3
votes
0answers
61 views

Why not defining a measure as a function on functions?

A measure $\mu$ is a function to $\left[0,\infty\right]$ on the sets belonging to a $\sigma$-algebra. Then for integrable functions $f$ the integral $\int fd\mu$ comes in, having nice properties ...
0
votes
1answer
25 views

contact point and point of intersection

I am just unable to understand the definitions of contact point and point of intersection.My doubts can be summed up into the following two questions : 1) Suppose $f(x)=x^2$ and $g(x)=0$ are two ...
1
vote
1answer
39 views

A special kind of metric-spaces

Is there a special name for those metric-spaces or topological spaces in which every non-empty open set is uncountable ?
1
vote
0answers
35 views

A question about the names of the components of an interval

Suppose you have the interval $[n,10n]$. How do you call the $n$ in this interval? I think the $n$ can be called the "independent variable of the interval" (since the interval $[n,10n]$ can be written ...
0
votes
0answers
28 views

Uniformly bounded function?

For all I know, the term "uniformly bounded" only makes sense when applied to a family of functions, sequences of functions, sets of functions with one parameter and such. But I've seen in several ...
2
votes
3answers
93 views

How to define the magnitude of a rotation in $\mathbb{R}^n$?

Is there some well established way on how to quantify rotations in $\mathbb{R}^n$? To say which rotation is greater and which is smaller? In $\mathbb{R}^2$ the rotation is characterized by a single ...
0
votes
0answers
35 views

The term $rank$ in methematics

Reading wikipedia's disambiguation page about the "rank" word I see many concept of rank of many different matematical object. I only know about the rank of a graded poset and the rank of a set that ...
1
vote
1answer
143 views

Have any definitions in mathematics been redefined

Based on certain intuitions and motivations we make certain definitions and then proceed to use these concepts in further developing our intuition. For example, we have an intuition that a line has ...
0
votes
2answers
130 views

Rigorous mathematical definition of “much greater than” symbol

What does $f(x) \gg g(x)$ mean mathematically? How can we characterize "much greater than"?
2
votes
3answers
122 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 ...
2
votes
1answer
36 views

Why do we want to define a $k$-scheme to be birational if the rational map (and its inverse) to $\Bbb A_k^n$ is over $k$?

Two varieties $X,Y$ are said to be birational if there exist rational maps in each direction such that either composition is the identity on a open dense subset. Note that here the morphisms aren't ...
3
votes
2answers
77 views

What is an isomorphism?

I'm familiar with the concepts of group isomorphism, ring isomorphism, and graph isomorphism, but it's never been presented to me what an isomorphism is in general: given any X, what is an X ...
19
votes
5answers
564 views

$\epsilon, \delta$…So what?

Over the course of my studies I often encounter phrases in reference material of the type "and this avoids the need for using $\epsilon$, $\delta$ definitions" or "by this we can omit those ...
2
votes
5answers
294 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 ...
4
votes
2answers
143 views

Why do differential geometry textbooks bother with equivalence classes of smooth structures?

In contemporary textbooks on differential geometry, the definition of smooth manifolds is given in a (IMHO) awkwardly obfuscated way, by saying that a smooth manifold is a topological space endowed ...
1
vote
2answers
122 views

Infinite sum with 0 terms: comparison to infinite product

Depend on what text you read, an infinite product with an infinite number of terms that are 0 is either divergent, or diverge to 0. Even though, obviously, the partial product is still a convergence ...
6
votes
3answers
281 views

How can people understand complex numbers and similar mathematical concepts?

In mathematics, how does something like complex numbers apply to the real world? Why do complex numbers exist? How can we comprehend addition of complex numbers? For example, addition of natural ...
0
votes
2answers
106 views

Why do we use “if” in the definitions instead of “if and only if”? [duplicate]

I often write my notes as logical statements and constantly wonder why people use only the "if" direction in the definitions instead of the "if and only if". Consider: "A homomorphism $\phi$ is said ...
1
vote
0answers
17 views

Does the notion of “commutative algebraic theory” generalize to many-sorted signatures?

I think that the notion of commutative algebraic theory only makes sense for unisorted signatures, and cannot sensibly be generalized to the case of more than one sort. Does anyone know of a ...
0
votes
0answers
46 views

What does the sentence “every element of $S$ has a unique colour” mean?

Does the statement mean that each element of $S$ has exactly one colour, or that no two elements of $S$ share the same colour? Or could either interpretation be valid, depending on the context?
3
votes
2answers
124 views

Should axioms be viewed as part of the signature?

