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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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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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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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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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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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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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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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 ...
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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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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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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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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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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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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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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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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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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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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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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 ...
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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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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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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 ...
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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.
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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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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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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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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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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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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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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 ...
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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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Is math just a description of space and its properties? [closed]

Is all math just a description of space and it's properties? Every mathematical concept I have learned can be represented in space. There is a lot left for me to learn in the field of mathematics but ...
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Examples of theorems that are true conceptually, but false computationally

I am currently reading Theorem Proving in Lean, a document dealing with how to use Lean (an open source theorem prover and programming language). My question stems from chapter 11. The particular ...
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Are Mathematicians Pluralists About Math?

This has been rangling around my head for awhile. With the death of Hilbert's program via Gödel's Incompleteness Theorems (and the prior damage done to Logicism via Russell's Paradox), have ...
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Why is Markov's Principle an axiom?

Markov's Principle: Let $ x \in \mathbb{R}$. Then the following holds: \begin{align*} \neg (x = 0) \Longrightarrow \vert x \vert >0. \end{align*} In constructive mathematics (no law of excluded ...
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Metaphysical/psychological aspects of describing a formal language (mentioned in Bourbaki)

In the introduction to Bourbaki vol. 1, we read: "It goes without saying that the description of the formalized language is made in ordinary language, just as the rules of chess are. We do not ...
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Is there any “real” number that may not actually be a real because we haven't found its Dedekind cut?

I just watched a video that shows how real numbers are constructed using Dedekind cuts, and what I understood was that a real number is a subset of Q which, among a few other conditions, contains no ...
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What does it mean when we say a mathematical object exists?

I learned recently that there are mathematical objects that can be proven to exist, but also that can be proven to be impossible to "construct". For example see this answer on MSE: Does the existence ...
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Does the existence of a mathematical object imply that it is possible to construct the object?

In mathematics the existence of a mathematical object is often proved by contradiction without showing how to construct the object. Does the existence of the object imply that it is at least possible ...
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The use of square in maths

The variance of a random variable is defined as $E[(X-E[X])^2]$. In machine learning and linear regressions, loss is sometimes calculated with the squared error. In both cases, the main function of ...
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Minimal requirements for standard model of set theory leading to inconsistency?

A purported existence of a standard model of such theory as ZFC has been a cause of discomfort for a number of experts. Taking a strong Platonist view of entities and their collections, etc., in a ...
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Are there still mathematicians who don't accept proof by contradiction?

When I was a kid, I read popular scientific texts about the different philosophies of mathematics; formalism, intuitionism, constructivism and many others. I learned that there existed mathematicians ...
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When a structure is a mathematical structure? [closed]

We know that a group or a topology, for example, are purely mathematical objects. But what, for example, about a turing machine, or a normal-form game? They are also structures, but one could say that ...
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Increase in conditional probability for contradictory hypotheses in bayesian confirmation theory?

Although this question has a philosophical slant and my motivations for asking it are philosophical, I'm going to justify asking this in the mathematics stack exchange in two ways: 1) I've asked ...
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3answers
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A friendly introduction to Hilbert's sixth problem with special focus in its meaning and issues

While I was reading the Wikipedia's article dedicated to Hilbert's problems I've known the so-called Hilbert's sixth problem from this page and the corresponding Wikipedia's entry dedicated to this ...
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Link to a paper

I remember seeing, in one of the comments to one of the top-voted questions here, or on MO, a link to a paper. It contained, in addition to other things, a part about the dangers of doing mathematics....
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Why did mathematician construct extended real number systems, $\mathbb R \cup\{+\infty,-\infty\}$?

I know some properties cannot be defined with the real number system such as supremum of an unbounded set. but I want to know the philosophy behind this construction (extended real number system ($\...
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Since model theory must be founded in set theory, do you get different model theoretic results depending on your choice of set theory?

As I understand it, model theory tells you a lot about set theories (plural) but is also founded upon set theory, which means you need to pick one set theory to do model theory within. But different ...