Definitions in Mathematics

Introduction

Definitions per se are not something specific to the field of mathematics, but are rather prevalent in almost all subjects (for example English Grammar). An obvious need for definitions is to express a new term using already existing vocabulary. In mathematics however they have much more importance than in any other subject. Definitions in mathematics are rather precise and clearly express what a thing is or is not. From now on we will restrict ourselves to the form of definition used in mathematics.

What is a good definition?

As we remarked above a definition has to be precise. This is first test of a good definition. To understand what a precise definition is we need to contrast it with some vague idea. For example a vague notion of limit is as follows:
A function $ f(x)$ is said to have a limit $ l$ as $ x \to a$ when value of $ f$ approaches $ l$ as $ x$ approaches $ a$.

The definition above is not wrong, but it is not precise enough. The reason is the appearance of word "approaches" which has different connotations. The above definition can be made precise by giving an explicit meaning to the word "approaches". When this is done we arrive at the following usual definition of limits:
A function $ f(x)$ is said to have a limit $ l$ as $ x \to a$ if corresponding to any given number $ \epsilon > 0$ it is possible to find another number $ \delta > 0$ such that $ |f(x) - l| < \epsilon$ whenever $ 0 < |x - a| < \delta$.

As we can see above having precise definitions almost always leads to some extra complexity but we will see that this extra complexity is worth our while.

Another requirement of a definition is that it must be closely related to the intuitive idea of the thing being defined. In other words it should not be mysterious. Let's have a look at an artificial example of a mysterious definition (sorry for not being able to think of a better example):
An integer $ p > 1$ is said to be a prime number if $ (p - 1)! + 1$ is divisible by $ p$.

This is really mysterious for a student in secondary classes. The term $ (p - 1)!$ is simply out of place here. The above statement is actually true but does not seem to be related to the conception of a prime number as known by students in primary and secondary classes.

For this particular example we have two more options:
An integer $ p > 1$ is prime if it does not have any factors other than $ 1$ and itself.
An integer $ p > 1$ is prime if whenever $ p \mid ab$ (for positive integers $ a, b$) we also have either $ p \mid a$ or $ p \mid b$.

Here the first statement is the usual one given in primary classes and represents the conception of a prime number very successfully. The second definition above is somewhat mysterious as far as primary and secondary classes are concerned but is still far far better than the one which uses $ (p - 1)!$. This second definition serves as a good definition because of another requirement that a good definition must be useful.

Without using the above property ($ p \mid ab \Rightarrow p \mid a\,\,\text{or}\,\,p \mid b$) it is impossible to prove the unique factorization property of integers. Sometimes the usefulness of a definition can be its proper justification even if the definition is somewhat mysterious.

As another example of the usefulness let see the definition of a derivative of a function: $$f'(a) = \lim_{h \to 0}\frac{f(a + h) - f(a)}{h}$$ In most cases it turns out that the ratio $ \{f(a + h) - f(a - h)\}/{2h}$ gives a better approximation to the actual derivative $ f'(a)$ than the ratio $ \{f(a + h) - f(a)\}/h$. In fact the following is true:
If $ f'(a)$ exists then $ \displaystyle f'(a) = \lim_{h \to 0}\frac{f(a + h) - f(a - h)}{2h}$

To see why the above result holds we can write $$f(a + h) = f(a) + h\{f'(a) + A\} = f(a) + hf'(a) + hA$$ where $ A$ tends to $ 0$ with $ h$ and therefore we also have $$ f(a - h) = f(a) - h\{f'(a) + B\} = f(a) - hf'(a) - hB$$ where $ B$ tends to $ 0$ with $ h$.

Thus we have $$\frac{f(a + h) - f(a - h)}{2h} = f'(a) + \frac{A + B}{2}$$ and our result holds when $ h \to 0$.

So why not define the derivative as the limit of the ratio $ \{f(a + h) - f(a - h)\}/\{2h\}$? Because although it is connected with the intuitive notion of derivative it is not useful enough. It fails for simple functions like $ f(x) = |x|$ where this ratio tends to $ 0$ at point $ x = 0$, but the graph does not have a tangent at $ x = 0$.

