Questions tagged [philosophy]
Questions involving philosophy of mathematics. Please consider if Philosophy Stack Exchange is a better site to post your question.
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Gödel Incompleteness theorems - gap between first order logic and arithmetic
I have a doubt concerning Gödel's incompleteness theorem which might be stupid but that makes me uneasy. If I'm not mistaken, Gödels proof is broadly undertaken using first order formulas, numbering ...
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(When) are recursive "definitions" definitions?
This is a "soft" question, but I'm greatly interested in canvassing opinions on it. I don't know whether there is anything like a consensus on the answer. Under what conditions (if any) are ...
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What are the criteria for a subject to be under the domain of mathematics [duplicate]
It is to my understanding that mathematics is in some way the domain of all logical systems. However unconventional, as long as certain criterias are met, they could be considered as part of ...
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Proof by Contradiction: "Bad Form" or "Finest Weapon"? Reconciling Perspectives [duplicate]
G.H. Hardy famously described proof by contradiction as "one of a mathematician's finest weapons." However, I've encountered claims that some schools of thought consider proof by ...
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Confused about abstract models for axiomatic systems
I am studying axiomatic systems and I have a hard time understanding how one is supposed to come up with an "abstract" model for an axiomatic system.
I will use the following example taken ...
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statements that can be accepted by finitists.
"On the finitist view, the formula $\exists n P(n)$ is meaningful only when it is used as a statement specifying how to calculate an $n$ for which $P(n)$ is true".
It is mentioned as above ...
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Why do we use real numbers for (for example) masses in physics and how do we verify product axioms? [closed]
I have studied the definition of real numbers as V.A.Zorich explains it in his first book Mathematical Analysis I. Basically, he says that any set of objects that respects a certain list of properties ...
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Why Does This Proof Hold?
I'm currently reading "Mathematics Without Numbers" by Hellman, G., and I'm on pages 26-27. It seems like Hellman is discussing opposition to viewing mathematical proofs solely through the ...
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Why do we need a metatheory if we can include self-referencing language in the object theory?
I am wondering why we need to have a metatheory in order to talk about a theory- why can't we just add self-referencing terms to the language of the formal system on which the theory itself is based, ...
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What is the formal system when we are using many different sets of axioms?
I am just starting to learn about formal systems, and have learnt that the many axiom systems in Mathematics, such as those of plane geometry, Peano's axioms, vector axioms, etc. can each be used to ...
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Definite description in homotopy type theory
I asked this question there and I have been suggested to ask it here.
In a paper by David Corfield, we have an account of definite description in homotopy type theory. The author gives the following ...
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Are quasi-sets (and therefore Schrödinger logic(s)) studied by mathematicians or are they purely in the domain of philosophers?
Context:
I'm a fan of different kinds of logic. I'm conflicted about whether different logics actually exist beyond, say, a philosophical oddity.
The Question:
Are quasi-sets (and therefore ...
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Creating larger structures from smaller ones without an explicit construction
I'm asking this question as a replacement for my previous one, which I admit isn't clear, and which I am voting to close. Hopefully I'll be clearer now.
Admittedly, I'm not sure if this question ...
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Is everything an object in Math, just like in Objected-Oriented Programming? (Tao's Analysis I)
I am reading Tao's Analysis I, and there are a number of passages which seem to suggest an object-oriented point of view of mathematics reminiscent of the object-oriented programming with which I, as ...
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Why is addition not completely defined here?
Say for the natural numbers, we define addition this way:
$0 + 0 = 0$, and if $n+m = x$, then $S(n) + m = n+S(m) = S(x) $
Say we have the regular Peano axioms, except we delete the axiom of ...
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Which forcing technique implies "every set is countable from some perspective"? Which notion of "the same set" is used between models?
https://plato.stanford.edu/entries/paradox-skolem/ contains this claim:
Further, the multiverse conception leads naturally to the kinds of conclusions traditional Skolemites tended to favor. Let $a$ ...
