Questions involving philosophy of mathematics

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3
votes
2answers
255 views

Leibniz' Law and that good old riddle

There exists a Theory of Identity in mathematical logic. I've encountered it for the first time in Principia Mathematica by Alfred North Whitehead and Bertrand Russell (1910). Quote: "This definition ...
0
votes
2answers
119 views

General questions about theorems and laws

I have doubts about the construction of mathematical elements. There are proofs, that are proven using other theorems (corollaries) and/or axioms or definitions, such as Fermat's Last Theorem, the ...
-4
votes
2answers
284 views

Do irrational numbers really exist?

Isn't it possible that an irrational number is in reality the quotient of two infinitely long integers that even if there were repeating sections in it, it would take infinite digits to find the first ...
3
votes
2answers
165 views

Understanding: Axiom of Specification and Russell's Paradox: there is no universe?

Following Halmos's Naive Set Theory, Russell's Paradox emerges from using the axiom of specification (that for every set $A$ and property $\phi$ there exists a set $Y$ whose elements are those ...
5
votes
1answer
179 views

Explain/illustrate Goedel's theorems and possible implications to non-mathematicians

I am asked to give a talk about (a) mathematical practice, (b) axiomatization, (c) Gödel's theorems and (d) possible antimechanist arguments based on the incompleteness theorems (as mentioned in P ...
1
vote
1answer
159 views

Explain mathematical practice and axiomatization to non-mathematicians

I am asked to give a talk about (a) mathematical practice, (b) axiomatization, (c) Gödel's theorems and (d) possible antimechanist arguments based on the incompleteness theorems (as mentioned in P ...
4
votes
2answers
206 views

Why do imaginary numbers work (somewhat philosophical question)?

Asking as a layman, I've always puzzled over imaginary numbers and how they can be used to solve problems involving real numbers or quantities only (e.g. contour integration methods or Fourier ...
11
votes
1answer
321 views

What would qualify as a valid reason to believe there is a closed form?

I noticed that almost every non-homework-level integral posted on this site prompts somebody to ask "Do you have any reason to believe there is a closed form?" (some recent examples here and here) I ...
5
votes
1answer
731 views

How much math does one need to know to do philosophy of math?

I'm looking for advice from mathematicians who also study philosophy of math (PoM). Due to interest I'd like to study PoM as a hobby, but I'm worried if I don't understand math well enough from a pure ...
2
votes
2answers
193 views

Is there a proper term and/or symbol for an “agnostic” conclusion?

My question stems from the material conditional: $p \rightarrow q\\p\\\therefore\space q$ However, if $\bar p$ then the conditional is silent. I would like a way to represent this fact using, if ...
0
votes
2answers
136 views

Does infinity have a limit?? [closed]

Infinity being the far extent that the numerical system can stretch,can we say that infinity is actually a limit or infiity has another limit?
65
votes
22answers
6k views

Is math built on assumptions?

I just came across this statement when I was lecturing a student on math and strictly speaking I used: Assuming that the value of $x$ equals <something>, ... One of my students just rose ...
6
votes
5answers
318 views

Are there more real numbers than we can actually imagine?

I mean, if we could imagine all the real numbers then we could assign each number a finite sentence (or a finite book). Since the set of the finite books is countable then the set of real numbers ...
5
votes
2answers
300 views

Dogmas and Mathematics

What are the dogmas that restrict or promote the development of mathematics? I know that a dogma is a set of beliefs that is accepted by the members of a group without being questioned or doubted. ...
0
votes
5answers
333 views

Why doesn't $1/x=0$ have any solution?

Just out for curiosity ! Why $1/x=0$ doesn't have any solution? Or is it that the solution takes you to $1=0$ situation which would nullify mathematical principle that we stood for years Educate ...
89
votes
17answers
12k views

Is 10 closer to infinity than 1?

