Questions tagged [philosophy]

Questions involving philosophy of mathematics. Please consider if Philosophy Stack Exchange is a better site to post your question.

7
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1answer
314 views

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 ...
2
votes
3answers
150 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 ...
-1
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1answer
58 views

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 ...
2
votes
1answer
890 views

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 ...
2
votes
2answers
169 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\...
0
votes
1answer
48 views

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'...
-2
votes
3answers
139 views

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 ...
2
votes
2answers
114 views

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 ...
3
votes
2answers
237 views

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 ...
-5
votes
1answer
129 views

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 ...
7
votes
1answer
1k views

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 (...
1
vote
1answer
50 views

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 ...
6
votes
2answers
540 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 ...
1
vote
2answers
224 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 ...
1
vote
1answer
44 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 ...
-1
votes
1answer
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 ...
3
votes
5answers
403 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 ...
1
vote
2answers
37 views

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 ...
7
votes
3answers
329 views

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 ...
0
votes
4answers
108 views

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 = ...
10
votes
4answers
186 views

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$$a\oplus b:=\frac{1}{\frac{1}{a}+\frac{1}{b}}$$ $$\...
1
vote
1answer
64 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 ...
2
votes
1answer
89 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 ...
5
votes
1answer
146 views

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 ...
6
votes
2answers
149 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 ...
2
votes
0answers
95 views

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 ...
8
votes
4answers
174 views

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 ...
11
votes
2answers
509 views

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 ...
9
votes
1answer
450 views

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 ...
2
votes
1answer
649 views

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 ...
36
votes
6answers
8k views

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 ...
1
vote
4answers
66 views

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 ...
3
votes
1answer
236 views

Why are we using first-order logic and how to fix PA?

First-order logic (FOL) is pretty bad at pinning down specific structures for which we have no problem to think about intuitively. The classical example for me is $\Bbb N$, for which the first-order ...
-1
votes
1answer
121 views

Different set-theoretic constructions of ordinals

My background: I have been studying set theory for a couple of months, via Paul Halmos's book «Naive Set Theory» (which, as it turns out, does not present Cantor's naïve and inconsistent set theory, ...
2
votes
2answers
402 views

Self-verifying theory! But what can we learn from it?

There is this seemingly surprising result that no consistent and sufficiently expressive theory can prove its own consistency. But what would be the actual benefit from knowing that a theory $T$ can ...
1
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0answers
178 views

Is Introduction to Mathematical Philosophy by Bertrand Russell a good read? [closed]

I am thinking about my next math book, I'm really interested in the philosophy of math, so could anyone give some opinion on the book ?
21
votes
4answers
2k views

How do we know what natural numbers are?

Do I get this right? Gödel's incompleteness theorem applies to first order logic as it applies to second order and any higher order logic. So there is essentially no way pinning down the natural ...
3
votes
2answers
721 views

What is the empty graph?

There are often "empty objects" in maths. For example, sets are vaguely thought of as "objects that contain things", and the empty set $\emptyset$ is the "set containing nothing". However, something I ...
3
votes
3answers
85 views

How do we say that two logically equivalent definitions are not semantically equivalent?

This is a question of terminology (and also relates to philosophy). Very often we have a certain term, such as "differentiable function", and then we have a definition: (1) def. A differentiable ...
1
vote
1answer
408 views

Is category theory “conceptual”? [closed]

Warning: This question is in part philosophical in nature. Several prominent authors in category theory (CT) have claimed that CT is 'conceptual'. It is my impression that this sentiment is widely ...
0
votes
2answers
128 views

Law of excluded middle (Realist vs Intuitionistic mathematics) clarification question

I'm currently reading Thinking about Mathematics by Stewart Shapiro. In chapter 1, it says: "For an intuitionist, (statement 1) The content of a proposition stating that not all natural numbers have ...
3
votes
2answers
143 views

Number of symbols required to represent a number in unary?

I was thinking about the different number systems, and realised that technically binary is not the simplest. The simplest is unary - i.e. powers of 1. Wikipedia confirms this view: https://en....
2
votes
0answers
63 views

How to apply rigor to proof sketches

When doing analysis, I have difficulties to recite the proofs given in the lectures in oral examinations. I am able to do calculations with the theorems that I encountered. (Note that in central ...
3
votes
2answers
154 views

Is there any areas of mathematics that depend on the continuum hypothesis being true? (Or not)

I've been looking up a little bit about the Continuum Hypothesis, I understand that it is independent of $ \textbf{ZFC} $, that is, cannot be proved or disproved using the current axioms in $ \textbf{...
19
votes
8answers
4k views

What's your explanation of the Raven Paradox?

The Raven Paradox starts with the following statement (1) All ravens are black. which is equivalent to the following statement (2) Everything that is not black is not a raven. In all the ...
1
vote
2answers
398 views

Understanding Hilbert's program

Upon covering Hilbert's program in class and consulting https://plato.stanford.edu/entries/hilbert-program/, there were some questions I had on Hilbert's work, and was trying to find answers to them. ...
2
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4answers
190 views

How are primitive mathematical objects chosen?

On page of 22 Terry Tao's Analysis I, he writes (after presenting the Peano axioms): Remark 2.1.14. Note that our definition of the natural numbers is axiomatic rather than constructive. We have ...
0
votes
1answer
49 views

a modal logic question

I am looking for help understanding a problem of modal logic. In his 1963 essay 'Hume on evil', the philosopher Nelson Pike raises the following inconsistent triad of propositions (where S, W, and L ...
0
votes
0answers
147 views

Intuitive Randomness, Uniform PDFs, and Bertrand's Paradox

Suppose that somebody asks us the following question: Consider a straight line segment that goes from 0 to 1 (inclusive). Suppose that a point is chosen at random on this line segment. What is ...
3
votes
3answers
446 views

Natural language examples for failure of double negation elimination

I am trying to explain why double negation elimination $\neg \neg \phi \vdash \phi$ is invalid in intuitionistic logic, but introduction $\phi \vdash \neg \neg \phi$ is valid. The latter is easy: ...