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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Should “not P” be interpreted as “P implies a contradiction”?

From this answer, (...) "not P" should be interpreted as the assertion "P implies a contradiction". Is this the (only/widespread/mainly) accepted definition of ...
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26 views

Do you know a good book or study on the philosophical background of the ZFC axioms?

I'm especially interested in how the Axiom of Choice might be derived from nature through philosophy. After studying basic set theory at university, I could convince myself of the ZF axioms being true,...
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Does Category Theory only apply to mathematical structures or does it also study other mathematical undefined but logical propositions? [on hold]

https://en.wikipedia.org/wiki/Category_theory It is evident that Category Theory studies mathematical structures formalizing it in terms of a labeled directed graph called a "category". But there ...
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what does “ calculus” mean when used in logic? and more broadly in mathematics? ( not a question dealing specially with derivatives, integrals, etc.)

Sentence logic is sometimes called " propositional calculus". I'd be interested in knowing what the word " calculus" means precisely/ technically here, and what are the ( historical) roots of this ...
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Is Seth Lloyd's cosmological model background independent?

I have been interested in Seth Lloyd's cosmological model (which proposes that the universe is a some kind of quantum computer or at least similar to it): https://en.wikipedia.org/wiki/...
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1answer
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Why should we accept Cantor theorem philosophically? [closed]

Just reading Terence Tao's book. If we consider Cantor's theorem which states that given any set X, there does not exist any bijection between that set and its power set, P(X). But why should we ...
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3answers
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The language of mathematics is not absolute [closed]

I know that all the representation,the symbols,equations all this is a sort of expressing some universal fact mathematically. Understanding the universe doesnt mean learing those symbols and ...
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43 views

Coarsenings In Deutsch Et Al's Constructor Theory

Disclaimer: I posted a questions on constructor theory here a few days ago but received two closing votes, I guess because it consisted of several subquestions, so I deleted it and now try to focus on ...
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55 views

What is this process/action called in English?

it is a fairly generate question regarding a terminology. People without science or engineering discipline makes an unfounded claim X, but people with such discipline start with proven facts A, B, C, ...
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63 views

Negative introspection axiom and Euclidean property of accessibility relation

Revising the modal logic principles, I have encountered an negative introspection axiom: $$ \vDash \neg \square \alpha \longrightarrow \square \neg \square \alpha $$ with additional information, that ...
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2answers
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Is “everything is true unless the opposite is proven” a fundamental math philosophy principle?

I am not a mathematician, though I am aware that: Any forall-statement about empty set is (vacuously) true because $\neg{(\forall x \in \{\}: P)} \rightarrow \exists x \in \{\}: \neg P$, where $\...
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About Tegmark's mathematical universe

I'm not sure anyone else than Tegmark himself can answer this, but why not give it a try. Would Tegmark consider a cellular automata a mathematical structure? If nature is mathematical, isn't it also ...
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What qualifies as a geometry? [duplicate]

My question really is: What qualifies as a geometry? As in a Euclidean geometry; a Riemannian geometry, a hyperbolic geometry, etc. To give a sense of the depth of answer that I am seeking; consider ...
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1answer
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A Probability Thought Experiment

Scenario: Lets say you have 100 trillion unique locks and their corresponding 100 trillion unique keys. You scramble them up, and then place all the locks and all the keys in two separate boxes. ...
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3answers
88 views

What is the real being(entity?) the word set or class denotes?

It is somewhat philosophical(at least to me). The question is as above. What is the 'substance' referred to as by the word set or class? Especially how is the thing called class defined? I cannot ...
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31 views

Is it possible to generalize without abstracting?

According to Wikipedia, Abstraction: Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which ...
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4answers
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Should math logic reflect “real” logic

In math we use logic. However, it seems mathematicians were free to define some of its rules. Say the OR. It is true, if either of arguments is true - or both. Now we use math to prove some facts ...
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How many solvable and unsolvable problems exist

I am unable to quantify it as there are many problems which are in polynomial time and certain problems can be reduced to polynomial time.How exactly to quantify them?
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92 views

Why mathematicians use natural language? [closed]

This might be more on the philosophy of math side, but in the same way that lots of math is formalized, and everybody use the symbols in the same way, why not go further and only use an artificial ...
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1answer
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Equation of Tango dancing [closed]

Good evening! I have been encouraged to ask my question on this forum, even though it might be perceived as a pure subjective and open-ended question, but I am 100% sure there is a perfectly ...
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4answers
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How to prove A → (B ∨ C) given A → B

How to prove A → (B ∨ C) given A → B I know this is a valid argument, I'm just terrible at fitch-style proofs and have no idea how to start, let alone finish.
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Idempotent monads and dialectical materialism. [closed]

This question is part category theory and part philosophy. Lawvere claims that a Hegelian dialectic is an adjunction between idempotent (co)monads. The dialectical materialism of Marx and Engels is ...
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1answer
50 views

What makes a good mathematical theory?

I recently read about Galois connections, and that they show up in a lot of different places in mathematics. Given there apparent ubiquity, I thought they might have a rich theory. However, when ...
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2answers
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The “correct” standard deviation

This may end up being a question more about scientific best practice than anything else, but I think this is the right community to ask it in to get the insight I'm looking for. Say I have two little ...
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1answer
128 views

Are axioms truly the foundation of mathematics?

It is said that the ZFC axiom system is a foundation of mathematics. In my understanding, for something to truly be a foundation, if you gave this system to an entity without any intuition or ...
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about two differrent opinions in mathematics

My question is: what is the name of mathematicians who ignore the proofs by contradiction and say all of the proofs should be constructive, and what is the name of opposite opinion?
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Given axioms, how do we know it defines a geometry?

