Questions tagged [philosophy]

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

Filter by
Sorted by
Tagged with
37
votes
2answers
2k views

Difference between provability and truth of Goodstein's theorem

I have been thinking about the difference between provability and truth and think this example can illustrate what I have been wondering about: We know that Goodstein's theorem (G) is unprovable in ...
4
votes
2answers
2k views

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. ...
3
votes
3answers
1k views

What exactly does tautology mean?

I'm struggling to understand the actual meaning of tautology. I know how to determine tautology with a truth table. But I don't understand what it actually means, so consider the following reasoning: ...
45
votes
6answers
6k views

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 ...
2
votes
3answers
829 views

Why is negative minus negative not negative? Why is negative times positive not directionless?

What I understand is that the only difference between plus and minus is direction. I've never understood this: That +1 + +1 is ...
38
votes
1answer
4k views

$e^{e^{e^{79}}}$ and ultrafinitism

I was reading the following article on Ultrafinitism, and it mentions that one of the reasons ultrafinitists believe that N is not infinite is because the floor of $e^{e^{e^{79}}}$ is not computable. ...
0
votes
0answers
165 views

Why linear congruential generator is called random number generator?

As far my understanding linear congruential generator is used to produce pseudo random number. But the algorithm requires four parameters like, ...
0
votes
1answer
83 views

Is it possible to construct a formal system such that all interesting statements from ZFC can be proven within the system?

I'm familiar with Godel's incompleteness theorem, which very basically states that there exists a statement that can be neither proved or disproved within a formal system powerful enough to include ...
6
votes
4answers
789 views

Decidability and “truth value”

One can read in the Wikipedia page for "Gödel's incompleteness theorems": Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of ...
1
vote
1answer
178 views

Where do I go wrong with Presburger “multiplication”?

This is a speculation about the Presburger arithmetics, the Peano axioms with only addition added, vis-à-vis the same axioms with both addition and multiplication added. In the first case I understand ...
18
votes
3answers
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 ...
9
votes
3answers
399 views

The nature of infinities

I have been thinking about the nature of infinity lately. I have no experience with higher mathematics or theorems regarding infinity, so please forgive me if my ideas on this topic are extremely ...
1
vote
3answers
148 views

Should axioms be seen as “building blocks of definitions”?

This question is about the difference between a definition and an axiom. However, it does not address the following point: Whenever we define something, this is often written as a series of axioms. ...
1
vote
1answer
114 views

Does Planck length contradict math? [closed]

I have a general question about math and infinity which really bothers me as a math student - can we actually divide every length by two? I would like to believe the answer is yes, because it ...
463
votes
36answers
59k views

Do complex numbers really exist?

Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...
7
votes
4answers
2k views

Good Sources for Lecture Movies in Set Theory, Logic and Philosophy of Maths

Of course as any other researcher I'm not able to attend any scientific event in my research area. But it is always interesting and useful to watch the lecture movies of these events. I will ...
4
votes
2answers
355 views

Is mathematics aprioristic? [closed]

Is mathematics aprioristic? I do not know. Some axioms of arithmetic and geometry arose clearly inspired by the observation of Nature. After that, those areas of mathematics were often developed with ...
4
votes
1answer
777 views

Why did we settle for ZFC?

I understand the need for a solid foundation of mathematics. But it seems to me that we have definitely settled for ZFC and I would like to know if there are strong reasons for this or not. I am not ...
8
votes
1answer
2k views

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 ...
0
votes
2answers
139 views

What happens to a point when you rotate a line?

If I have a line on an $XY$ grid: o---o---o---o---o a b c d e And I rotate it like so: ...
3
votes
3answers
528 views

Implications and Ordinary language

I studied propositional logic, and everyday I see applications of what I learned on the internet, in mathematical books and miscelaneous resources. One particular case is sentences in the form $...
1
vote
3answers
137 views

Good resources on the intersection of probability theory and logic from a foundations/philosophical perspective?

What are some good books, courses, or online resources for probability theory that highlights differences between classical, frequentist, Bayesian, epistemic etc.? I majored in philosophy and am now ...
2
votes
3answers
154 views

Are the laws of mathematics 'absolute' in this universe? [closed]

We observe that almost all physical phenomena (which has been explained) can eventually be explained by the laws of mathematics. Mathematics seems ubiquitous- for example the form of the differential ...
0
votes
0answers
48 views

Are there examples of theorems incompatible with ZF that have proven useful in the sciences?

I was thinking about "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" and I realized all the mathematics I know about in science are compatible with ZF (even if they sometimes ...
-1
votes
3answers
99 views

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 ...
11
votes
2answers
553 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 ...
1
vote
1answer
96 views

'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 ...
2
votes
1answer
1k 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 ...
-5
votes
2answers
301 views

$\sqrt{-1}$ is both a positive and a negative number [closed]

I contend that there is a third category of number (in addition to positive and negative numbers), which are neutral. For the sake of expression, let us call these numbers neutral numbers. Zero, for ...
0
votes
1answer
58 views

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 ...
14
votes
1answer
1k views

Why is CH true if it cannot be proved?

Continuum hypothesis (CH) states that there can be no set whose cardinality is strictly between that of integers and real numbers. Godel, 1940 and Paul Cohen,1963 showed that CH can neither be proved ...
1
vote
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 ...
8
votes
4answers
1k views

Why is homeomorphism understood as stretching and bending?

A function $f: X \to Y$ between two topological spaces $(X, T_X)$ and $(Y, T_Y)$ is called a homeomorphism if it has the following properties: $f$ is a bijection (one-to-one and onto) $f$ is ...
14
votes
1answer
1k views

Mac Lane and Eilenberg's motivations for category theory

I'm looking to understand the conceptual process that brought Eilenberg and Mac Lane in developing the basic concepts of category theory. I quote Mac Lane's book "Category theory for working ...
5
votes
2answers
979 views

What is the meaning/purpose of finding the “foundations of mathematics”?

I've read in a lot of places how there was a "foundational crisis" in defining the "foundations of mathematics" in the 20th century. Now, I understand that mathematics was very different then, I ...
0
votes
1answer
442 views

Study of all published works of Bertrand Russell on foundations of mathematics: Please recommend his works. [closed]

Study of all published works of Bertrand Russell on foundations of mathematics: Please recommend his works. I think Bertrand Russell was a special mind and I set a goal for myself to study all his ...
6
votes
1answer
554 views

Principle of mathematical induction

In his book “Introduction to Mathematical Philosophy” Bertrand Russell seems to reach the conclusion that mathematical induction is a definition and not a principle. In essence he states that ...
1
vote
2answers
392 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 ...
4
votes
3answers
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 ...
0
votes
1answer
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 ...
7
votes
4answers
4k views

Abductive vs. inductive reasoning

To me, abductive reasoning and inductive reasoning are very very similar, in that they both go from the specific to the general and they are distinguished only through the examples which are provided ...
4
votes
2answers
214 views

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 "...
1
vote
0answers
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, ...
2
votes
2answers
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 ...
24
votes
10answers
11k views

How can Zeno's dichotomy paradox be disproved using mathematics?

A brief description of the paradox taken from Wikipedia: Suppose Sam wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must ...
1
vote
2answers
690 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 ...
7
votes
1answer
326 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
1answer
546 views

Difference between Deduction and Induction

I would like to know what is the difference between deduction and induction. Mathematical induction I know well, but now I would like to look at these from a philosophical point of view. All help is ...
1
vote
0answers
89 views

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 ...
0
votes
1answer
26 views

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 ...