About Mathematics and Real World Mathematics Applications
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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
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'
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
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
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
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
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.
Important things you can learn from mathematics are not about counting only,
but also about mathematics’ methods of discovering new truths about numbers.
With the term mathematical proof we
want to indicate a logical proof, i.e. proof using logical inference rules, in
the field of mathematics, as oppose to other disciplines or area of human
activity. Hence, it should really be “a proof in the field of mathematics”.
Also, we have to assume, and be fully aware, that proof must be “logical”
anyway. There are really no illogical proofs. Proof that appears to be obtained
(whatever that means) by any other way, other than using rules of logic, is not
a proof at all.
Assumptions and axioms need no proof. They are starting points and their
truth values are assumed right at the start. You have to start from somewhere.
If they are wrong assumptions, axioms, the results will show to be wrong. Hence,
you will have to go back and fix your fundamental axioms.
Often when you have first encountered a need or a task for a mathematical
proof, you may have asked yourself "Why do I need to prove that, it's so obvious!?".
We used to think that we need to prove something if it is not clear enough
or when there are opposite views on the subject we are debating. Sometimes,
things are not so obvious, and again, we need to prove it to some party.
In order to prove something we have to have an agreement which things we
consider to be true at the first place, i.e. what are our initial, starting
assumptions. That’s where the “debate” most likely will kick in. In most cases,
debate is related to an effort to establish some axioms, i.e. initial truths, and
only after that some new logical conclusions, or proofs will and can be obtained.
The major component of a mathematical proof is the domain of mathematical
analysis. This domain has to be well established field of mathematics, and
mathematics only. The proof is still a demonstration that something is true,
but it has to be true within the system of assumptions established in mathematics. The true statement, the proof, has to
(logically) follow from already established truths. In other words, when using
the phrase "Prove something in math..." it means "Show that it
follows from the set of axioms and other theorems (already proved!) in the
domain of math..".
Which axioms, premises, and theorems you will start the proof with is a matter of art, intuition, trial and error, or even true genius. You can not use
apples, meters, pears, feelings, emotions, experimental setup, physical
measurements, to say that something is true in math, to prove a mathematical
theorem, no matter how important or central role those real world objects or
processes had in motivating the development of that part of mathematics. In
other words, you can not use real world examples, concepts, things, objects, real
world scenarios that, possibly, motivated theorems’ development, in
mathematical proofs. Of course, you can use them as some sort of intuitive guidelines
to which axioms, premises, or theorems you will use to start the construction of a proof. You can use your intuition,
feeling, experience, even emotions, to select starting points of a proof, to
chose initial axioms, premises, or theorems in the proof steps, which, when
combined later, will make a proof. But, you can not say that, intuitively, you
know the theorem is true, and use that statement about your intuition, as an
argument in a proof. You have to use mathematical axioms, already proved mathematical
theorems (and of course logic) to prove the new theorems.
The initial, starting assumptions in mathematics are called fundamental
axioms. Then, theorems are proved using these axioms. More theorems are proved
by using the axioms and already proven theorems. Usually, it is emphasized that
you use logical thinking, logic, to prove theorems. But, that's not sufficient.
You have to use logic to prove anything, but what is important in math is that
you use logic on mathematical axioms,
and not on some assumptions and facts outside mathematics. The focus of your
logical steps and logic constructs in mathematical proofs is constrained (but
not in any negative way) to mathematical (and not to the other fields’) axioms
Feeling that something is "obvious" in mathematics can still be a
useful feeling. It can guide you towards new theorems. But, those new theorems
still have to be proved using mathematical concepts only, and that has to be
done by avoiding the words "obvious" and "intuition"!
Stating that something is obvious in a theorem is not a proof.
