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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Priest's nonstandard $N$: showing $\not\vdash_N \square p\supset p$.

I'm reading up on nonclassical-logic. In Priest's nonstandard $N$ of his "Introduction to Nonclassical Logic [. . .], Second Edition", it is an exercise to show $$\not\vdash_N \square p\...
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3answers
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What is Statistics? [on hold]

In Mathematics, what is Statistics? For myself I just want to know what Statistics mean in it's purest form? also I need a much simpler definition or explanation for kids aged 4-10. Need to explain ...
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0answers
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Can you explain “well-posing strategy” vs. “distinction strategy”?

In the paper "Bertrand’s Paradox and the Principle of Indifference." Nicholas Shackel uses the terms "well-posing strategy" and "distinction strategy" to describe the two avenues to resolve the ...
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1answer
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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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1answer
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Propositional Logic and Redundancy

There are Philosophical problems with the Material conditional. The Dutch philosopher Emanuel Rutten has written an article about it, titled: Dissolving the Scandal of Propositional Logic? From that ...
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1answer
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Choice of closed monoidal structure

This might be a somewhat philosophical question in category theory. I sometimes have trouble understanding with some monoidal structures defined, why the one we choose are the "good ones". For example,...
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2answers
44 views

Do all mathematical and logical axiomatic systems implicitly ground natural numbers?

Maybe this question is more suitable for Philsophy SE, but I want to hear mathematicians' opinions. Suppose that we have an axiomatic system $\mathcal{A}$ with axioms $A_1, A_2, A_3,\dots,A_n,\dots$ ...
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Why can almost all ordinary mathematics be formalized by sets?

there must exists a reason of why the idea 'collection' is so powerful that it can formalize nearly all mathematics. subquestion: is there any which can not be formalized by this perspective? if so, ...
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Law of Excluded Middle in Logic Proof

I'm having some difficulty doing a proof for the following: $$\neg A \vee \neg(\neg B \wedge (\neg A \vee B))$$ It is said that you could use the law of excluded middles. Any help or guidance would ...
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1answer
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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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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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2answers
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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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What do people mean by “finite”?

Many arguments about the foundations or philosophy of mathematics centre on the question of whether or not there exist objects or entities (such as certain sets) which are not "finite". (For instance,...
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What is the correct reading of $\bot$?

I have some doubts about the "natural" interpretation of $\bot$ in Natural Deduction and sequent calculus. In Prawitz (1965) $\bot$ (falsehood or absurdity) is called a sentential constant [page 14] ...
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1answer
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What is Mathematics? [closed]

I study electronic engineering at university, 3rd course. I had to use mathematics a lot, from basic algebra to analysis. Yesterday, after watching some mathematics-related videos and reading some ...
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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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634 views

Is there an introduction to probability and statistics that balances frequentist and bayesian views?

Perhaps, roughly, I might be described as advanced undergraduate regarding mathematics. However, I have not learned statistics and have only learned elementary probability. Does there exist a book or ...
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3answers
71 views

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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7answers
461 views

What exactly is real number?

This question may sound philosophy, but it has been bothering me for a very long time, therefore I have to ask it here. The story goes back when my first time reading Apostol's Calculus, then I had ...
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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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2answers
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
101 views

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
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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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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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What's the difference between a proof and a derivation?

I did a BSc in Theoretical Physics, meaning that a lot of my time was spent deriving equations, making hand-wavy arguments, and arriving at solutions with a distinct lack of rigour. I'm now doing an ...
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Infinitely many axioms of ZFC vs. finitely many axioms of NBG

It is known that ZFC needs infinitely many axioms, but NBG (Neuman-Bernays-Gödel set theory) is finitely axiomatizable (as first-order theories of course). But both theories agree completely on the ...
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2answers
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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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How to interpret material conditional and explain it to freshmen?

After studying mathematics for some time, I am still confused. The material conditional “$\rightarrow$” is a logical connective in classical logic. In mathematical texts one often encounters the ...
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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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5answers
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How does one understand and resolve Zeno's paradox?

Zeno, a follower of Parmenides, reasoned that any unit of space or time is infinitely divisible or not. If they be infinitely divisible, then how does an infinite plurality of parts combine into a ...
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Why isn't finitism nonsense?

This is a by product of this recent question, where the concept of ultrafinitism came up. I was under the impression that finitism was just "some ancient philosophical movement" in mathematics, only ...
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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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25answers
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Can a coin with an unknown bias be treated as fair?

This morning, I wanted to flip a coin to make a decision but only had an SD card: Given that I don't know the bias of this SD card, would flipping it be considered a "fair toss"? I thought if I'm ...
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Interviews of famous modern mathematicians

I was wondering, are there any good collections of interviews of famous modern mathematicians? It can be text interviews, or audio or video recordings. I am not sure what exactly I mean by "modern". ...
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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
138 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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If all sets were finite, how could the real numbers be defined?

An extreme form of constructivism is called finitisim. In this form, unlike the standard axiom system, infinite sets are not allowed. There are important mathematicians, such as Kronecker, who ...
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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 ...