3
votes
0answers
44 views

How does one create “good” math problems?

As lifelong students of mathematics, we find problem-solving to be absolutely essential to enhance our understanding of the subject. Teaching others what we know serves to reinforce our existing ...
1
vote
1answer
43 views

What exactly is a property?

How is a property $P$ formally defined in mathematics? I mean for example if $f$ is a morphism from an object $X$ to $Y$ in some category, then somehow I feel that "has codomain $Y$" is too broad to ...
3
votes
4answers
128 views

Soft question: Examples where implications derived from mathematical models failed to describe reality

I have always been fascinated by how well conclusions drawn from mathematical models could fit reality, so I wondered if there are any counter examples. In "Gödel, Escher, Bach" I could already find ...
3
votes
1answer
58 views

What is gained by internalizing LST (the language of set theory)?

I'm reading up on Gödels constructible universe L in the book "Constructibility" by Devlin, and by comparing his text with texts like Kunen and Jech, there is one thing in particular that he's doing ...
5
votes
5answers
329 views

Are there finitely many interesting theorems? [closed]

I'm not a logician, I read, a long time ago, about Gödel etc... A theorem is a provable proposition under a system of axiom. Let's take the usual system of ZFC. Of course, the notion interesting is ...
0
votes
1answer
32 views

What is the “lowest” set of axioms that can be used in proofs?

What is the most basic set of axioms that one can use in proofs? As in, the axioms are irreducible. The most basic set of irrefutable rules in mathematics. I assume it has something to do with number ...
1
vote
2answers
126 views

What problems are there left to solve? [closed]

From the ancient Greek mathematicians (Archimedes, Pythagoras) before Christ to Issac Newton to George Birkhoff, these mathematicians have made huge strides in mathematics, developing theorems and ...
5
votes
3answers
103 views

On trusting the mathematical process [closed]

In studying math we are, at least partially, interested in making abstraction of real world problems and solving them through rigorous techniques and methods, and then interpreting the result. Let us ...
5
votes
7answers
233 views

What is the right interpretation of the axiom of extensionality

A set $a$ can be called extensional if it has the following propery: $$\forall b\left[\forall x\left[x\in b\iff x\in a\right]\Rightarrow a=b\right]$$ Based on this the axiom of extensionality can be ...
7
votes
1answer
261 views

Why can almost all ordinary mathematics be formalized by sets?

there must exists a reason of why the idea 'collection' is so powerful that it can formalize nearly all mathematics. subquestion: is there any which can not be formalized by this perspective? if so, ...
14
votes
3answers
237 views

Mathematical Notation and its importance

You can see how mathematical notation evolved during the last centuries here. I think everyone here knows that a bad notation can change an otherwise elementar problem into a difficult problem. Just ...
2
votes
1answer
72 views

Provocations on the existence of mathematical objects

The few Mathematics I have been studying so far is pure Mathematics. I happen to have some discussions with philosophers of Mathematics, but as they know I totally ignore their subject, we do not ...
9
votes
1answer
138 views

When the mathematical community consider the inclusion of a new axiom?.

At first I was thinking about the axiom of choice, but let's keep it general. What motivates the inclusion of new axioms (or change the ones we already have in an already defined axiomatic theory?. It ...
3
votes
3answers
249 views

How come mathematics is applicable to the real world?

Before the dam breaks, namely, the one that holds the waters of accusations, I want to specify that the question I'm asking is a "reference-request", and therefore does have an answer. Often in ...
3
votes
0answers
64 views

Is there a link between level of abstraction and use of numbers?

One of my friend who stopped studying maths in high school told me once You study maths, can you help me fill my tax forms? In her mind, advancing in maths studies implied manipulating an ...
-1
votes
5answers
240 views

What is maths? “Maths is the study of ______”? [closed]

I can fill in the blank by just listing the different fields of maths but my goal is to define all of mathematics. An answer that I would've accepted a few years ago is "Maths is the study of ...
-1
votes
1answer
132 views

Why is the number Pi more popular than any other constant? [closed]

What is so special about the number $\pi$? There are many more interesting constants, such as e, $\gamma, \sqrt{2}$ or the catalanian number. $\pi$ has been calculated to more digits than any other ...
11
votes
1answer
308 views

What underlies formal logic (or math, generally)?