Again, proving means to show that the statement is true by demonstrating it
follows, by logical rules, from established truths in mathematics, as oppose to
established truths and facts in other domains to which mathematics may be
As another example, we may say, in mathematical analysis, that something is
"visually" obvious. Here "visual" is not part of
mathematics, and can not be used as a part of the proof, but it can play important
role in guiding us what may be true, and how to construct the proof.
Each and every proof in math is a new, uncharted territory. If you like to
be artistic, original, to explore unknown, to be creative, then try to
construct math proofs.
No one can teach you, i.e. there is no ready to use formula to follow, how
to do proofs in mathematics. Math proof is the place where you can show your
true, original thoughts.
For instance, let's take a look at the cars on a highway, apples on a table,
coffee cups in a coffee shop, apples in the basket. Without our intellect
initiative, our thought action, will, our specific direction of thinking, objects
will sit on the table or in their space, physically undisturbed and conceptually
unanalyzed. They are and will be apples, cars, coffee cups, pears. But then, on
the other hand, we can think of them in any way we wish. We can think how we
feel about them, are they edible, we can think about theory of color, their
social value, utility value, psychological impressions they make. We can think
of them in any way we want or find interesting or useful, or we can think of
them for amusement too. They are objects in the way they are and they need not
to be members of any set, i.e. we don’t need to count them.
Now, imagine that our discourse of thought is to start thinking of them in
terms of groups or collections, what whatever reason. Remember, it's just came
to our mind that we can think of objects in that way. The fact that the apples
are on the table and it looks like they are in a group is just a coincidence. We
want to form a collection of objects in our mind. Hence, apples on a table are
not in a group, in a set yet. They are just spatially close to each other.
Objects are still objects, with infinite number of conceptual contexts we can
put them in.
Again, one of the ways to think about them is to put them in a group, for
whatever reason we find! We do not need to collect into group only similar
objects, like, only apples or only cars. Set membership is not always dictated
by common properties of objects. Set membership is defined in the way we want
to define it! For example, we can form set of all objects that has no common
property! We can form a group of any kind of objects, if our criterion says so.
We can even be just amused to group objects together in our mind. Hence, the
set can be specified as “all objects we are amused to put together”. Like, one
group of a few apples, a car, and several coffee cups. Or, a collection of
apples only. Or, another collection of cars and coffee cups only. All in our
mind, because, from many directions of thinking we have chosen the one in which
we put objects together into a collection.
Without our initiative, our thought action, objects will float around by
themselves, classified or not, and without being member of any set! Objects are
only objects. It is us who grouped them into sets, in our minds. In reality, they are still objects,
sitting on the table, driven around on highways, doing other function that are
intrinsic to them or they are designed for, or they are analyzed in any other
way or within another scientific field.
Since, as we have seen, we invented, discovered a direction of thinking
which did not exist just a minute ago, to think of objects in a group, we may
want to proceed further with our analysis.
Roughly speaking, with the group, collection of objects we have introduced a
concept of a set. Note how arbitrary we even gave name to our new thought that
resulted in grouping objects into collections. We had to label it somehow.
Let's use the word set!
Now, if we give a bit more thought into set, we can see that set can have
properties even independent of objects that make it. Of course, for us, in real
world scenarios, and set applications, it is of high importance whether we
counted apples or cars. We have to keep tracks what we have counted. However,
there are properties of sets that can be used for any kind of counted objects.
Number of elements in a set is such one property. If we play more with counts
and number of elements in a set we can discover quite interesting things. Three
objects plus six objects is always nine objects, no matter what we have counted!
The result 3 + 6 = 9 we can use in any set of objects imaginable, and it
will always be true. Now, we can see that we can deal with numbers only,
discover rules about them, in this case related to addition that can be used
for any objects we may count.
