if we assume
... that for a given property P, there is a set of all and only the unordered pairs {x,y} such that x satisfies P and y doesn't satisfy P?
Clearly, to avoid Russell's paradox, we would refrain from ...
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What is the difference between the complex numbers i and -i?
First, I begin by making some possibly dubious assumptions with unclear definitions.
An object is defined by its properties. (Two objects with the same properties are the same.)
Numbers are objects.
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Is math (only) a language?
"Math is the langauge in which God has written the universe" ~ Galileo Galilei (no less)
I recall vaguely, dovetailing with Galileo's words supra, reading math is a language. I recognize &...
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Thomson's lamp: a useless paradox?
Thomson's lamp was mentioned at How to understand numbers that become really large? (as well as a number of posts elsewhere on SE). I have mentioned elsewhere that in addition to Cantorian infinite ...
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Is the the standard model of physics related to the fibonacci sequence?
I know this is a physics question, however I am unable to ask questions, on that site because of confused questions I ask, when I am unwell and the moderators will not let me ask anything further.
So ...
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Is there a difference between "set" and "collection"?
I thought of asking this on the math stack exchange, but I think this stack exchange is better suited. Is there a difference between sets and collections? Some people say they mean the same, while ...
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Is it even possible to define "set" non-circularly?
Is it possible to define the notion of "set" in mathematics in a non-circular way? All the informal definitions I have seen basically use a synonym for "set", like "collection&...
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Some questions about the material conditional and entailment in intuitionist math
In an excellent answer to a question about the history of material implication, @Bumble notes:
Unfortunately, the word ‘implies’ is ambiguous between these meanings. In particular, mathematicians are ...
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What is something that math cannot be applied to and doesn't involve math?
I have been asked this question, yet I am unable to answer it.
The issue with this question is that I have given all that I know, therefore I too am at a loss.
What I do know is it has no concept of ...
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Is it fair to say truth is used more in logic than in math? If so, what are the reasons for doing so?
Right out of the gate in logic we see sentences (propositions) like "all men are mortal" and we say both that it is a true proposition (e.g. independent of being a premise) and that in a ...
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Objection to indirect proof in Intuitionism
From my understanding, Brouwer's conception of intuitionism is that mathematical objects only exist in the mind once they have been constructed. And we can create constructions using computable ...
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Has anyone ever studied which proof types are feasible for which theorems in mathematics? If not, why not?
For instance, when asked to prove that sqrt(2) is irrational, we go straight for the proof by contradiction where we assume it’s equal to a/b in lowest terms and end up with a and b not being in ...
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Intuitionist perspectives on Greek mathematics
My question pertains to how intuitionist perspectives on the philosophy of mathematics might apply to Greek geometry and number theory. It seems that the standard examples given to justify the ...
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Can Internal Set Theory provide a complete system of arithmetic?
Internal Set Theory (IST) is a conservative extension of ZFC that adds three axioms that serve to define a predicate standard such that all numbers are either standard or not. There are finitely many ...
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Is infinity a number?
So I've been on a number of math fora, part of learning some calculus (not much of set theory, no). To my surprise I found what I would describe as strong resistance from some folks against (using) ...
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Mathematical Realism and 0=1
Mathematical Realism is the notion that mathematical truth exists, and is not subjective or merely a mental construction.
Inspired by Noah Schweber’s recent post on Math Stack Exchange: https://math....
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Could mathematical truths have been otherwise?
Could at least some mathematical truths have been otherwise? Or are all mathematical truths necessarily true? For example, is it the case that "there exists only one complete ordered field up to ...
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What papers or books should I read in order?
I have been reading literature on modal set theory and am currently reading Putnam's "Mathematics Without Foundations," which is known for being one of the earliest presentations of this ...
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Must constants denote exactly one thing? [closed]
Consider the algebraic equation
C2 - 1 = 0
C is a constant.
When we solve this equation we report it's solution as C = 1 or C = -1. But that can't be an exclusive or. Thus, we can say C = 1 and C = -...
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Why does this proof hold?
I'm currently reading Mathematics Without Numbers: Towards A Modal-Structural Interpretation by Geoffrey Hellman, and I'm on pages 26-27. It seems like Hellman is discussing opposition to viewing ...
