# 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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### Could the Relativity of Time Impact the P vs NP Problem?

Given that time behaves differently under relativistic conditions (such as near black holes or in high-speed motion), could the relative nature of time influence the complexity classes P and NP? For ...
1 vote
180 views

### Is there a mathematical concept that could serve as an "anti-zero", representing an absolute maximum or the complete counterpart to zero?

I’ve been thinking about the concept of zero in mathematics, particularly its role as the absolute absence of quantity (or as a global minimum, in a sense). Edited: In my limited mathematical ...
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### Why Pythagorean theorem is all about 2?

We all know $a^2 + b^2 = c^2$ on a right angle triangle. Yes. It works. It can be proven using area of square and everything. But my question is: why? What makes the number 2 so special, that it ...
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### Real life examples of pseudo-metrics

Every mathematician knows that, given a set $X$, a metric is a function $d:X\times X \to \mathbb{R}$ that verifies (i) $d(x,y)=0 \iff x=y$ (ii) $d(x,y) = d(y,x)$ (iii) $d(x,y)\leq d(x,z) + d(z,y)$ ...
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### Are Compactness for FOL and the Pumping Lemma for RL/CFL two instances of the same phenomenon?

As title states, I'm curious whether my intuition for the Compactness result for FOL and the Pumping Lemma for RL/CFL being two expressions expressions of the same phenomenon (that is: an attempt to ...
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### what's means V is model of zfc?

I thought that set of formulas S has a model iff there is an interpretation satisfying that all the formula in S make true. so I understand zfc's model is a kind of assignment that make all the zfc's ...
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### How does extensionality demand the existence of objects for which it does not supply a definition?

From Moschovakis, Notes On Set theory, p. 111: The Axiom of Choice is different in form from the earlier “constructive” axioms (II) – (VI), because it postulates directly the existence of a set for ...
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### Question regarding Fitch's paradox of knowability

Section 2 of Fitch's paradox of knowability states the following. Let $K$ be the epistemic operator 'it is known by someone at some time that.’ Let $\diamond$ be the modal operator ‘it is possible ...
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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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### Why can't three-valued logic (ternary logic) simply have only two truth values?

Consider the statement: P ∧ ¬P ⊢ Q where: P is any proposition. -¬P is the negation of P. Q is another proposition. Wouldn't proving both P and ¬P to be true simply lead to a new proposition Q, ...
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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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1 vote
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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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1 vote
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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 ...
1 vote
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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 ...
1 vote
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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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1 vote
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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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1 vote
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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 ...
1 vote
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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 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 ...