Every real world example for mathematics can generate mathematical concepts,
mainly sets, numbers, sets of numbers, pair of numbers. Once obtained, all
these pure math concepts can be, and are, analyzed independently from real
world and situations. They can be analyzed in their own world, without
referencing any real world object or scenario they have been motivated with or
that might have generate them, or any real world example they are abstracted
from. How, then, conception of the math problems come into realization, if the
real world scenarios are eliminated, filtered out? Roughly speaking, you will
use word “IF” to construct starting points. Note that this word “IF” replaces
real world scenarios by stipulating what count or math concept is “given” as the
But, it is to expect. Since a number 5 is an abstracted count that
represents a number of any objects as long as there are 5 of them, we can not,
by looking at number 5, tell which objects they represent. And we do not need
to that since we investigate properties of sets and numbers between themselves,
like their divisibility, which number is bigger, etc. All these pure number
properties are valid for any objects we count and obtain that number! Quite
Moreover, even while you read a book in pure math like "Topology
Fundamentals" or "Real Variable Analysis" or "Linear
Algebra" you can be sure that every set, every number, every set of
numbers mentioned in their axioms and theorems can represent abstracted
quantity, common count, and abstracted number of millions different objects
that can be counted, measured, quantified, and that have the same count denoted
by the number you are dealing with. Hence you can learn math in the way of thinking
only of pure numbers or sets, as a separate concepts from real world objects,
knowing they are abstraction of so many different real world, countable objects
or quantifiable processes (with the same, common count), or, you can use,
reference, some real world examples as helper framework, so to speak, to
illustrate some of pure mathematical relationships, numbers, and sets, while
you will still be dealing, really, with pure numbers and sets.
There may be, also, a question, why it is important to discover properties
of complements, unions, intersections, of sets, at all? These concepts look so
simple, so obvious, how such a simple concepts can be applied to so many
Let’s find out! Looking at sets, there is really only a few things you can
do with them. You can create their unions, intersections, complements, and then
find out their cardinalities, i.e. sizes of sets, how many elements are there
in a set. There is nothing else there. Note how, in math, it is sufficient to
declare sets that are different from each other, separate from each other. You
don’t have to elaborate what are the sets of, in mathematics. You do not even
need to use labels for sets, A, B, C,… It’s sufficient to imagine two (or more)
different sets. In mathematics, there are no apples, meters, pears, cars,
seconds, kilograms, etc. So, if we remove all the properties of these objects,
what properties are left to work with sets then? Now, note one essential thing
here! By working with sets only, by creating unions, complements, intersections
of sets, you obtain their different cardinalities.
And, in most cases, we are after these
cardinalities in set theory, as one of the major properties of sets, and
hence in mathematics. Roughly speaking, cardinality is the size of a set, but
also, after some definition polishing, it represents a definition of a number
too. Hence, if we get a good hold on union, complement, intersection
constructions and identity when working with sets, we have a good hold on their
cardinalities and hence counts and numbers. And, again, that's what we are
after, in general, in mathematics!
As for real world examples, you may ask, how distant is set theory or pure
mathematical, number theory from real world applications? Not distant at all.
Remember the fact how we obtained a number? A number is an abstraction of all
counted objects with the same count, of all sets of objects with the same
number of elements (apples, cars, rockets, tables, coffee cups, etc). Hence,
the result we have obtained by dealing with each pure, abstracted number can be
immediately applied to real world by deciding what that count represents or
what objects we will count that many times. Or, the other way is, even if we
dealt with pure math, pure numbers all the time, we would've kept track what is
counted, with which objects we have started with. There is only one number 5 in mathematics, but in real world
applications we can assign number 5 to as many objects as we want. Hence, 5
apples, 5 cars, 5 rockets, 5 thoughts, 5 pencils, 5 engines. In real world math
applications scenarios it matter what you have counted. But that fact and
information, what you have counted (cars, rockets, engines, ..) is not part of
math, as we have just seen. Math needs to know only about a specific number
obtained. Number 5 obtain as a number of cars is the same as number 5 obtained
from counting apples, from the mathematical point of view. But, it can and does
represent sizes of two sets, cars and apples. For math, it is sufficient to
write 5, 5 to tell there are two counts, but for us, it is practical to drag a
description from the real world, cars, apples, to keep track what number 5
I want to show methods how we can conceptualize a number in
order to actually understand what it is and how it is used in real life or in
This introductory book should provide multiple benefits for
anyone interested in a deeper, more fundamental understanding of mathematics or
for a reader who wants to refresh or upgrade her knowledge in mathematics at
school or at workplace. If you ever have wondered where all those theorems come
from, can I create a theorem, how come math is so wide in scope, what are the
roots of mathematical thinking, this book is for you. The book will, hopefully,
put all your previous mathematical knowledge on firm and correct footing. It
can provide answers about what exactly is the subject of investigation and
research in mathematics and what mathematics is all about. The book should
provide firm, in depth intellectual tools for understanding a quantification
process that can be used virtually in any area of human activity such as
economics, finance, engineering, space sciences, physics, sales, planning, etc.