It is known that besides using coordinates and algebra, there are axiomisation of geometry such as Tarski, Hilbert and Euclid. However looking at the axioms of Tarski for example: Betweeness $B(\...
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Variables and Language

I've been thinking lately about the kind of language we use when doing math involving variables. Consider a typical variable defining statement: "Let x = 2." If we try to parse this statement ...
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Why do we call complex numbers “numbers” but we don’t consider 2-vectors numbers?

We refer to complex numbers as numbers. However we refer to vectors as arrays of numbers. There doesn’t seem to be anything that makes one more numeric than the other. Is this just a quirk of history ...
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1answer
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How Brouwer think about mathematics as a non linguistic phenomenon?

I have a course in mathematical logic and i heard some argument about intuitionism math. I'am curious about it and i look at some book and i am just confused about phenomenon. But i realized brouwer ...
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Does Mereological Logicism fit the Logicism program of philosophy of mathematics?

It appears that Ackermann's set theory can find a nice interpretation of its primitives in a theory that has logicistic genre. This is a personal work of mine of this issue, its present here. I ...
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1answer
78 views

Can we see all mathematical concepts as (possibly uncountable-time) algorithms?

Note: I am interpreting “algorithms” broadly, to include algorithms that require an infinite and even uncountable number of computational steps. It seems to me that any definition can be seen as a ...
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1answer
109 views

What's the difference between predicate logic and model theory?

I studied model theory a little bit, and now that I'm reading the Wikipedia aricle on predicate logic, it seems to me that this is precisely model theory in that there is also the notion of signature (...
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2answers
330 views

What are the other foundations of math?

I know there are at least three foundations of math: Zermelo-Frankel set theory. (Sp?) ECTS Category theory Are there any others? Experimental foundations are welcome.
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Different math in physical multiverses or black holes?

Are there thoughts that different physical multiverses or black holes have different math, I.d. physical events follow the mathematics that is not discovered yet, whose logic may be different from ...
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Can a mathematical proof always be objectively determined as correct or incorrect?

Fields medalist Michael Atiyah claimed a simple proof of the Riemann hypothesis, but many mathematicians rejected his proof. Am I right in saying that Atiyah's proof is either objectively correct (...
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233 views

Formalist understanding of metalogical proofs

How can a formalist understand a metalogical proof such as the completeness theorem? These proofs exist exclusively outside of a logical system where the "rules of the game" are undefined. These sorts ...
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1answer
132 views

Is there criticism in literature of Euclid's fifth common notion (“The whole is greater than the part”)?

In Book I of Euclid's Elements, the fifth common notion says "The whole is greater than the part". For Euclid, magnitudes are objects that can be compared, added, and subtracted, provided they are of ...
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1answer
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Why is it possible to use logic? [closed]

First of all: there is a philosophy of mathematics tag in this SE, so this could be the proper site for my question. I do not know how to evaluate this. When we assume an axiom to prove something, ...
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129 views

What is a fair coin?

The title of this question is almost a retorical question. My point is that there is no way to define probability in a non circular manner. Let's say the probality of getting a tail when tossing a ...
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Arithmetic systems without Induction

It's often said that AC is a controversial axiom and so often in my math classes when it is used a brief comment is made to the effect of "we can prove this without Zorn's Lemma but it's more work". ...
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1answer
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What is the minimum number of things needed to declare you have a variety? [closed]

When people say things like "we have a wide variety of products" or "product x can run in a variety of modes", what is the lowest number of modes or products which one can comfortably call a variety? ...
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41 views

Proof in mathematics vs everyday [closed]

Does the word "proof" has meaning only in mathematics ? when in conversations someone ask someone else to prove something what exactly does he mean ?
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Sorites paradox natural deduction problem

I'm new to natural deduction and am attempting a question which sounds like the Sorites paradox: 1 million grains of sand is a heap. If 1 million grains of sand is a heap, then 999,999 grains is a ...
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1answer
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What are the arguments of the mathematicians who objected against the ontological proof Gödel offered?

Q: What are the arguments of the mathematicians who objected against the ontological argument/proof Gödel offered? $$ \begin{array}{rl} \text{Ax. 1.} & \left\{P(\varphi) \wedge \Box \; \forall ...
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101 views

Hilbert Program: Consistency vs Soundness

I have been wondering why exactly Hilbert asked for a decidable, complete, and consistent set of axioms for mathematics, rather than a decidable, complete, and sound set of axioms. Consistency being: ...
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1answer
112 views

What formal proof systems are capable of proving $\forall x \exists y x = y$ without needing to apply $\forall$-I to $\exists y x = y$?

I am interested in some philosophical questions that depend on whether the open formula $\exists y x = y$ is a logical truth. I'm making the assumption that some logical systems are intended, in the ...
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Is there any “good” definition for what constitutes “applied mathematics”?

Is there any "good" definition for what constitutes "applied mathematics"? Wikipedia lists stuff such as statistics, optimization. However, e.g these have certainly "pure mathematical" aspects to ...
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What is the relationship between the common concept of “model” and “model” as used in Model Theory? [duplicate]

To my understanding, a model in Model Theory is an interpretation (in a form of a set or other algebraic structures) for a certain sentence S which makes S true. In everyday language, and also in ...
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Is there an accurate view on the distinction between “what mathematics can model” and what it cannot? [closed]

Is there an accurate view on the distinction between "what mathematics can model" and what it cannot? Not just in hard sciences. What about social, political questions? What's the accuracy of ...