Another important requirement of a good definition is that it should be general enough or should be capable of substantial generalization.

For example the last two definitions of prime numbers can be generalized for number fields and in doing so we find that these two definitions are not equivalent. Therefore we have two conceptions in the general context: 1) irreducibility i.e. unable to reduce into smaller factors and 2) primeness ($ p \mid ab \Rightarrow p \mid a\,\,\text{or}\,\,p \mid b$). In general number fields these are two distinct concepts which may or may not be equivalent. Only in very specialized contexts (like the case of usual integers) these concepts are equivalent. And this generalization forces us to study the beautiful problem of finding out when these concepts are equivalent and when they are not. Such generalizations are the key to advancement of mathematics.

Fuss over definitions

Sometimes we see that same concepts are defined differently by different authors. Many times this leads to confusion especially in higher secondary classes. To complicate the matters the teachers themselves advocate that one of these definitions is the correct one and all others are wrong.

Let's consider an example of an increasing function. Some authors define the concept as follows: A function $ f(x)$ defined on some interval is said to be an increasing function if $ f(x) < f(y)$ whenever $ x, y$ are points of that interval with $ x < y$.

Some authors prefer to use the following definition:
A function $ f(x)$ defined on some interval is said to be an increasing function if $ f(x) \leq f(y)$ whenever $ x, y$ are points of that interval with $ x < y$.

The argument in favor of first definition is that an increasing function should actually have increasing set of values whereas the second definition treats even a constant function as increasing. The proponents of second definition say that their point of view is more general and in addition they define strictly increasing functions as those which satisfy the first definition.

I remember having a discussion specifically on this example with some person on an online forum. He was simply not ready to accept the second definition as correct and used arguments based on authority (i.e. he quoted some famous Russian authors in support of the first definition). On the other hand I maintained that there is no need of preferring one definition over another. We can chose any of them and stick to it while proving further results. Both of these definitions are equally useful (but note that they are not equivalent) and it really does not matter which one is chosen by some author while writing a book. A better approach is to also discuss the alternative definition in a footnote so that the reader is aware of the variety of definitions.

Another familiar example is from the theory of logarithmic and exponential functions. Since these functions cannot be defined without recourse to the methods and theories of calculus, most of the higher secondary books don't define them at all and assume that the student can work with them using some vague notions about them. This is the worst (and sadly the heavily used) approach towards these beautiful functions.

We can classify the approach to these functions in two main categories:

1. Define exponential function first and treat the logarithm as its inverse.
2. Define logarithm function first and treat the exponential function as its inverse.

In each of these approach we have many variations and I will point out some of them:
1A) $ \displaystyle \exp(x) = \lim_{n \to \infty}\left(1 + \frac{x}{n}\right)^{n}$
1B) $ \displaystyle \exp(x) = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots$
2A) $ \displaystyle \log x = \lim_{n \to \infty}n(x^{1/n} - 1)$
2B) $ \displaystyle \log x = \int_{1}^{x}\,\frac{dt}{t}$

Any of these four definitions (there are many more which I have omitted) can be used to establish all the properties of logarithmic and exponential functions. The choice should be based on the elegance and ease with which the theory can be developed and as I mentioned earlier the alternative approaches should also be mentioned (in brief at least). For example personally I like the approach of defining $ \log x$ as an integral as it leads to very easy proofs of all the properties of logarithmic and exponential functions.

Another example in this context is the definition of a rank of matrix where two approaches are used commonly. One of these is based on the elementary row operations which is far simpler and is closely related to the usefulness of the concept of rank (solving linear equations). The other approach uses determinants and offers some practical value in calculating the rank of the matrix. In this case it is important to establish the equivalence of both approaches (just mentioning in a footnote is not sufficient in my opinion).

Whenever we have alternative definitions for a concept it is better to present them so that the useless fuss over correct definition can be avoided. Moreover the philosophy behind definitions is to have precise meanings for various terms therefore it makes sense to avoid confusion arising out of multiple definitions.

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