This may be considered a philosophy but is the number "10" closer to infinity than the number "1"?
3
votes
0answers
42 views

looking for good book on the history of formalism

In 1868 Beltrami published a paper ""Saggio di interpretazione della geometria non-euclidea" that seems to have led to the formalist philosophy of mathematics. But what was written exactly what were ...
1
vote
1answer
105 views

Analogue of prime numbers in addition? [closed]

What is the analogue of prime numbers in addition?
2
votes
1answer
188 views

True and provably true sentences in a model. Are they the same thing?

In logic, it is said that each sentence in a (consistent) theory is either true or false in a given model. Checking the truth of a sentence in a finite model amounts essentially to finite enumeration ...
3
votes
0answers
115 views

Can one define informational content of a mathematical expression?

At least in physicist's thinking, information, vaguely, is something that allows one to select a subset from a set. Say, a system can be in states A and B, we have done a measurement on it ...
1
vote
3answers
110 views

how can we write abstract algorithms?

Writing pseudo-code for algorithms is common practice in the applied mathematics literature. It is also often the case that the ideal input of an algorithm is an infinite set, for example it could be ...
3
votes
1answer
126 views

Why Is It Rational to Bet on the Most Probable Event?!

Suppose that someone is going to bet in a game. A dice is rolled, and there are only these two options for betting: Option 1. Give 1 dollar and bet on 6. Option 2. Give 1 dollar and bet on 1, 2, 3, ...
10
votes
0answers
336 views

What Do Mathematicians Do?

The American Mathematical Society maintains a web page entitled "What Do Mathematicians Do?" which references two interesting surveys. (One of the reference links is broken, but this one works: What ...
2
votes
1answer
210 views

Is there a geometrical proof of the impossibility of squaring the circle?

The impossibility of certain constructions in Euclidean geometry, such as squaring the circle with straight-edge and compass is usually shown by using algebraic methods. I am wondering if there are ...
7
votes
7answers
599 views

What is a number?

A dictionary I consulted said a 'number' is a 'quantity', so I looked up what quantity means and the same dictionary said it is an amount or number of some material or thing. Since quantity and ...
9
votes
1answer
468 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 ...
2
votes
1answer
61 views

Recommendation request: Reasoning behind statistics

It seems, to me at least, that most Statistics textbooks focus on the Statistical methods and techniques, or on the mathematics behind them. Would you recommend me some textbooks (or any online ...
2
votes
3answers
132 views

Optimal Solution in Natural Deduction

Does there exist an optimal solution for derivations in natural deduction, which is to say that the derivation in question requires the least amount of steps to arrive at the desired conclusion?
4
votes
0answers
219 views

How much are mathematics driven by applications?

At some point this provocative question came to my mind: Are mathematics mostly driven by applications? I am taking into account some of the comments made to my original question so I want to ...
12
votes
7answers
550 views

Is it possible to alternate the law of mathematics?

I am freelance writer. Recently I have been planning a science fiction - just planning, nothing solid yet - and I was wondering would it be possible for some other universes that have different set of ...
1
vote
0answers
69 views

If most of the mathematics needs a context to be not subject of interpretation, what part of mathematics doesn't need a context at all, if any?

In the past, I have asked this question here: Is mathematics the only language that is not subject of interpretation? And one of the answer started with: First, mathematics notation is subject ...
1
vote
0answers
74 views

Stylish Academic Writing [duplicate]

I don't know that this question belongs here, but I'd like to know of any references out there anyone here might recommend for writers of mathematical ideas, be it a book, an article, a dissertation, ...
4
votes
2answers
55 views

Are there any natural measures that can be put on the space of models?

I've already asked this question is Philosophy.SE and is a 'soft' question. An undecidable proposition, as in Gödel's Incompleteness theroem, is one whose truth value cannot be determined, because it ...
3
votes
0answers
86 views

Understanding how to vs. Knowing how to do something in Math?

