Questions on the usage and meaning of words in mathematics, the names for mathematical entities, and other such questions.
56
votes
5answers
3k views
What is the Riemann-Zeta function?
In laymen's terms, as much as possible: What is the Riemann-Zeta function, and why does it come up so often with relation to prime numbers?
49
votes
1answer
4k views
Why are rings called rings?
I've done some search in Internet and other sources about this question. Why the name ring to this particular object? Just curiosity.
Thanks.
46
votes
5answers
1k views
Why does mathematical convention deal so ineptly with multisets?
Many statements of mathematics are phrased most naturally in terms of multisets. For example:
Every positive integer can be uniquely expressed as the product of a multiset of primes.
But this ...
40
votes
11answers
4k views
Is it wrong to tell children that 1/0 = NaN is incorrect, and should be ∞?
I was on the tube and overheard a dad questioning his kids about maths. The children were probably about 11 or 12 years old.
After several more mundane questions he asked his daughter what 1/0 ...
40
votes
4answers
2k views
Why “characteristic zero” and not “infinite characteristic”?
The characteristic of a ring (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. Otherwise, we ...
34
votes
8answers
2k views
33
votes
12answers
4k views
I need mathematical proof that the distance from zero to 1 is the equal to the distance from 1 to 2 [closed]
I didn't know how to phrase the question properly so I am going to explain how this came about.
I know Math is a very rigorous subject and there are proofs for everything we know and use. In fact, I ...
32
votes
6answers
1k views
Why is compactness in logic called compactness?
In logic, a semantics is said to be compact iff if every finite subset of a set of sentences has a model, then so to does the entire set.
Most logic texts either don't explain the terminology, or ...
24
votes
3answers
496 views
Where does the word “torsion” in algebra come from?
Torsion is used to refer to elements of finite order under some binary operation. It doesn't seem to bear any relation to the ordinary everyday use of the word or with its use in differential geometry ...
22
votes
2answers
763 views
How did “one-to-one” come to mean “injective”?
How did a "one-to-one" function come to mean an injective one? I find it so non-intuitive that I often have to backtrack when reading texts that use "one-to-one" because I suddenly discover that I ...
21
votes
3answers
432 views
Why is the topological pressure called pressure?
Let us consider a compact topological space $X$, and a continuous function $f$ acting on $X$. One of the most important quantities related to such a topological dynamical system is the entropy.
For ...
20
votes
5answers
802 views
Is the square root of -1 rational?
This is not a deep question, but if there is a definite answer then here is the place where I will find it.
Is it justified to say that $i =\sqrt{-1}$ is rational?
The origin of this question lies ...
20
votes
2answers
1k views
What do Algebra and Calculus mean?
I sometimes see phrases like 'the relational algebra' or 'the lambda calculus'. What is the difference between an algebra and a calculus?
19
votes
1answer
377 views
An interesting topological space with $4$ elements
There is an interesting topological space $X$ with just four elements $\eta,\eta',x,x'$ whose nontrivial open subsets are $\{\eta\},\{\eta'\},\{\eta,\eta'\}, \{\eta,x,\eta'\}, \{\eta,x',\eta'\}$. This ...
18
votes
3answers
966 views
difference between class, set , family and collection
In school I have always seen sets. But I was watching a video the other day about functors and they started talking about any set being a collection but not vice-versa and I also heard people talking ...
18
votes
2answers
623 views
Does integration by parts with “deja vu” have a name?
In some integration by parts problems, such as evaluating the integral of $e^x \cos x$ or $\sec^ 3 x$, one performs integration by parts (possibly more than once, and possibly together with algebraic ...
17
votes
6answers
1k views
Lemma vs. Theorem
I've been using Spivak's book for a while now and I'd like to know what is the formal difference between a Theorem and a Lemma in mathematics, since he uses the names in his book. I'd like to know a ...
17
votes
2answers
455 views
A place to learn about math etymology?
I was recently wondering where the word `kernel' comes from in mathematics. I am sure the internet must know. I did manage to find
http://www.pballew.net/etyindex.html#k
which contains the origin ...