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Trying to understand how numbers themselves (s0, ss0, sss0, etc) are represented in Gödel numbering
Problem solved: I did not actually read the table given on page 70 of nagel and newman. s does have a Godel number. It's 7. So ss0 would be broken down into 7, 7, and 6, since 0 is given the number 6. ...
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On motivations of continuous geometry
The development of continuous geometry as an abstract field seems to be following a trend of removing the significance of low-dimensional entities from geometry. As classical treatments of geometry ...
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Can the fundamental theorems of real analysis be proven/developed without proof by contradiction?
I've been reading about philosophical debates between mathematicians, and some seemed to reject the ideas of real analysis (such as the extreme value theorem) based on a school called "...
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Mathematical Induction: Strong vs Weak Form
I have a rather naive question: The usual mathematical induction works by the same scheme: Let $n_0 \in \mathbb{N}$ a pos integer and $A(k), n_0 \le k \in \mathbb{N}$ family of statements. Then the &...
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Are there any problems about the difference between set theoretic definitions of polynomials?
I am a novice about this question, so if there is a misunderstanding then I apologize for it.
As for Peano axioms, if I choose Zermelo natural numbers, and you choose von Neumann ones, then this doesn'...
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Why pullback of ideal sheaf should be the conormal sheaf?
I'm sorry that this isn't really a math question, but this gap between my intuition and the truth bothers me. For closed subvariety (for simplicity) with ideal sheaf $\mathcal{I}$, the pullback $i^*(\...
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For every object, is there a unique notion of isomorphism?
Do you think that, according to most mathematicians, the following claim holds?
(Claim) For every object, there is a unique notion of isomorphism.
Perhaps one might think that for some sets, such as $(...
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doesn't the independency phenomenon make a case for non-classical logic? [closed]
alright, this question is philosophical and somewhat fuzzy. i also admit to knowing little about logic. all in all, this question can possibly be easily resolved by either pointing to (perhaps even ...
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What is the logical system of Tractatus Logico-Philosophicus?
Tractatus Logico-Philosophicus states simply that
6 The general form of the truth function is: $[\bar p, \bar\xi, N(\bar \xi)]$. This is the general form of the sentence.
Wikipedia and other sources ...
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Finitists reject the Axiom of Infinity - are there groups who reject the others?
I've seen rejections of the Axiom of Infinity. This is called finitism. Some ultrafinitists even add the negation of the Axiom of Infinity. Definitely doable.
I've seen rejections of the Axiom of ...
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Is there a term for the idea that mathematical objects are defined by their relationships?
In a recent Veritasium video discussing Euclid's Elements, Alex Kontorovich comments that Euclid's definitions of primitive objects (e.g. "A point is that which has no part.") are absurd and ...
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Conceptual Question regarding Shannon Entropy and bits
It is said that the number of "information bits" contained in a certain piece of information can be roughly translated as the number of yes/no-questions that would have to be answered in ...
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Do we ever reason about a non-associative algebra without embedding it in an associative algebra?
This question most certainly contains some errors in phrasing. It is on the subject of the philosophy of mathematics, and it is hard to stay precise when reaching towards the fundamentals of math.
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A philosophical question on the nature of mathematics [closed]
I had a seemingly simply question today, that goes as following.
What do we need for a mathematics to exist in a universe, or a system, more broadly speaking?
Is it a matter of having the ability to ...
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Zero-dimensional space with multiple objects
I am unsure if this belongs to math or philosophy.
Let's say there's 0-dimensional space, however multiple objects exist within in, occupying the same "spot". If multiple objects exist, is ...
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What exactly makes the ordinals an indefinitely extensible concept?
I understand the principles of generation that cantor used to create the ordinals but I cannot see what exactly is the property that makes the ordinals an indefinitely open plurality and not the ...
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What did Richard Dedekind mean exactly by his statement about generality?
But—and in this mathematics is distinguished from other sciences—these
extensions of definitions no longer allow scope for arbitrariness; on
the contrary, they follow with compelling necessity from ...
