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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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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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0answers
29 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
85 views

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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1answer
37 views

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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2answers
90 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
52 views

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
47 views

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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0answers
84 views

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
47 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
51 views

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
114 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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1answer
46 views

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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69 views

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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78 views

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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8answers
4k views

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
92 views

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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0answers
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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
73 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 ...
2
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1answer
101 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
320 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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1answer
77 views

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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2answers
83 views

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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1answer
220 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 ...
3
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1answer
119 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
65 views

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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0answers
95 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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0answers
86 views

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
30 views

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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2answers
45 views

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
79 views

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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1answer
98 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
110 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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2answers
136 views

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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1answer
43 views

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 ...
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0answers
60 views

Why we need such a restrictions in logics?

Note:I am not competent is logic so this question may look weird to you. So as I know there are different types of logics (first-order logic, second-order...), and the difference between them is that ...
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1answer
144 views

Why do we need/use proof theory?

Note that my knowledge of both proof theory and model theory is incredibly weak. I just started learning about them using Kleene's "Mathematical logic". If I understand it correctly then one of the ...
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1answer
403 views

Choosing formal system for mathematics

I always wondered, we have many choices for choosing what kind of postulates - axioms, deduction rules, we choose for our formal system. For example, there are Hilbert style systems where there are ...
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1answer
154 views

Is mathematics a syntax?

I have read that syntax is symbol and semantics is meaning those symbols convey. Is mathematics a syntax? Where is semantics in mathematics? What gives meaning to mathematics, to those symbols I ...
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1answer
95 views

Understanding a quote from G. H. Hardy in 'A Mathematician's Apology'

I recently learned about the philosophy of constructive mathematics. In several discussions of the topic, I keep seeing a quote from G. H. Hardy's book A Mathematician's Apology; Reductio ad ...
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1answer
59 views

Negation-incompleteness, Godel's theorems and interpretations

So, as far as I understand, Godel's theorem says that sufficiently strong theories such as Peano Arithmetic are negation-incomplete, which means that there exists a formula $\phi$ such that neither $\...
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I'm having difficulty understanding this problem to do with statistical inference (specifically, Bayesian) in scientific investigation.

I'm over here from the philosophy page since a very similar question that I asked there a couple times wasn't ever properly answered and I think statisticians here might be able to provide a helpful ...
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1answer
65 views

Why we can safely treat objects like mathematical entities?

In the study of number systems we learn the axioms of real numbers. For example: The commutative axiom $X.Y = Y.X$ The distributive axiom $X.Z + Y.Z = (X+Y).Z$ Well, the thing is that we are ...
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1answer
367 views

Proving a Mathematical hypothesis using Physics

We know that mathematical models of physical phenomena need to becomes more and more sophisticated as our observations become more precise and comprehensive by utilizing more advanced instruments that ...
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100 views

Motivation of the Continuum Hypothesis

Why do mathematicians care about the Continuum Hypothesis? Does it have philosophical implications? If it was true or false, would it have had some sort of implications in mathematics? Does the ...
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1answer
65 views

Why did Frege need to use Courses-of-Value in his number Concept

I am studying Frege's work and Russell's Paradox. I can't understand why did Frege need to use Courses-of-Value in his number Concept, wouldn't it be enough to state "Any concept F is equal to Zero, ...
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1answer
99 views

Induction on complexity, upside down?

I have to prove that in a formation sequence of a formula F, all formulas that appear are sub-formulas of F. The proof that the text (Boolos, Computability and Logic) suggests is by induction on ...
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2answers
256 views

Why and how is logic related to set theory?

I am learning set theory on my own at the moment, and I realised I can't avoid not to learn logics. There is a strong connection between these two. such that, proofs for sets are based on logics. I ...
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The understanding of category of groups

When we say the category of groups, do we refer to the same category of groups? For example, in Tom's mind, there is a category of groups; in Jack's mind, there is a category of groups. However, one ...