16
votes
4answers
1k views
What do you call numbers such as 100, 200, 500, 1000, 10000, 50000 as opposed to 370, 14, 4500, 59000
There are different categories of numbers that we use every day.
Integers that written in decimal notation have 1, 2 or 5 as the leading figure, followed by none, one or more zeros. These are very ...
16
votes
3answers
832 views
Why algebraic topology is also called combinatorial topology?
I remember reading somewhere(at least more than once) that algebraic topology is also known by the name "Combinatorial Topology" which essentially tags the subject fundamentally with some counting ...
16
votes
3answers
545 views
How to answer a student objection to the use of “of” in pronouncing f(x)?
Once upon a time in elementary school, a student learned how to translate certain English words into math. For example, 'and' usually means 'plus' such as "If John has 3 oranges AND 5 apples, how ...
16
votes
3answers
607 views
When do I use “arbitrary” and/or “fixed” in a proof?
In many proofs I see that some variable is "fixed" and/or "arbitrary". Sometimes I see only one of them and I miss a clear guideline for it. Could somebody point me to a reliable source (best a ...
16
votes
3answers
391 views
How do you pronounce the inverse of the $\in$ relation? How do you say $G\ni x$?
If I am talking about sets $G$ and $H$ and I want to say in words that $G\subset H$, I, like everyone else, will say that $G$ is contained in $H$, or that $H$ contains $G$.
But if I am talking about ...
16
votes
3answers
900 views
where does the term “integral domain” come from?
Self-explanatory title really! A student today asked me why they were called integral domains -- and I realised that the word "integral" seems to be being used in a way totally unlike any other way I ...
16
votes
1answer
1k views
Where did the word “logarithm” come from?
Where did the word logarithm come from? Any relation to the word algorithm?
15
votes
5answers
1k views
How fundamental is the fundamental theorem of algebra?
Despite its name, its often claimed that the fundamental theorem of algebra (which shows that the Complex numbers are algebraically closed - this is not to be confused with the claim that a polynomial ...
15
votes
2answers
3k views
“A proof that algebraic topology can never have a non self-contradictory set of abelian groups” - Dr. Sheldon Cooper
In the current episode "The Big Bang Theory", Dr. Sheldon Cooper has a booklet titled "A proof that algebraic topology can never have a non self-contradictory set of abelian groups". I'm still an ...
14
votes
8answers
3k views
Why does “convex function” mean “concave *up*”?
A function $f : \mathbb{R} \to \mathbb{R}$ is convex (or "concave up") provided that for all $x,y \in \mathbb{R}$ and $t \in [0,1]$,
$$f(tx + (1-t)y) \le tf(x) + (1-t)f(y).$$
Equivalently, a line ...
14
votes
5answers
582 views
Is 'no solution' the same as 'undefined'?
Today in class my teacher wrote something along the lines of:
$6^x = 0$
And proceed to heed a response from the class. A few people shouted undefined.
So the teacher then writes:
no solution ...
14
votes
4answers
3k views
What exactly does it mean for a function to be “well-behaved”?
Often in my studies (economics) the assumption of a "well-behaved" function will be invoked. I don't exactly know what that entails (I think twice continuously differentiability is one of the ...
14
votes
3answers
928 views
Why doesn't 0 being a prime ideal in Z imply that 0 is a prime number?
I know that 1 is not a prime number because $1\cdot\mathbb Z=\mathbb Z$ is, by convention, not a prime ideal in the ring $\mathbb Z$.
However, since $\mathbb Z$ is a domain, $0\cdot\mathbb Z=\{0\}$ ...
14
votes
4answers
2k views
Use of “without loss of generality”
Why do we use "without loss of generality" when writing proofs?
Is it necessary or convetion? What "synonym" can be used?
Thanks.
14
votes
5answers
671 views
What does “formal” mean?
I know the definition of formal power series, power series and polynomials. But what does the adjective "formal" mean? In google English dictionary, does it mean "9. Of or relating to linguistic or ...