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why learning math in elementary school was harder for me rather than upper grades? [closed]
When I was an elementary student, I'd suffered from understanding basic things like multiplication table and other simple things and I had to memorized them. Last hours I was searching for genesis of ...
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When mathematicians say "true" do they mean "true in all models"?
According to the comments to this question,
Truth is ordinarily defined by reference to models.
If so, even axioms and theorems are not true without reference to a model.
However, when ...
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Arithmetization of Turing machines
Refer to Turing's 1936 paper, page 248, last paragraph. I present the paragraph in verbatim below :
The expression "there is a general process for determining..." has been used throughout ...
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Benefits/uses of non-base 10 number systems?
For reference, I'm studying math and anthropology at university, and I've been dying to find some overlap of math theory and ethnomathematics (math uses/tools/systems/etc in other cultures). I'm ...
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Formally how do we view finite sets
This might be silly, but I have been thinking about how we would work with finite sets very formally.
So, $\{1,2,3,\cdots,n\} = \{k \in \mathbb{Z}^+ \mid k \leq n\}$ gives a representee for which any ...
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What are fun mathematical facts for non-mathematicians? [duplicate]
I like to spend my life with mathematics. I think it is the best thing I can do in my life. However, I have great difficulty explaining what I am doing to non-mathematicians, even educated ones. For ...
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Why is it important to prove that some particular set is a vector space as opposed to just asserting such objects exist?
In Axler's Linear Algebra Done Right Example 1.24, we are asked to prove that the set of all functions from some set S to the set of real (or complex) numbers is a vector space.
I proved this by using ...
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How to get LNC as a theorem using Frege's Prop Calculus?
So Im using axioms from,Frege propositional calculus and is there any way to derive Law of non contradiction as theorem from them.
The axioms
A → (B → A) | THEN-1
(A → (B → C)) → ((A → B) → (A → C)) ...
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What are the differences between equality, equations and identities? [duplicate]
What are the differences between the followings:
Identity
$$
\sin^2(\alpha) + \cos^2(\alpha) = 1
$$
Equation
$$ 4x = 16 $$
Equality - $x,y$ are mathematical objects.
$$ x = y $$
All of the three ...
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Is there a mathematical or physical, real world use for numbers passed I? Who determines what can be conceptualized or not? [closed]
I is the Square root of -1, such that I * I = -1. Through this, I can be considered like a "Half Negative."
Why hasn't this been taken further? Why don't we make a quantity such that I^...
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Set theory and model theory: which set is ZFC?
I have yet another post about what is model theory doing and why is it valid; I hope I can be coherent.
(1) https://mathoverflow.net/questions/23060/set-theory-and-model-theory
(2) What exactly is the ...
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Probability - Interview Question - Hidden Assumptions and Phrasing Issues
I’ve encountered the following seemingly simple probability interview question in my workplace:
Two reviewers were tasked with finding errors in a book. The first had found 40 errors and the other ...
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How to interpret what a set is to see how it could be infinite?
Currently, 'infinite set' sounds oxymoronic to me, so my question is how to interpret what a set is such that it is consonant with it being infinite. I understand that we take it as axiomatic that ...
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Are opinions considered sentences in Logic?
I am beginning to read the book titled forallx An Introduction to Formal Logic by P.D. Magnus. This is an open source book. On page 4 Magnus states:
In this open source book found here: https://math....
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Is there a mathematical notion of "why"?
Is there a mathematical notion of "why"? That is, are there reasons behind the truth of certain mathematical statements? Personally, my belief is that true mathematical statements just are ...
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I don’t know what a natural number actually is
For some context I did a course on set theory where I was taught about ZFC, and the construction of the natural numbers, integers etc.
I think I was far too young to take the course because it’s left ...
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Why do constructive mathematicians claim that mathematical truth is temporal?
It seems to me (and correct me if this is a misconception) that the traditional divide in the interpretation and practice of mathematics is between platonists, who believe that mathematical objects ...
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