So here's my question, in math, there are several places in which understanding why something works helps for a deeper understanding in mathematics. However, there are also other ways in which knowing ...
19
votes
1answer
480 views

Is there any mathematical meaning in this set-theoretical joke?

Recently I heard a joke: If an object exists, mathematicians call it a set and study it. But if an object does not exist, mathematicians call it a proper class and study it anyway. I wonder, ...
2
votes
1answer
182 views

Can there be two different math?

As per usual, let PA denote Peano Arithmetic and ZFC denote Zermelo-Fraenkel set theory with choice. Furthermore, ZFC 'validates' PA, in the sense that it proves that the PA axioms hold for the ...
2
votes
2answers
199 views

Books about modal logic?

I've just approached modal logic reading "An Introduction to Non-Classical Logic" of Graham Priest. I am looking for some books that treat this argument in a more extensive way than the book I am ...
4
votes
5answers
571 views

Books on logic, proof theory and set theory?

I graduated in Computer Science at University of Bologna in Italy some years ago. For various reasons now I am discovering a back interest in mathematic logic higher than I was a student. I have only ...
10
votes
6answers
302 views

Axiomatic Foundations

I am trying to deduce how mathematicians decide on what axioms to use and why not other axioms, I mean surely there is an infinite amount of available axioms. What I am trying to get at is surely if ...
26
votes
4answers
2k views

Is mathematical history written by the victors?

The question is the title of a recent piece in the Notices of the American Mathematical Society, by twelve authors (of which I am one). The contention is that traditional history of mathematics is ...
11
votes
9answers
963 views

Mathematicians talking about their identity as a person and as a mathematician?

I was wondering if any of you know of any books, articles, interviews, youtube videos, ... (etc) where a mathematician talks about his or her identity as a person and as a mathematician? Thank you for ...
4
votes
1answer
103 views

Trustworthiness of foundational systems

Naively, we might think that if a foundations of mathematics is consistent, then its fair game. Then we learn a bit more, and we realize that even if a foundations of mathematics is consistent, it may ...
3
votes
2answers
378 views

Don't Gödel's completeness and incompleteness theorems contradict each other? [duplicate]

Gödel's completeness theorem: Given a set of axioms, if we cannot derive a contradiction, then the system of axioms must be consistent. Gödel's incompleteness theorem:'Given any consistent, ...
4
votes
3answers
577 views

Why is the well ordering principle counter-intuitive?

I read here that while 'The Axiom of Choice agrees with the intuition of most mathematicians; the Well Ordering Principle is contrary to the intuition of most mathematicians'. I don't understand why ...
13
votes
2answers
518 views

Proofs from the “Ugly Book”

There is a famous saying in mathematics from Paul Erdős: "You don't have to believe in God, but you should believe in The Book." "The Book" is an imaginary book in which God had written down the best ...
4
votes
0answers
74 views

Constructivism implied or not

Let me take up some details in the answer of another question. Submitted by user hyg17: Heading: All real numbers can be expressed as a limit of rational numbers? The question was: Let $C$ be a set ...
4
votes
3answers
236 views

Yablo's paradox? a paradox without self-reference [closed]

Yablo's paradox arises from considering the following infinite set of sentences: $$(S_1): \mbox{for all }k > 1, S_k\mbox{ is false} \\ (S_2): \mbox{for all }k > 2, S_k\mbox{ is false} \\ ...
1
vote
2answers
164 views

What leads us to believe that 2+2 is equal to 4? [closed]

My professor of Epistemological Basis of Modern Science discipline was questioning about what we consider knowledge and what makes us believe or not in it's reliability. To test us, he asked us to ...
1
vote
1answer
101 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 ...
5
votes
1answer
237 views

Is there an introduction to probability and statistics that balances frequentist and bayesian views?

Perhaps, roughly, I might be described as advanced undergraduate regarding mathematics. However, I have not learned statistics and have only learned elementary probability. Does there exist a book or ...