# Tag Info

### 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.

### 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 ...

### 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".

### 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

### Nonequivalent definitions in Mathematics

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

### 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 ...

### 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 ...

### 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 ...

### 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) ...

### 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, ...

### 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.
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 ...
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### 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 ...

### Nonequivalent definitions in Mathematics

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

### Nonequivalent definitions in Mathematics

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

### 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.

### Nonequivalent definitions in Mathematics

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

### Nonequivalent definitions in Mathematics

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

### 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 ...

### 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 ...

### 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.
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### 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; ...

### 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 ...

### Nonequivalent definitions in Mathematics

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

### 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 ...

### 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 ...
$(a,b)$ 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 ...