Questions tagged [philosophy]

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

779 questions
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 ...
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 ...
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 ...
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 ...
169 views

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$$a\oplus b:=\frac{1}{\frac{1}{a}+\frac{1}{b}}$$ \...
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 ...
89 views

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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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, ...
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 ...
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 ?
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 ...
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 ...
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 ...
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 ...
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 ...
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....