The explanation of a number concept, of a number definition can
be a useful and effective starting point for all those creative minds who ask
“why mathematics?” “what for we have to calculate all that?”, and “what is the
So, let’s start. Let’s say that we can conceptually,
visually, in our minds, distinguish objects among themselves, and that we can,
and then that we want, to count those distinct objects.
When we, for whatever reason, group objects into some
collections (they don’t have to be of the same kind nor similar at all), we can
be in a position to determine which collection has more objects, if we want to.
Don’t forget, we don’t have defined yet any concept of numbers, i.e. labels,
names for counts that reflect the size of a collection, its number of elements.
How, then, we can determine which set, which collection has
more elements, objects?
The essential method is to compare collections of objects by
pairing, matching, elements from one collection, with elements, objects from
another collection. Pairing has to be, obviously, one to one.
Let’s say, we have a set of pens, set of apples, and a set
of bananas, as shown in the Picture 1.
We can start first by determining which, if any, set has
In order to do that,we need to match, to pair, one object from one set with only one objectfrom another set. If no objects are left
unmatched then two sets have the same number of elements. As we can see in the
Picture 1, by pairing apples and pens, and apples and bananas, no objects are
left unmatched, hence these three sets have the same number of elements. We do
not have the name yet for that count, for that number, but, the good thing is
we know what we are talking about! We are talking about certain number of
elements, defined by the exact match of these
three collections. That property is what we are after! That “numberness” is
what we are looking for to capture and define.
Now, note one very interesting thing. If we replace bananas
with cars, and pair, match every car with every pen, we can see that the match
is again achieved! The pairs are again complete and no cars or pens or apples
are left unmatched. Hence, these two collections, two sets, have the same
number of elements.
If we add a car then we can see that sets of cars has more
elements than the set of apples or set of pens. Or, if we add a pencil, we can
see that set of pens has more elements than the other two sets.
It is this property, this “numberness”, common in pairing
two or more sets, that we call a number.
You see, no matter what kind of objects are in two sets, if they match, that
property, is the actual number. That
is the concept of a number! It is, at this point, completely arbitrary how we
are going to label this numerical concept we have just discovered and defined.
Word, symbol, reference for it is completely arbitrary. We talk about number three
here. In English, it is called number three,
and the numeric symbol is 3. In
other languages it can have different name and nomenclature for it may be
different as well. Label for our new concept is really of very minor
significance at this point. It is the concept of “numberness” that adjective,
that property the sets have, the property sometimes used as a noun number, is way more important than the tag, than that curvy
trail of ink on the paper we use to reference it. What is important is what the
trail of ink on paper represents and not ink itself.
Let’s go back to our quantification adventure.
Of course, we could start with three object and obtain
number five, or seven objects, and obtain number seven, etc. Notice how we, now,
have this set property to work with. set property related to its number of
elements, the quantity of objects, the pure count of objects, that can be
abstracted (because that count is the same for all sets having that number of
elements). We have abstracted a property, quantity from any two sets of
objects, that we can call a number, or a count! That’s the actual concept of a
Notice, also, how “number” in its essence, is not even a
noun, but more like an adjective, that describes “quantitative” aspect of two
sets, the number of paired elements of two collections.
