179
votes

### Nonequivalent definitions in Mathematics

Perhaps the mother of all examples is "natural number". You can start an internet flame war by asking whether zero is a natural number.

Community wiki

152
votes

### Nonequivalent definitions in Mathematics

Linear functions:
In high-school algebra (sometimes called "pre-calculus"), we are taught that linear functions are those of the form $y=mx+b$, because they are graphed by a straight line in the ...

Community wiki

121
votes

### Nonequivalent definitions in Mathematics

Not a word but a piece of notation: Sometimes I have seen $\subset$ used to mean "is a proper subset of" while other times I have seen it used to mean "is a subset of".

Community wiki

72
votes

### Nonequivalent definitions in Mathematics

Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions.
— F. Klein

Community wiki

68
votes

### Nonequivalent definitions in Mathematics

Another kind of answer. There is
increasing; strictly increasing
and there is
nondecreasing; increasing

Community wiki

64
votes

Accepted

### How can I improve my formal definitions?

I don't know about a definition-checking service, but I can give some general advice which I think will help.
Let me begin by rewriting your definition (hopefully correctly!):
Suppose I have a set $S$...

59
votes

### Nonequivalent definitions in Mathematics

Polygons!
Is a polygon a sequence of vertices together with the edges that connect consecutive vertices? If so, can two distinct vertices be the same point? How about two consecutive vertices? Or ...

Community wiki

54
votes

### Nonequivalent definitions in Mathematics

Positive definite matrix (and related terms). Most authors require these to be hermitian (or symmetric in the real case), but not all.
EDIT: also, "positive" can be ambiguous even for numbers: most ...

Community wiki

51
votes

### Nonequivalent definitions in Mathematics

Tensor: for some people, a tensor is an element of a tensor product of vector spaces. For others it’s a section of a tensor product of tangent and cotangent bundles on a manifold. Members of the first ...

Community wiki

51
votes

### Nonequivalent definitions in Mathematics

"Function", prior to about 1910 always meant the $y$ in $y=f(x)$ (Look up any definition form that period). Since roughly 1920 it's officially the $f$. Physicists, engineers and many applied ...

Community wiki

48
votes

### Nonequivalent definitions in Mathematics

The Fourier transform is defined in at least 3 different ways depending on which subject and school one comes from:
$$
\hat{f}(\xi)=\int_{\mathbb{R}^n} f(x)e^{-2\pi ix\cdot\xi}dx
$$
or
$$
\hat{f}(\xi) ...

47
votes

### Nonequivalent definitions in Mathematics

How about "algebra"? Usually an algebra over a field is assumed to be associative by default, but sometimes it is not.
Not to mention the various category-theory uses of "algebra" (over a monad, ...

Community wiki

44
votes

### Examples of advance via good definitions

Not sure if this qualifies since it is a whole theory, but distribution theory is in essence the consequence of a particularly well chosen definition.

Community wiki

44
votes

Accepted

### Why are there so many fractional derivatives?

The reason is that the fractional derivative is not a local operator. The usual derivative is a local derivative in the sense that the value of the derivative at one point only depends on the value of ...

42
votes

### What definitions were crucial to further understanding?

A famous definition which led to a completely new point of view is the definition of Schwartz distribution. It changed the understanding of what a "function" is, even among engineers.
Actually, the ...

Community wiki

41
votes

### Nonequivalent definitions in Mathematics

To many authors a "category" is necessarily locally small, but to others it need not be.

Community wiki

41
votes

### Nonequivalent definitions in Mathematics

What is a "set"? ZFC has one answer; ETCS has another; Bishop had another; HoTT has yet another.

Community wiki

40
votes

### Nonequivalent definitions in Mathematics

Some authors formulate separability axioms T3/T4 as normality/regularity plus T1. Some other authors do not require T1, on the contrast, they define normality/regularity as T3/T4 plus T1.

Community wiki

35
votes

### Nonequivalent definitions in Mathematics

Simply connected : for some authors, such a space is necessarily path-connected, for others, not.

Community wiki

34
votes

### Nonequivalent definitions in Mathematics

"Topos" sometimes means "elementary topos" and sometimes "Grothendieck topos".

Community wiki

33
votes

### Examples of advance via good definitions

A really good example is the definition of a scheme by Grothendieck. Previous attempts (e.g. "Foundations of Algebraic Geometry" by André Weil) were extremely complicated in comparison and didn't have ...

Community wiki

32
votes

### What definitions were crucial to further understanding?

I am surprised no-one has mentioned Weierstrass's $\epsilon,\delta$ definition of limits. This enabled mathematicians to rigorously reason about convergence and eliminate numerous apparent ...

Community wiki

31
votes

### Main statement as theorem or corollary

Another approach is to present both as theorems, but to present only A as a `marquee result' in the introduction, mentioning that it follows from the stronger but more technical B.

29
votes

### Nonequivalent definitions in Mathematics

A tree can be a very different thing in different parts of mathematics. It might be a certain kind of acyclic graph; or a partial order such that the predecessors of every node are linearly ordered; ...

Community wiki

27
votes

### Nonequivalent definitions in Mathematics

I would say the word "kernel" is probably among the most overloaded terms in mathematics. You've got kernels of linear operators, convolutional kernels, distribution kernels, Markov kernels, and ...

Community wiki

27
votes

### Nonequivalent definitions in Mathematics

Topology: Some authors use the term neighborhood while other use open neighborhood instead.

Community wiki

26
votes

### What definitions were crucial to further understanding?

A good example comes from the definition of manifolds, although it's less of a single definition and more of an evolving notion.
The background to modern differential geometry lies in trying ...

Community wiki

25
votes

### Nonequivalent definitions in Mathematics

Perhaps a prominent example is the definition of a smooth manifold. Some authors require the underlying locally Euclidean Hausdorff space to be 2nd countable, while others require it to be only ...

25
votes

### Nonequivalent definitions in Mathematics

$(a,b)$
Is that a coordinate pair representing a point in the plane? or,
The open interval from $a$ to $b$? or
The greatest (highest) common factor (divisor) of $a$ and $b$? or
The ideal ...

Community wiki

25
votes

### Nonequivalent definitions in Mathematics

The definition of a Turing Machine is a great example, where multiplied all together there are at least hundreds of possible definitions.
Is the tape doubly infinite or singly infinite? (If singly ...

Community wiki

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