After some initial counting and some thinking put into it, you may have asked yourself, what is there more to investigate about numbers? A number is a number, there are a few operations on it, I have just seen that, a clean and dry concept, a quite straightforward count of objects you have been dealing with. Five apples, seven pears, six pencils. The number five is common to all of them. We have abstracted it, and together with other fellow numbers (three, four, seven,, 128, 349, ...) it is a part of a number system we are familiar and we work with.
From our everyday encounters with mathematics, we may have a feeling there are only integers present in the world of math, and that it is not really clear where and how those mathematicians find so many exotic numerical concepts, so many other kinds of numbers, like rational, irrational, algebraic ... Moreover, you may even think that, without some real objects to count or to measure, there would be no mathematics, and that mathematics is, actually, always linked to a real world examples, that numbers are intrinsically linked to the quantification of things in the real world, to the objects counted, measured, that they are inseparable. You may think that a number, despite its "mathematical purity", somehow shares other, non mathematical properties, of the objects it represents the count of.
In this article I will discuss these thoughts, assumptions, maybe even misconceptions. But, no worries, you are on the right track by very action that you want to put a thought about math and numbers.
Before I go to the exciting world of basketball and poker, as an illustration, let me discuss a few statements. A famous mathematician, Leopold Kronecker, once said that there are only positive integers in the mathematical world, and that everything else, i.e. definition of other kinds of numbers, is the work of men. I support that view. Essentially, many mathematicians do as well. Here is the flavour of that perspective. Negative numbers are positive numbers with a negative sign. Rational numbers are ratios of two integers, m/n, (where n is not equal 0). Real numbers (rational and irrational) are limiting values of rational numbers’ sums and sequences (which are in turn ratios of integers), convergent sums of rational numbers, where rational numbers are smaller and smaller as there are more and more of them. As we can see that all these numbers are, fundamentally, constructed from positive integers.
As for "purity" of a number here is a comment. Number has only one personality! Take number 5, for instance. It's the same number whether we count apples, pears, meters, cars...That's why we need labels below, or beside, numbers, to remind us what is measured, what is counted. For real world math applications that’s absolutely necessary, because by looking at the number only, we can not conclude where the count comes from. When you write 5 + 3 = 8, you can apply this result to any number of objects with these matching counts. So, numbers do not hold or hide properties of the objects they are counts of. As a matter of fact, you can just declare a number you will be working with, say number 5, and start using it with other numbers, adding it, subtracting it etc, without any reference to a real life object. No need to explain if it is a count of anything. Pure math doesn't care about who or what generated numbers, it doesn't care where the numbers are coming from. Math works with clear, pure numbers, and numbers only. It is a very important conclusion. You may think, that properties of numbers depend on the objects that have generated them, and there are no other intrinsic properties of numbers other than describing them as a part of real world objects. But, it is not so. While you can have a rich description of objects and millions of colourful reasons why you have counted five objects, the number five, once abstracted, has properties of its own. That's why it is abstracted at the first place, as a common property! When you read any textbook about pure math you will see that apples, pears, coins are not part of theorem proofs.
Now, you may ask, if we have eliminated any trace of objects that a number can represent a count of, that might have generated the number, what are the properties left to this abstracted number? What are the numbers' properties?
That's the focus of pure math research. Pure means that a concept of a number is not anymore linked to any object whose count it may represent. In pure math we do not discuss logic or reasoning why we have counted apples, or why we have turned left on the road and then drive 10 km, and not turned right. Pure math is only interested in numbers provided to it. Among those properties of numbers are divisibility, which number is greater or smaller, what are the different sets of numbers that satisfy different equations or other puzzles, different sets of pairs of numbers and their relationships in terms of their relative differences, what are the prime numbers, how many of them are there, etc. That's what pure math is about, and these are the properties a number has.
In applied math, of course, we do care what is counted! Otherwise, we wouldn't be in situation to "apply" our results. Applied math means that we keep track what we have counted or measured. Don't forget though, we still deal with pure numbers when doing actual calculations, numbers are just marked with labels, because we keep track by adding small letters beside numbers, which number represent which object. When you say 5 apples plus 3 apples is 8 apples, you really do two steps. First step is you abstract number 5 from 5 apples, then, abstract number 3 from 3 apples, then use pure math to add 5 and 3 (5 + 3) and the result 8 you return back to the apples’ world! You say there are 8 apples. You do this almost unconsciously! You see the two way street here? When developing pure math we are interested in pure numbers only. Then, while applying math back to real world scenarios, that same number is associated with a specific object now, while we kept in mind that the number has been abstracted from that or many other objects at the first place. This is also the major advantage of mathematics as a discipline, when considering its applications. The advantage of math is that the results obtained by dealing with pure numbers only, can be applied to any kind of objects that have the same count! For instance, 5 + 3 = 8 as a pure math result is valid for any 5 objects and for any 3 objects we have decided to add together, be it apples, cars, pears, rockets, membranes, stars, kisses.
While, as we have seen, pure math doesn't care where the numbers come from, when applying math we do care very much how the counts are generated, where the counts are coming from and where the calculation results will go. We even have invented mechanical, electrical, electronic devices to keep track of these counted objects. We have all kinds of dials that keep track of fuel consumption, temperature, time, distance, speed. Imagine that! We have devices which keep track of counted objects so when we look at them and see number five, or seven, or nine, we will know what that number represents the count of! Say, you have several dials in front of you, and they show all number 5. It is the same number 5, with the same numerical, mathematical, properties, but represents counts of different objects or measurements. We can say that the power of mathematics is derived from noticing that number 5 is the same for many objects and abstracting that number 5 from them, then investigating number 5 properties. After mathematical investigation we can go back, from pure number 5 to the real world!