I read a useful definition of the word understanding. I can't recall it verbatim, but the notion was that understanding is 'data compression': understanding happens when we learn one thing that ...
0
votes
2answers
98 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 ...
10
votes
1answer
201 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 ...
62
votes
22answers
5k 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
287 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
260 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
320 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 ...
87
votes
17answers
11k 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"?
1
vote
1answer
97 views

Analogue of prime numbers in addition? [closed]

What is the analogue of prime numbers in addition?
3
votes
0answers
107 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
100 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 ...
10
votes
0answers
264 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 ...
4
votes
0answers
200 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 ...
9
votes
4answers
355 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 ...
17
votes
1answer
376 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, ...
4
votes
1answer
98 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
234 views

Why demonstrations are important in mathematics? [closed]

Good evening, I'm studying math and would like to know how important are mathematical proofs in the world and particularly in a school of mathematics Thanks for your help
40
votes
3answers
926 views
25
votes
4answers
960 views

Is $\mathbb{N}$ impossible to pin down?

I don't know if this is appropriate for math.stackexchange, or whether philosophy.stackexchange would have been a better bet, but I'll post it here because the content is somewhat technical. In ZFC, ...
3
votes
2answers
655 views

Philosophy of a Mathematician.

Introduction I don't study Mathematics at university, and probably I don't have any chances to have a little understanding of what mathematics in all its aspects. But I love to find structures and ...
139
votes
22answers
11k views

Is mathematics one big tautology?

Is mathematics one big tautology? Let me put the question in clearer terms: Mathematics is a deductive system: it works by starting with arbitrary axioms, and deriving therefrom "new" properties ...
1
vote
2answers
149 views

Big Topics in Mathematics [closed]

My question is as follows: It is now the year 2013 as we know it, and I'm wondering what the "big topics" in mathematics are. What fields are of utmost interest and foundation in the modern era? How ...
20
votes
5answers
567 views

What does it mean for a set to exist?

Is there a precise meaning of the word 'exist', what does it mean for a set to exist? And what does it mean for a set to 'not exist' ? And what is a set, what is the precise definition of a set?
7
votes
2answers
214 views

Does $\mathsf{ZFC} + \neg\mathrm{Con}(\mathsf{ZFC})$ suffice as a foundations of mathematics?

I've heard people make the argument that: $\mathsf{ZFC}$ suffices as a foundations of mathematics because almost all theorems in the mathematics literature can be proven using $\mathsf{ZFC}$, so ...
7
votes
6answers
940 views

What is this physicist saying?

I do not want to poison this forum with politics. But I want to understand, precisely, what is meant by the bolded statement. It is made by a physicist who used to work at Harvard regarding the ...
10
votes
3answers
334 views

What have been some of the most revolutionary philosophical shifts in perspective in mathematics?

Often times, great revolutions in mathematics come from shifts in philosophical perspective. The shift from extrinsic to intrinsic geometry yields manifolds (and much else). The shift in focus from ...
32
votes
7answers
2k views

Does mathematics require axioms?

I just read this whole article: http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf which is also discussed over here: Infinite sets don't exist!? However, the paragraph which I found most ...
7
votes
3answers
175 views

Measure of how much information is lost in an implication

In an implication like $p \implies q$, is there some measure of how much information is lost in the implication? For example, consider the following implications, where $x \in \{0,1,\ldots,9\}$: ...
1
vote
2answers
143 views

Just a thought… defining “competition”?

Let's think about galaxies and animals. At first, they seem completely different. But their behavior seems to be governed by (or at least arise from) the same rules. Think about competition. ...
25
votes
5answers
845 views

What's the best way to measure mathematical ability?

Very soft question I admit, but it's something that's been bothering me for a while. I've been thinking that being self taught has the problem of accreditation. You can't evaluate a mathematician ...
40
votes
13answers
3k views

Is there such a thing as proof by example (not counter example)

Is there such a logical thing as proof by example? I know many times when I am working with algebraic manipulations, I do quick tests to see if I remembered the formula right. This works and is ...
3
votes
1answer
137 views

Formality fades away in the air

I'm trying to study by myself mathematics, but I realized that I have only a naive notion of certains building blocks of mathematics; certain parts of the formalism. So I tried to start with logic, ...
1
vote
1answer
752 views

Is mathematics considered a science [duplicate]

Possible Duplicate: what is the definition of Mathematics ? I would like to know if mathematics is considered a science? I've searched the internet and asked many people for insight to no ...