Thursday, February 28, 2013

From Basketball, Financial Math to Pure Math and Back

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.

Wednesday, February 27, 2013

Mathematical Proof for Enthusiasts - What It Is And What It is Not

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 and theorems.

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 applied to.

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.

More About the Concept of a Set and the Concept of a Number

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 starting point.

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 amazing!

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 complex fields?
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 represents.

Sunday, February 24, 2013

Abstract Nature of Geomatrical Figures

One of the fascinating points observations about a circle is that the circle is a pure abstraction. It does not exist really anywhere but in our minds as a perfect abstraction of all points equally distant from a one single point, the circle center. No perfect circle can be found in nature, only approximations of it, and each one will have some imperfections, yet the major theories are based of this unexcited in nature geometrical figure. The same can be said for triangle, square, and most of other geometric figures.

Extrapolating these thoughts to electrical engineering, for example three-phase power systems are built around electric fields that by construction are with phase difference of 120 degrees, no ideal voltage is produced that calculates exactly sin and cos functions for the circuitry analyses (this includes complex numbers, that translates to active and reactive power in electric power systems).

Saturday, February 9, 2013

Function, map, pair in mathematics

As an illustration of a mathematical function concept a teacher can write arbitrary numbers, each on a separate, rectangular piece of paper, and then let the students pair them arbitrarily, on the table, and then investigate numerical, mathematical, properties of the those pairs of numbers. The properties may be what sequence the pairs they can be put in, or the order of numbers magnitudes in different pairs.

In the same way we can pair different fruits with, say, CDs (for whatever  reason!), we can pair numbers together, and even fruits with numbers! Fruits and numbers are paired when an exchange of fruits for money takes place in an open fruit market! Note how is a third agent present when we pair fruits and numbers. It is the exchange "agent" that motivates pairing and that gives sense to the pairing action.

These examples should reinforce main concept that a function is a pair of numbers and not necessarily a formula that gives y for given x. Function is not always output for a given input. Function is not a formula. Function is a map or pairs of numbers.

[ math, mathematics, mathematical function, function, map ]