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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.
