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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Is 0 the absence of something? [closed]

In computer programming, null and 0 values are two different things. I was wondering if the same applied to just mathematics in general. The reason I ask this question is because I don't understand ...
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1answer
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'The Computer and the Brain' - The mathematical language of the brain

I couldn't decide whether this question is more appropriate to post here or at the philosophy SE, but I thought I'd give people with a mathematical perspective the opportunity to help me decide. I'm ...
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Why and how Mathematicians define unity?

I hope this one (pun intended) post won't get ripped by the community. I wondered what are the most abstract ways to define unity element? Why is there a need for unity element in general? Is it just ...
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Does advanced math “power” more rudimentary math?

I came across this quote by Eric Weinstein that "when things got supposedly more advanced, they actually got simpler because mathematicians started revealing what was powering all the things that you ...
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1answer
68 views

Supposing that numbers are reducible to sets, how would one go about reducing a complex number to one?

I'm aware of the Benacerraf's identification problem, but suppose that numbers are reducible to sets such that the empty set is identified with zero, the power set of the empty set is identifiable ...
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1answer
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Are category-theory and set-theory on the equal foundational footing?

Set-theory is widely taken to be foundational to the rest of mathematics. So is category-theory. My question is: Are they two alternative, rival candidates for the role of a foundational theory of ...
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57 views

Logarithm, exponentiation, addition, and multiplication,

If the logarithm converts multiplication to addition, thus simplifying mathematics, does exponentiation, converting addition to multiplication, "complicate" mathematics? I've only ever seen arguments ...
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391 views

Axioms in Gödel's ontological proof are inconsistent?

So, my problem is with Axiom 5 of the proof, where Gödel considers necessary existence as a property. However, by his own definition, a 'property' applies to objects including those whose necessary ...
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785 views

The boundaries of mathematics?

There are five possible answers to a multiple-choice question. Given that the student does not know the answer, what is the probability that the student chooses the first answer? That is not a well ...
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300 views

Are numbers real, in a metaphysical sense? [closed]

I live and work with numbers almost all the time and have done so for most of my 77 years. I can almost feel them. But it is only almost. I have to believe, contra Plato, that they and moreover all of ...
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What kind of alternative mathematics systems exist? [closed]

What kind of alternative mathematics systems exist? What I mean is, mathematical systems that use a different sort of "basic premises" or e.g. logic(s) than the contemporary "mainstream" mathematics. ...
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Axiomatic formal language theory

I am struggling with this problem of philosophy of mathematics, that is to apply the axiomatic method to give foundations to the theory of formal languages, without mention explicitly the concepts of "...
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96 views

How can one think conceptually about Type Theory when one explains the differences between ZFC and Type Theory?

It's a big question, this I am quite aware of, so please excuse my little understanding on the subject but with reference to the following question, to which I would rather like to extend a little, ...
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100 views

Russell's “On Denoting,” formalization

So, I was reading Russell's paper 'On Denoting' and stumbled upon the (in)famous paraphrase\analysis of the definite description "x was the father of Charles II." As it is known, Russell's paraphrase ...
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267 views

Can we hope for an elementary proof of a conjecture of Goldbach?

It is written on Wikipedia: "During the 20th century, the theorem of Hadamard and de la Vallée-Poussin also became known as the Prime Number Theorem. Several different proofs of it were found, ...
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2answers
686 views

Is a set that consists of a single point connected?

If $S=\{a\}$ then surely for every two points from that set there is a path that joins them, choose $x_1=a$ and $x_2=a$ and define continuous function on $[0,1]$ such that $f(x)=a$. Or, one point ...
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What counts as a witness in constructive mathematics?

In order for a mathematical object to be accepted by a constructivist, a witness of a such object much be constructed. For example, we may let points be witnesses for natural numbers. For a real ...
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Expected Value Of A Process - Formalization / Foundations

Consider the question: Let $X$ be the random variable describing the number of rolls of a six-sided die needed till you see a $6$. What is $\mathbb{E}(X)$? Usually the answer given is $6$. What is ...
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Do mathematical realists believe that the continuum hypothesis is true?

The continuum hypothesis is known to be independent (neither provable nor disprovable) within the ZFC axioms. But as I understand it, mathematical realists (e.g. Platonists) believe that there is a ...
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156 views

How do mathematicians know that they're right? [closed]

How do mathematicians know that they're right? How do they know that there's no flaw in a proof, or know when something has been proved? Is this a welldefined concept, is it is some kind of intuition ...
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Random process and predictability

We are familiar with random process. For example the result of tossing a coin is considered a random process. So rolling a dice. In this logical scheme, the realization of a random process is ...
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1answer
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Things you can't prove in math

In science, we have laws through which we can explain various phenomena. It seems all of it can be reduced to a few basic laws. This is the idea of reductionism. It is also possible that we cannot ...
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2answers
174 views

How can we define something and have it translate to the real world?

I'm looking at a definition in my textbook (although my question applies to most definitions) Let A, B be two events. Define $P(B|A)$ i.e. the probability of B given A, by: $$P(B|A) = \frac{P (A\...
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Is there a term(s) for considering a larger context?

