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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, I have just seen that, a clean and dry concept, a quite straightforward count of objects I have been dealing with. Five apples, five pears, the number five is common to all of them. We have abstracted it, and together with other fellow numbers (three, four, seven, ...) it is a part of a number system we are familiar and we work with. We may have a feeling there are only integers present, and that, really, it is not clear where those mathematicians find so many exotic concepts, so many other numbers, like rational, irrational, and others. Moreover, you may think that, without some real objects to count or 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 real world, to the objects counted, measured, that they are inseparable, that a number, despite its "purity", somehow shares 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 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 positive integers only and that everything else is the work of man. I support that view and essentially many mathematicians do. Here is the flavor of that perspective. Negative numbers are positive numbers with a negative sign. Rational numbers are ratios if two integers. Real numbers (rational and irrational) are limiting values of rational numbers (which are in turn ratios of integers), a sequence of ratios that are smaller and smaller and there are more and more of them, that converge to one value. So, essentially, all these numbers are 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...That's why we need labels below, or beside, numbers, to remind us what is measured, what is counted, because by looking at the number only we can not conclude where the count comes from. I have written about this in my previous posts. When you write 5 + 3 = 8, you can apply this result to any number of objects with these counts. So, numbers do not hold or hide properties of the objects they are counts of. As a 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.. 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. It works with clear, pure numbers, and numbers only. It is a very important conclusion. You may think, that properties of numbers depends on the objects that have generated them, and there are no other properties of numbers other than describing them as a part of real world objects. But, it is not so. Properties of numbers don't depend on the objects or processes that have generated them. While you can have a rich description of objects and millions of colorful reasons why you have counted five objects, 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, 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 why we have counted apples, or why we have turned left on the road and then drive 10 km. Pure math is only interested in numbers provided to it. Among those properties of number 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. 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 calculations, they 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 eight 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 number 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 has 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.
While, as we have seen, pure math doesn't care where numbers come from, when applying math we do care very much. We care so much what we have counted that we have invented devices to keep track of these counted objects. We have dials that keep track of fuel, temperature, time, distance, speed. Imagine, we have devices which keep track of kind of counted objects so when we look at them and see number five, we will know what that number five represent the count of! Say you have four dials in front of you, and they show all number 5. It is the same number five, with the same properties, and we can say that the power of mathematics is derived from noticing that number 5 is the same for many object and abstracting that number 5 from them, then investigating its properties. Now, here, we went back! We used that universal number five, and keep track in our dials, which exact objects the number 5 represents the count of.
In applied math it is so important to keep what is counted that we have invented dials for those countable objects or measures. There are dials in car, 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, like, 5, 2300, 120, 35, 2.78 without knowing what they represent until we assigned them a proper dial units. This example shows the essence of 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.
Now, we may go to great extent to investigate all kinds of properties of all kinds of numbers and sets of numbers. Hence, linear algebra,calculus, real analysis. They are all useful and sometimes very elegant parts of mathematics. But, frequently, we do not need all those properties to apply or use pure math results in everyday situations. Excelling in some business endeavor 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, but also, I mean, for instance, as we will see soon, basketball, and even poker.
When playing basketball we also need to know some math, at least dealing with positive integers and zero. However, knowledge of basketball rules are way more important than math, in basketball domain.
Those basketball rules are mostly non mathematical, which doesn't make them at all less significant. Moreover, they are way more important ingredient, and more complex part, for that matter, of a basketball game, than adding the numbers. You can have knowledge of adding integers, but without knowing basketball rules, and know how to play basketball, you will not move anywhere in a basketball team or in the game. Moreover, basketball rules are actual axioms of a basketball game. And, every move in the basketball court, any 30 seconds strategy developed by one team or the other, corresponds to theorems of the 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 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 it (like points in basketball) you need to know areas other than math, and to develop, logic, creativity, even intuition in those 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! 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 "winning game" inside math too, you have to show creativity in math as well, as you would in basketball game!
In business it is often more important to know where the numbers are coming from than to know in detail their properties. For instance, in poker. again, only integers are involved. 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, we have to use that abstracted number 5 and put it back to the objects it may have been abstracted from. 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 (players, suits, strategies, scenarios).
Another example is finance. Any contract you have signed, say for a credit card, is actual detailed definition what belongs to a set, i.e. 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 a rational number 23.789,32, but what sets it belongs to is outside mathematics, it's a domain of financial definitions. Are you going to pay the bill of $23,789.32, or someone else, is a non mathematical question, while mathematics involved is quite simple, it's number 23,789.32.
Now, note, when you are paid for your basketball game, suddenly you have math from two domains put together! It may be that the number of points you scored are directly linked to a number of dollars you may 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 after the set of games.
A theorem in one system is usually an axiom in another. Which systems to link in this way is in the center of innovative thinking.