There are dials in cars, for instance, for fuel consumption, speed, time, engine temperature, ambient temperature, fan speed, engine shaft speed. If it was not up to us, those numbers would float around, enjoying their own purity like, 5, 23, 120, 35, 2.78 without knowing what they represent until we assigned them a proper dial units. This example shows the essential difference between applied and pure math, and how much is up to our thinking and initiative, what are we going to do with the numbers and objects counted or measured. Pure math deals with numbers only, while in applied math we drag the names of objects, associated them with numbers. In other words, we keep track what is counted.
Now, when dealing with pure numbers, we may go to a great extent to investigate all kinds of numerical, mathematical properties of all kinds of numbers and sets of numbers. Hence a spectrum of mathematical areas like linear algebra, calculus, real analysis, etc. These mathematical disciplines are all useful and there is, frequently, a beauty and elegance in their results. But, often, we do not need to apply or use all those mathematical properties, and pure math results, in everyday situations. Excelling in some business endeavour frequently depends on actually knowing what and why something is counted, while, at the same time, mathematics involved, can be quite simple. When I say business, I mean business in usual sense, like finance, trading, engineering, but also, I mean, for instance, as we will see soon, basketball, and even poker.
Let’s go now into a basketball game. When playing basketball we also need to know some math, at least working with positive integers and zero. However, in the domain of basketball game, knowledge of basketball rules are way more important than math,
Those basketball rules are mostly non mathematical. Most of basketball rules do not deal with any kind of quantification, which doesn't make them at all less significant. Moreover, they are way more important ingredient, and represent more complex part, for that matter, of a basketball game, than adding the numbers.
You can posses knowledge of adding integers, but without knowing basketball rules, and without knowing how to play basketball, you will not move anywhere in a basketball team or in a game. Moreover, basketball rules are actual axioms of a basketball game. And, every move in the basketball court, any 30 seconds strategy development by one team or the other, corresponds to theorems of a basketball game! Any uninterrupted part of the game, without fouls or penalties, is an actual theorem proof, with basketball rules as axioms. We can say that basketball rules are those statements that define what belongs to a set "number of scored points"! You see here how we have whole book of basketball game rules that serve the purpose just to define what belongs to a set (of scored points). Compare that to those boring, and sometimes, ridiculous examples, in many math texts, with apples, pears, watermelons (although they may illustrate the point at hand well). With ridiculing the importance of rules of what belongs to a set, belittling their significance and logic associated to obtain them, those authors, unintentionally, pull you away from an essential point of "applied" math. In order to define what belongs to a set, and then, count its elements (like points in basketball) you need to know areas other than math, and to develop logic, creativity, even intuition in those non mathematical areas, in order to decide what really belongs to a set and what needs to be counted. Because, accuracy of rules and logic to determine what belongs to a set dictates the set's cardinality, the size of the set, the number of its elements. And this is the number you will enter in all your calculations later! That number has to be accurate!
Note, also, that only knowing rules of basketball game doesn't make you a first class player, nor your team can be a winner just knowing the rules. You have to develop strategies using those rules. You have to play within those rules a winning game. The same is in math. Knowing the fundamental axioms of math will not make you a great mathematician per se. You have to play the "winning game" inside math too, as you would in basketball game. You have to show creativity in math as well, mostly in specifying theorems, and constructing their proofs!
In business, it is often more important to know where the numbers are coming from than to know in detail the numbers’ properties. For instance, in poker. again, only integers and rational numbers (in calculating probabilities) are involved ( we will skip stochastic processes and calculus for now). You have to remember that the same number 5 can be any of the card suits, and, in addition, can belong to one or more players. Note how abstracting number 5 here and trying to develop pure math doesn't help us at all in the game. We have to go back to the real world rules, in this case world of poker,, we have to use that abstracted number 5 and put it back to the objects it may have been abstracted from, in this case cards and players. You have to somehow distinguish that pure number 5, and associate it with different suit, different player. And strategy you develop, you do with many numbers 5, so to speak, but belonging to different sets, suits, players, game scenarios. Hence, being a successful poker player, among other things, you need to memorize, not exotic properties of integers and functions, but how the same number 5 (or other number) can belong to so many different places, can be associated, linked to different players, suits, strategies, scenarios.
Let’s consider another example, in finance. Any contract you have signed, for instance contract for a credit card, is actual detailed list of definitions what belongs to a certain set. For example, whether $23,789.32 belongs to your account under the conditions outlined in the contract. Note how even your signature is a part of the definition what belongs to a set, i.e. are those $23,789.32 really belong to your account. You see, math here is quite simple, it is just a matter of declaring a rational number 23.789,32, but what sets it belongs to is extraneous to mathematics, it's in the domain of financial definitions, even in the domain of required signatures. Are you, or someone else, is going to pay the bill of $23,789.32, is a non mathematical question (it’s even a legal matter), while mathematics involved is quite simple. It's a number 23,789.32.
Note, when you are paid for your basketball game, suddenly you have math from two domains fused together! It may be that the number of points you scored are directly linked to a number of dollars you will be paid. Two domains, of sport and finance, are linked together via monetary compensation rules, which can have quite a bit of legal background too, and all these (non mathematical in nature!) rules dictate what number, of dollars, may be picked and assigned to you, as a basketball player, after the set of games.