1
vote
2answers
74 views

How can we explain the discrepancy between $\rightarrow$ (IF-THEN) and $\setminus$ (A-BUT-NOT-B)?

Let $\mathbb{B} = \{0,1\}$ denote the Boolean domain, ordered in the usual way. Then $\mathbb{B}$ is a lattice. It has a join operation $\vee$ that coincides with "OR," a meet operation $\wedge$ that ...
-2
votes
2answers
176 views

Is Infinity Needed in Maths? Does Infinity Actually Exist? [closed]

I'm asking this question as I have been having an on going online debate with a friend of mine. I claimed that Infinity does in fact exist in Maths and in Reality, as there's a whole plethora of ...
2
votes
2answers
138 views

Three-valued logic as foundation

Isn't it more natural to use Three-valued logic(false-true-unknown) as the foundation of mathematics? It is a better model for natural languages. And it also can model sentences like the lair paradox ...
13
votes
4answers
250 views

Which mathematical ideas most influenced the way you think?

This is not a question about how you use a formula or mathematical method to solve quantitative problems - that is applied mathematics. Rather, I'd like to hear how deeper ideas gained through the ...
7
votes
6answers
365 views

Interviews of famous modern mathematicians

I was wondering, are there any good collections of interviews of famous modern mathematicians? It can be text interviews, or audio or video recordings. I am not sure what exactly I mean by "modern". ...
26
votes
8answers
2k views

Complex analysis is more “real” than real analysis

In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself ...
0
votes
1answer
30 views

Evolution of Relations

In Frege, one finds relations treated as predicates in complex terms. However, modern set theory appears to treat them as two-place relation. Is this correct? If so, when did this shift occur and to ...
1
vote
2answers
102 views

Abstract Objects in Logic

I am confused on the concept of extensionality versus intensionality. When we say 2<3 is True, we say that 2<3 can be demonstrated by a mathematical proof. So, according to mathematical logic, ...
21
votes
10answers
2k 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 ...
2
votes
1answer
160 views

Why should we accept the existence of subsets $A$ such that neither $A$ nor $A^c$ are recursively ennumerable? And how can we persuade others?

Encode every pair $(t,x)$ (where $t$ is a Turing machine and $x$ is an input string) as a distinct natural number. Then the halting subset $H$ fails to be recursive. $$H := \{(t,x) \in \mathbb{N} ...
1
vote
2answers
79 views

How an axiomatic system is made?

An axiom is a sentence that is taken to be true without a proof. A set of (well organised) axioms is called an axiomatic system. As consequence of these axioms we get a lot of results that we call ...
5
votes
4answers
147 views

“The enrapturing discoveries of our field systematically conceal (…) the analogical train of thought that is the authentic life of mathematics…”

In the preface of the book Discrete Thoughts, Gian-Carlo Rota writes: Sometime, in a future that is knocking at our door, we shall have to retrain ourselves or our children to properly tell the ...
3
votes
1answer
163 views

Errors of Euler interpretation?

To complement the recent post on Euler's errors, I would pose the following question: what common errors of Euler interpetation appear in the literature? What errors are attributed to Euler's work in ...
3
votes
0answers
52 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
51 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
159 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 ...
5
votes
2answers
85 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 ...
0
votes
1answer
58 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
157 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
116 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
6answers
261 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
316 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
325 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 ...
3
votes
3answers
164 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 ...
10
votes
2answers
207 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
281 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
73 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 ...
2
votes
5answers
338 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
143 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 ...
12
votes
1answer
373 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
118 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 ...
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 ...
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
317 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"?
1
vote
1answer
105 views

Analogue of prime numbers in addition? [closed]

What is the analogue of prime numbers in addition?
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 ...
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 ...
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 ...
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, ...
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
261 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
42
votes
3answers
997 views
26
votes
4answers
1k 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
737 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 ...
151
votes
23answers
12k 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 ...