14
votes
1answer
396 views
When does variété mean manifold?
Following advice from this post, I am in the process of translating Ehresmann's 1934 paper "Sur la Topologie de Certains Espaces Homogènes" from French to English.
French-English dictionaries online ...
14
votes
1answer
368 views
Once and for all - “Rational numbers” - because of ratio, or because they make sense?
This is a question I'm sure was asked before but I can't find it. There are many sources claiming that the term "rational number" for the elements of $\mathbb{Q}$ comes from the word "ratio", since a ...
13
votes
5answers
1k views
Alternative ways to say “if and only if”?
There are some scenarios about which I would like to get some confirmation:
when defining a concept A,
We call A, if ... [definition of
concept A]
Does "if" here mean equivalence
instead of ...
13
votes
2answers
1k views
Why are invertible matrices called 'non-singular'?
Where in the history of linear algebra did we pick up on referring to invertible matrices as 'non-singular'? In fact, since
the null space of an invertible matrix has a single vector
an ...
13
votes
2answers
506 views
Why are harmonic functions called harmonic functions?
Are they related to harmonic series in any way? Or something else? Wikipedia didn't help.
13
votes
4answers
164 views
Is there a name for this kind of “betweenness structure”?
A homeomorphism $\mathbb R\to\mathbb R$ is almost the same thing as an order isomorphism, except that a homeophorphism can also be an order anti-isomorphism.
I'm wondering whether there is a natural ...
12
votes
14answers
1k views
Mathematical concepts named after mathematicians that have become acceptable to spell in lowercase form (e.g. abelian)?
I would like to collect a list of mathematical concepts that have been named after mathematicians, which are now used in lowercase form (such as "abelian"). This question is partly motivated by my ...
12
votes
7answers
2k views
Is $0$ a natural number?
Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number?
It seems as though formerly $0$ was considered in ...
12
votes
2answers
712 views
Why is lambda calculus named after that specific Greek letter? Why not “rho calculus”, for example?
Where does the choice of the Greek letter $\lambda$ in the name of “lambda calculus” come from? Why isn't it, for example, “rho calculus”?
12
votes
2answers
2k views
What happens if we remove the requirement that $\langle R, + \rangle$ is abelian from the definition of a ring?
Ever since I learned the definition of a ring, I've wondered why the additive group is required to be abelian. What happens if we allow $\langle R, + \rangle$ to be nonabelian as well as $\langle R, ...
12
votes
3answers
1k views
Domain, Co-Domain & Range of a Function
I'm a little confused between the difference between the range & co-domain of a function. Are they not the same thing (i.e. all possible outputs of the function)?
12
votes
5answers
841 views
What is a Markov Chain?
What is a intuitive explanation of a Markov Chain, and how they work? Please provide at least one practical example.
12
votes
3answers
420 views
Name for $(1-x)$?
The multiplicative inverse of $x$ is $\frac{1}{x}$,
and the additive inverse of $x$ is $-x$,
is there a similar term for $(1-x)$?
12
votes
3answers
265 views
Is the thingie/cothingie distinction absolute?
Is there some inherent quality of a mathematical object that marks it as being "naturally" a thingie or a cothingie?
Suppose, for example, that two mathematical concepts, say, doodad and ...
12
votes
1answer
638 views
Common English language mistakes in mathematical writing [closed]
Quoting from this excellent answer:
If you read enough math papers you'll find that there are certain linguistic ticks that people pick up from each other
So here's a question (primarily for you ...
12
votes
6answers
193 views
Good examples of Ansätze
Frequently, when talking to mathematicians, I have some trouble when I mention, use, or try to explain what an Ansatz is. (Apparently it is more of a physics term than a maths one, for some reason.) ...
11
votes
3answers
235 views
What are rational integer coefficients?
I have a question about the following excerpt from Atiyah-Macdonald (page 30):
“A ring $A$ is said to be finitely generated if it is finitely generated as a $\mathbb Z$-algebra. This means ...
