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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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0answers
31 views

Proof in mathematics vs everyday [on hold]

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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0answers
28 views

Standard Deviation in Mathematics [on hold]

Computation of Standard Deviation has sense because it produces a result which describes the degree of deviation from the 'mean' of observations. Mean computation procedure, P, in turn, without any ...
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2answers
38 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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0answers
24 views

Aesthetics in Mathematics [on hold]

Many mathematicians speak of elegance and art in mathematics. In Russell's words, "Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of ...
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0answers
19 views

Would it make sense to make a dichotomic tree of all semantic concepts know to humans? [closed]

Being there no semiotics in stackexchange I ask here but I'm no mathematician so be clement on my poor brain when you're about to use vocabulary I didn't know existed :D So I'm asking if there is a ...
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0answers
38 views

Mathematical Definition of 'Property' [closed]

Is it possible to formalise the definition of 'Property' ? Can we represent the constitution of the concept of Property in terms of mathematics (say set theory) or logic? Tarski's Theory of Truth ...
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1answer
67 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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0answers
85 views

Philosophy of mathematics

There is a question which made my mind busy for a while: assume that we are working with the function: $ y = \sqrt x$. when we draw it continuously, it means that for example the $x = \pi $ or $x = \...
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1answer
79 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: ...
1
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1answer
101 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
129 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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0answers
39 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
57 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 ...
1
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1answer
124 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 ...
5
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1answer
377 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 ...
2
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1answer
117 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
79 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
51 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 $\...
0
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0answers
11 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
62 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 ...
4
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1answer
357 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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0answers
96 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
64 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
68 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
150 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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0answers
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
240 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. ...
1
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1answer
55 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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0answers
61 views

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 ...
2
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1answer
44 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....
2
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0answers
80 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
71 views

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
63 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 ...
38
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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 ...
3
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2answers
66 views

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
75 views

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 ...
2
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1answer
104 views

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 ...
3
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1answer
131 views

What possible future mathematical methods are not considered rigorous math right now? [closed]

In the same way that in the past the use of irrational numbers, calculus, and transfinite numbers were considered to be "not rigorous math" which contemporary mathematical constructs are causing ...
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10answers
3k views

What exactly is circular reasoning?

The way I used to be getting it was that circular reasoning occurs when a proof contains its thesis within its assumptions. Then, everything such a proof "proves" is that this particular statement ...
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7answers
3k views

Does it follow from Gödel's theorem that this world cannot be fully described by math?

What are the flaws in the following reasoning? By Gödel's theorem, for any non-contradictory formal system $F$, at least as complex as ordinary arithmetic, there exists a true statement $G(F)$ (or ...
2
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1answer
97 views

Fitch Logic Proof

I am stumped on this proof. I have attached a link with my proof so far. I'm not sure how to derive a contradiction from WeakPref(a,b) on line 12.
2
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0answers
57 views

Very General question on the idea of categories [closed]

First I have to say that I am a physics student, I don't know very much about mathematics but I am really intrested and fascinated by it. Anyway, my question is a very General one, and even a ...
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1answer
46 views

Fitch Arrow Proof C10 Help

I am having a hard time finishing this proof. Here is how far I've gotten. I was stuck at almost the end of the proof. The first thing is that, I am pretty confused why the step 23 isn't checked out ...
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1answer
112 views

Using Fitch to proof ∀x Indiff(x,x). Help

I am having a hard time solving this Fitch Proof. Goal: ∀x Indiff(x,x) I have to proof this goal using the following four premises: (might not need all of them) P1: ∀x∀y(WeakPref(x,y)∨WeakPref(y,x))...
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1answer
312 views

Fitch Proof Exercise 13.8

I am having trouble solving this Fitch Proof. Here is how far I’ve gotten Only the last step is not checked out in Fitch but I think the logic works well. Any help is appreciated. Thank you
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4answers
2k views

Is there a mathematical basis for the idea that this interpretation of confidence intervals is incorrect, or is it just frequentist philosophy?

Suppose the mean time it takes all workers in a particular city to get to work is estimated as $21$. A $95\%$ confident interval is calculated to be $(18.3, 23.7).$ According to this website, the ...
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1answer
46 views

Mathematical objects

If the essence of mathematical objects isn't important to mathematicians but rather what they do and how are they related is there a branch of science or mathematics itself that examines exactly this?
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0answers
44 views

Non-set-theoretical foundations

Nowadays most ideas of foundations are based on some set theories. But are there some non-set-theoretical foundations, I mean are there some ideas of creating a theory which can foundate other math ...