Say you have a conclusion, and you have drawn the parameters of your argument such that it bounds a certain set of hypotheticals. From the simple Bayesian perspective, it's basically just $A|B$. You'...
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How does statistics deal with calculating the odds of an event occurring that seems impossible. [closed]

So my thoughts on this lead me to believe that this may be a philosophical question as much as a mathematical question. That said, I find it hard to believe that no mathematician has ever ...
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Is there a way to explain or prove that there is not a single order of operations on the real numbers? [closed]

Given that addition and multiplication are both commutative and associative, Expressions containing only addition or multiplication can be evaluated in a any order. However, when the operation changes ...
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Books on the philosophy of geometry

I am looking for recent books ( say published after 2000) on the philosophy of geometry, most books on the philosophy of mathematics seem to ignore or bypass geometry at all or am I just looking with ...
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What finite results cannot avoid mentioning the reals? [closed]

I am interested in theorems whose. . . . . . hypotheses and conclusions are 'discrete' and mention only rational numbers. . . . most natural proofs invariably require an understanding of the ...
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Can you explain Lawvere's work on Hegel to someone who knows basic category theory?

I have worked my way through Simmons' intro to category theory. I also know what a subobject classifier/elementary topos is. Is there anyone who could explain what Lawvere did with Hegel's work (...
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Question about a Deduction in Modal Logic

Is the following deduction valid, and why? 1) A → B 2) ◇A ∴ ◇B Attempt: 1) in every world with A, you also have B. 2) there is at least one world with A. The conclusion, there is at least ...
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613 views

Why do people separate syntax and semantics in mathematical logic?

This is a rather vague or maybe philosophical question. Basically I want to have a deeper understanding on the motivation of syntax-semantics separation in mathematical logic, since it struck me when ...
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268 views

Set Description and their Context (Context of a Set)

When I was first introduced to sets the common "description" in place of a formal definition used to be something like a set is a collection of objects that do not have to be related in anyway prior ...
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1answer
53 views

Does there exist a finite false proposition $P$ such that assuming $P$ we can derive any given false proposition?

I'm unsure if this is an appropriate question to ask, in some sense it seems closer to philosophy, but it's something I've wondered for a while. If it is not an appropriate question say so and I'll ...
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61 views

Do we need implicit base 10 in order to represent a number in any given base?

We have been thought to notate a number n in base b as such, : e.g. 23 in base 10 $23_{10} =23$ , $3_2=11$ , n in base b as $n_b$ My problem is that both the number (n) and base (b) are written ...
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424 views

An “isomorphism” between continuous and discrete mathematics

First of all, I should inform everyone that I am not a mathematician and my question might sound not at all rigorous or maybe even absurd to many of you. But I have been thinking about this problem ...
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2answers
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Does it make more sense to say we've changed the vector, or we've merely changed the space in which we're looking at the vector?

Consider the vector $\vec{v} = (p, (x_1, y_1))$, where $p \in \mathbb{R}^2 = (p_1, p_2)$ is a coordinate point that denotes where the tail of the vector is, and $(x_1, y_1) \in \mathbb{R}^2$ denotes ...
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What does Betrand Russell mean when he says the mathematicians refuted the philosophers on the existence of infinity?

In the second-to-last chapter of his book "The Problems of Philosophy", Bertrand Russell writes that philosophers used to think that there can't be such a thing as infinity, but they were disproved by ...
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A lacking understanding of the basic axiomatic construction of $\emptyset$

I am truly embarrassed to admit my lacking knowledge of the Set Theory. Something always boggled my mind: Suppose that we have an empty set $\emptyset$. By extension, we can write it as $$\emptyset = ...
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What is the importance of $\frac{1}{(1/a)+(1/b)}$

I've noticed that this comes up in physics quite a bit, with resistors and has a part in the lensmaker's equation. It also comes up in math with$$a\oplus b:=\frac{1}{\frac{1}{a}+\frac{1}{b}}$$ $$\...
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65 views

What is “structure” and is it equivalent to its encoding?

I often come across a description of sets, as objects of "zero structure". I always intuitively understood Set Theory as a theory of size, meaning that the only information we get on it's objects of ...
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92 views

Is addition a 'sufficient' operation?

Is addition a sufficient operation? For instance, multiplication can be viewed as 'repeated' addition, and powers repeated multiplication (at least for rational powers), hence powers can also be ...
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Is mathematics done in an arbitrary model of ZFC?

Following up a previous thread I posted, I have tried to refine my questions. I would be happy with answers simply confirming that I have understood matters correctly, but of course I would also be ...
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151 views

Examples of non-invariant yet “useful” properties of mathematical objects

I am trying to find out whether there are mathematically important or useful properties (of some object(s)) that are nevertheless not invariant under some usual choice of isomorphism? Are there any ...
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Insight in the diagonal method as explained by Girard

I'm currently reading "The Blind Spot" by J.Y.Girard, and came across this passage about diagonalization: Is this way of describing diagonal arguments in general legitimate? If not is there another ...
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Balls and vase $-$ A paradox?

Question I have infinity number of balls and a large enough vase. I define an action to be "put ten balls into the vase, and take one out". Now, I start from 11:59 and do one action, and after 30 ...
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How to think about theories that prove their own inconsistency?

There are consistent first-order theories that prove their own inconsistency. For example, construct one like this: Assuming their is a consistent and sufficiently expressive first-order theory at ...
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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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Is i an integer? If so, i/1, which is i, is rational. 1 is an integer, at least.

There's this maths joke, where $i$ says to $π$, "get rational!" while $π$ says to $i$, "get real!" (I like to say that $e$ says to the both of them, "join me, and we will absolutely be one!" (don't ...
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How to create new mathematics? [closed]

How do scientists and mathematicians create new mathematics for describing concepts? What is new mathematics? Is it necessarily in format of previous mathematics? Can one person make (invent or ...
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What is meant by the following: This law is well motivated in cases where we may be ignorant of the facts …

The folllowing text is taken from a part in a book (reference below) which are about mathematical philosophy. "... the theorem of classical logic known as the law of excluded middle: for every ...