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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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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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2answers
77 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
84 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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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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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
89 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
310 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
75 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
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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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1answer
213 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
106 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
64 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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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
85 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
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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
77 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
92 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
108 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
135 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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42 views

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
137 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
392 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
130 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 ...
2
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1answer
89 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
55 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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0answers
12 views

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 ...
2
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1answer
64 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
363 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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99 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
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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
86 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
206 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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63 views

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 ...
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1answer
248 views

How do set theorists view this issue?

Since the question has changed significantly over the course of last few days, it was suggested to modify it in that light. I have removed those parts which aren't directly related to the main ...
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1answer
46 views

Are predicates merely convenient or are they necessary in logical language?

Model theory is typically based on a formal language whose production rules and syntax are designed to formalize the deductive process, and as such typically include predicate or relation symbols. ...
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1answer
56 views

Assumptions necessary to justify the method of proof by contradiction.

It seems to me, these assumptions are necessary to demonstrate proof by contradiction: i) Every proposition must belong to $T$ or $F$. ii) No proposition belongs to both $T$ and $F$ iii) If having $...
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Can we be sure proofs have no errors?

My current understanding is that work submitted to journals has mathematicians look over it for errors. Mathematics is deductive, yet with this being the burden of proof, how can we know for sure ...
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1answer
45 views

About proofs that we cannot verify every step by hand

For something I am planning to write, I need to clarify few issues with respect to computer-aided proofs that we cannot verify every step by hand. For example, the proof for the four-color map theorem....
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0answers
82 views

Intuitionism and theoretical physics

In the book by Kleene "Introduction to Metamathematics" I have read that Poincare was intuitionist. Nevertheless, due to the fact that I am an undergraduate student in physics, I know that Poincare ...
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2answers
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In rewiring systems do definitions creates new rewrite laws or an alias? And is this a meaningful question?

Lambda calculus is often introduced as a rewriting or substitution system. Where $\beta$ reduction is described as replacing bound variables with the value that variable is bound to. For example $(\...
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2answers
64 views

Definition of an axiom?

What would be some convenient and general enough definition of an axiom? If we go through mathematical theories we can observe that axioms either demand that something exists, or tell us how some ...
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8answers
5k views

Should a mathematical proof be 'convincing'?

I just read a description of what is a mathematical proof in my mathematical logic textbook, and I'm a bit puzzled by it. It goes like this: A mathematical proof is a finite sequence of mathematical ...
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2answers
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Analysis of Nested Quantifiers in $\epsilon$-Calculus:

(This is quite a small question, but also pretty specific so forgive the wall of text!) I'm trying to learn about Hilbert's $\varepsilon$-calculus (Bourbaki use a similar system in their volume on ...
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3answers
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Can we interpret sets of the same cardinality as distinct representations of the same set?

Usually mathematicians consider isomorphic fields as equal fields. That is, if the $(A,+,\cdot)$ is isomorphic to $(B,\oplus,\odot)$, then I can consider those fields as equals. Thinking about it, I ...
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1answer
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What if someone comes up with a proof of consistency of arithmetic under the following conditions?

Suppose that one day, someone comes up with a proof of consistency of Peano arithmetic (PA) within itself. Then, does it mean that PA is actually inconsistent? Because, by the Gödel's incompleteness ...