I included category theory in the tags in order to get feedback from the categorial logic community. It goes without saying that this isn't really category theory. A semigroup can be defined as a ...
2
votes
1answer
95 views

What is a good definition of Hilbert space?

Motivation of my question: in my opinion, in view of the common definition, the statement "$\ell_p$ is a Hilbert space if and only if $p=2$" makes no sense because there is no inner product in the ...
37
votes
8answers
5k views

Why is there never a proof that extending the reals to the complex numbers will not cause contradictions?

The number $i$ is essentially defined for a property we would like to have - to then lead to taking square roots of negative reals, to solve any polynomial, etc. But there is never a proof this cannot ...
0
votes
1answer
67 views

Why Integral domains haven't an unified definition? [duplicate]

We can define integral domains as: rings without zero divisors commutative rings without zero divisors commutative rings with identity and without zero divisors I don't know why integral domains ...
6
votes
4answers
254 views

What is the relationship between “recursive” or “recursively enumerable” sets and the concept of recursion?

I understand that "recursive" sets are those that can be completely decided by an algorithm, while "recursively enumerable" sets can be listed by an algorithm (but not necessarily decided). I am ...
1
vote
3answers
84 views

A question about root of a polynomial

If we plug in $x$ into a polynomial and we get the value of $0$ as a result, can we be certain that $x$ is the root of the polynomial? If that is the case, why in this Wikipedia article, it says ...
6
votes
5answers
442 views

What exactly is “approximation”?

There are a lot of great "approximations" that exist in the mathematical field:$$\dfrac{22}{7} \approx \pi$$ $$e \approx \left(1 + \dfrac{1}{n}\right)^n$$But the fact that I have yet to know what ...
5
votes
1answer
265 views

What is “algebra” in $\sigma$-algebra (or “field” in $\sigma$-field)?

I know that $\sigma$ in $\sigma$-algebra stands for the closure under countable union property. What about "algebra"? Surely it cannot algebra over a field or a ring as defined in algebra textbooks. ...
2
votes
1answer
63 views

Using the notation $m^{O(1)}$

$m^{O(1)}$ denotes the set of functions $\mathbb{N} \to \mathbb{N}$ which are polynomially bounded. (Is that what is usually means?) Now it used as follows: $$ f(m) \leq m^{O(1)}$$ To express that ...
3
votes
2answers
149 views

“Good” closure conditions

[Attention! This question requires some reading and it's answer probably is in form of a "soft-answer", i.e. it can't be translated into a hard mathematical proposition. (I hope I haven't scared away ...
8
votes
2answers
1k views

*Recursive* vs. *inductive* definition

I once had an argument with a professor of mine, if the following definition was a recursive or inductive definition: Suppose you have sequence of real numbers. Define $a_0:=2$ and ...
7
votes
1answer
138 views

When do modifiers denote sub or super? Pseudo-, quasi-, ultra-, strong-, well-, pre-, c0- …

One only needs to search MMA.SE, math journals, wikipedia, or god-forbid, n-cat lab, for keywords listed in the title, which can be extended with: uniform-, regular-, complete-, local-, partial-, non- ...
51
votes
9answers
3k views

What makes elementary functions elementary?

Is there a mathematical reason (or possibly a historical one) that the "elementary" functions are what they are? As I'm learning calculus, I seem to focus most of my attention on trigonometric, ...
17
votes
7answers
2k views

What is combinatorics?

I've tried to search the web and in books, but I haven't found a good definition or definitive explanation of what combinatorics is. Could anyone give me a definition/explanation of combinatorics, ...
1
vote
1answer
360 views

What does “a set of things” mean?

Suppose we defined some mathematical object $P$, where $P$ is natural number, polynomial, endofunction, geometric figure, etc. What does the expression “$A$ is a set of $P$s” mean: Set inclusion) ...
1
vote
2answers
111 views

Definition of identity in a monoid

I'm having trouble understanding the way the identity element is defined in Lang's Algebra. Below is the relevant information. Suppose we have a monoid G with elements $x_{1},...,x_{n}$. We can define ...
10
votes
3answers
608 views

Differentiable at a point

My roommates and I have an argument you guys can help to settle (peace is at stake, don't let us down!) In undergrad calculus courses, one usually explains what it means for a function to be ...
2
votes
1answer
80 views

a function of a dependent type, a section, a sheaf

I have defined this simple sheaf. Take $E:set, B:set, p:E\to B$. Let $P(B)$ be a set of subsets of $B$. Let $S$ be the set of sections of $p$. Let $F$ be a contravariant functor from $P(B)$ as a poset ...