Any time you have number in your mind first, and only then
you decide what you are going to count, you defined the number what it is.
As long as we know that these labels, symbols,
1,2,3,…represent that property of one to one pairing between the elements of two
sets (with the goal to determine if they have the same number of elements or
not) we are on a good path to work with numbers and quantification.
You may ask at this point “well, I don’t always see two sets
when I count objects of one set, I just count them without any pairing with the
elements of another set”. Good question! What you actually do, by, say,
counting CDs, or lemons, in your collection, is matching them with the set of
natural numbers in your mind, which is completely ok. But, note, you have
natural numbers at your disposal to use them for counting other objects, while
in our previous explanation we were actually just defining the numbers! We were
after very definition of a number. Once the numbers are defined, as we did for
number three, 3, you can use that number three and other numbers in counting
any kind of objects!
Look at that number five, say, the universal count of five,
for any kind of objects. One more interesting conclusion follows. You see, how
at this point, we can deal with counts only! We can deal with a count 3 and a
count 5, which we call a number 3 and a number 5, regardless which objects they
may represent count of! When we want to add them, it will always be 3 + 5 = 8,
no matter what we have counted! Apples, pears, cherries, lemons, their taste,
texture, color, cannot change the numerical result 3 + 5 = 8. That’s one of the
beautiful sides of mathematics. The search for truth about pure quantitative
relations. And this is exactly what pure math is about! Beauty of the applied
math, on the other hand, is in the challenge to find all kinds of relationships
between objects and concepts that need to be quantified.
From the application point of view, we, now, can use our generalized
knowledge that 3 + 5 = 8, and utilize it any real world situation. For
instance, if we have 3 cars and we buy 5 more cars, we will have 8 cars.
Note that “purity” of math is just related to the fact that
we do not care what we have just counted. We were only interested in adding,
subtracting, dividing pure counts, pure numbers we have abstracted from real
world objects counted.
Mathematicians have an exotic term for the number of
elements in a set, for the set’s size – it’s cardinality of a set.
You may ask “I can just start counting and continue counting
objects without putting them into any set. I can even stop counting at any
arbitrary time and still get a count, without specifying any set”. It is
actually completely true. You don’t have to have defined set first then count
elements within it to obtain a number. Technically, you are forming set “on the
go”, set whose elements can be defined as “anything I can see around me I can
put in set and count right then and there”.
This question is also interesting from another point of
view. In slightly different approach, you can start with number 5 and then
count any object you see around until you complete five counts. You see, at
this point, you used pure math in real world scenarios, perhaps even
unconsciously! You started with a pure number five, and it was up to you what
are you going to count. Of course in physics, engineering, economics, trading,
it matters what you count! That’s why we have to drag units beside pure numbers
to remind us what we have counted
and what we may want to count when we go back from pure mathematical calculations
to the real world. More about that in a second.
If we want to mix pure math and real world scenarios and
objects we are counting, it’s easy, but, of course, it has to be carefully done!
What we need to do is to put a small letter beside the number, to keep track
what we have counted. Hence, in physics we have 3m + 5m = 8m, for distance,
length in meters. Then, also we can put 3 apples + 5 apples = 8 apples in
agriculture studies. We essentially do two steps here, during the additions of
real objects. When we want to add 3m + 5m, we actually separate pure numbers
from the meters counted, we enter with these two numbers the world of pure
math, where we do calculation of pure numbers only, 2 + 3 = 8. Then we go back
to real world of meters (because we have those small letters to remind us what
we have counted) with the result 8, and associate the name of the object, in
this case it’s a physical unit of length, or distance, which is meter, (m), to
the number 8. And, voila, we have just used pure math in the real world