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Does a mathematical object that does not contradict itself have to exist?
I have recently finished the chapter on constructing the real numbers in my Analysis textbook (via Dedekind cuts). At first the natural numbers, then the whole numbers and the rational numbers were ...
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Is intuitionistic mathematics situated in time?
Mathematics, or at least classical mathematics (that is, mathematics based on classical logic), is thought to be timeless. A theorem that is proven to be true in classical mathematics was true always ...
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Logical Pluralism and Anti-Psychologism
Are there any forms of logical pluralism which reject psychologism? It seems that logical pluralism is predicated on the notion that the diversity of logic is attributed to the diversity of thought. ...
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Why do we have problem of concept of set?
I am reading George Boolos's "The Iterative Concept of Set," and in the first chapter of this paper, he criticizes Cantor's definition of a set as a whole or totality of objects, pointing ...
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How can we point to a specific element of an abstract mathematical space that has no distinctive elements with respect to the space's structure?
An example of such a space is a Euclidean affine space. Consider the statement "point O is the origin of the system". How could we clearly specify and convey what point O is supposed to be, ...
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"Entails" in probability propositions: Why are these claims made by the author true?
I've just began reading "Probability: A Philosophical Introduction" by D.H. Mellor. In chapter 1, section 8, he states:
... the epistemic probability of a proposition conditional on ...
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If a predicate doesn't determine a set, does that predicate even exist in the first place?
I thought of asking this in the Math Stack Exchange, but then I thought this stack exchange is better. Certain predicates define sets, such as "x is not equal to x". Other predicates do not, ...
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How do you prove mathematical induction without the notion of a set?
EDIT - Peano's axioms for N can't be used to answer this question, because they assume induction. So what axioms can be used? I am thinking the following:
P1. x ∈ N iff x=1 ∨ ∃y (x=y' ∧ y ∈ N)
P2. 0'...
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Who was the first philosopher to describe approximation?
Who was the first philosopher to describe what we now call curve fitting or approximation?
Pierre Duhem discusses this a bit in Aim & Structure of Physical Theory, pt. 2, ch. 3 "Mathematical ...
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Need help understanding how certain mathemetical statements across the landscape can seemingly contradict (e.g. Cantor-Hume vs Euclid)
Here are the main components to my understanding on this issue:
Almost all of math can be given in a foundation of set theory
Different math can seemingly contradict, e.g. in Euclidean geometry ...
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Is all rigor mathematical in nature?
Is all rigor, mathematical rigor? Or are there other forms of rigor besides mathematical rigor? For example, is there such a thing as philosophical rigor, or scientific rigor? Personally, I think all ...
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Is the B-theory of time only compatible with an infinitely renewing cyclical reality?
I'm not a mathematician and I may be misunderstanding some aspects of this concept.
According to the B-theory of time, the flow of time is an illusion, and every point in time exists equally. If this ...
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Can LLMs have intention?
In many movies, you have seen an AI robot moving here and there, doing this and that with an intention. Is it possible that a generative AI-like language model (e.g., ChatGPT) could ever do that?
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How do skeptics explain axioms not being arbitrary?
I get infinite regress but surely the axioms of ZFC or arithmetic were not so much chosen as discovered and intuited and thought about. They certainly didn't just grab whatever was around them and say ...
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Is there a paradox in the proof of Godel's incompleteness theorem? [closed]
The Gödel Incompleteness Theorem was a major discovery in modern logic that has consistently attracted the attention of scientific and philosophical circles.
However, since the Gödel Incompleteness ...
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How can objects be nonexisting?
A square circle. Obviously, this is contradictory, but i feel odd saying it doesnt exist as well. thats not the bestw ay to say it.
but, then again, whatg do we even mean in mathematics or logic by ...
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How do numbers and quantities relate in Aristotle's philosophy?
I'm trying to find out how numbers (and other mathematical objects) and quantities relate in Aristotles philosophy. For example if the distance between point A and point B is "287 miles", ...
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Idealization and Abstraction of Space
I found myself pondering about space and realized that aside from the general notion of dimension (that being the minimum number of coordinates needed to specify an entity in the space under ...
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