[ applied math, applied mathematics, math applications, poker, basketball, financial math, poker and math, math concepts, financial mathematics ]
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, I have just seen that, a clean and dry concept, a quite straightforward count of objects I have been dealing with. Five apples, five pears, the number five is common to all of them. We have abstracted it, and together with other fellow numbers (three, four, seven, ...) it is a part of a number system we are familiar and we work with. We may have a feeling there are only integers present, and that, really, it is not clear where those mathematicians find so many exotic concepts, so many other numbers, like rational, irrational, and others. Moreover, you may think that, without some real objects to count or 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 real world, to the objects counted, measured, that they are inseparable, that a number, despite its "purity", somehow shares 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 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 positive integers only and that everything else is the work of man. I support that view and essentially many mathematicians do. Here is the flavor of that perspective. Negative numbers are positive numbers with a negative sign. Rational numbers are ratios if two integers. Real numbers (rational and irrational) are limiting values of rational numbers (which are in turn ratios of integers), a sequence of ratios that are smaller and smaller and there are more and more of them, that converge to one value. So, essentially, all these numbers are 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...That's why we need labels below, or beside, numbers, to remind us what is measured, what is counted, because by looking at the number only we can not conclude where the count comes from. I have written about this in my previous posts. When you write 5 + 3 = 8, you can apply this result to any number of objects with these counts. So, numbers do not hold or hide properties of the objects they are counts of. As a 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.. 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. It works with clear, pure numbers, and numbers only. It is a very important conclusion. You may think, that properties of numbers depends on the objects that have generated them, and there are no other properties of numbers other than describing them as a part of real world objects. But, it is not so. Properties of numbers don't depend on the objects or processes that have generated them. While you can have a rich description of objects and millions of colorful reasons why you have counted five objects, 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, 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 why we have counted apples, or why we have turned left on the road and then drive 10 km. Pure math is only interested in numbers provided to it. Among those properties of number 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. 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 calculations, they 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 eight 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 number 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 has 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.
While, as we have seen, pure math doesn't care where numbers come from, when applying math we do care very much. We care so much what we have counted that we have invented devices to keep track of these counted objects. We have dials that keep track of fuel, temperature, time, distance, speed. Imagine, we have devices which keep track of kind of counted objects so when we look at them and see number five, we will know what that number five represent the count of! Say you have four dials in front of you, and they show all number 5. It is the same number five, with the same properties, and we can say that the power of mathematics is derived from noticing that number 5 is the same for many object and abstracting that number 5 from them, then investigating its properties. Now, here, we went back! We used that universal number five, and keep track in our dials, which exact objects the number 5 represents the count of.
In applied math it is so important to keep what is counted that we have invented dials for those countable objects or measures. There are dials in car, 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, like, 5, 2300, 120, 35, 2.78 without knowing what they represent until we assigned them a proper dial units. This example shows the essence of 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.
Now, we may go to great extent to investigate all kinds of properties of all kinds of numbers and sets of numbers. Hence, linear algebra,calculus, real analysis. They are all useful and sometimes very elegant parts of mathematics. But, frequently, we do not need all those properties to apply or use pure math results in everyday situations. Excelling in some business endeavor 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, but also, I mean, for instance, as we will see soon, basketball, and even poker.
When playing basketball we also need to know some math, at least dealing with positive integers and zero. However, knowledge of basketball rules are way more important than math, in basketball domain.
Those basketball rules are mostly non mathematical, which doesn't make them at all less significant. Moreover, they are way more important ingredient, and more complex part, for that matter, of a basketball game, than adding the numbers. You can have knowledge of adding integers, but without knowing basketball rules, and know how to play basketball, you will not move anywhere in a basketball team or in the game. Moreover, basketball rules are actual axioms of a basketball game. And, every move in the basketball court, any 30 seconds strategy developed by one team or the other, corresponds to theorems of the 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 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 it (like points in basketball) you need to know areas other than math, and to develop, logic, creativity, even intuition in those 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! 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 "winning game" inside math too, you have to show creativity in math as well, as you would in basketball game!
In business it is often more important to know where the numbers are coming from than to know in detail their properties. For instance, in poker. again, only integers are involved. 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, we have to use that abstracted number 5 and put it back to the objects it may have been abstracted from. 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 (players, suits, strategies, scenarios).
Another example is finance. Any contract you have signed, say for a credit card, is actual detailed definition what belongs to a set, i.e. 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 a rational number 23.789,32, but what sets it belongs to is outside mathematics, it's a domain of financial definitions. Are you going to pay the bill of $23,789.32, or someone else, is a non mathematical question, while mathematics involved is quite simple, it's number 23,789.32.
Now, note, when you are paid for your basketball game, suddenly you have math from two domains put together! It may be that the number of points you scored are directly linked to a number of dollars you may 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 after the set of games.
A theorem in one system is usually an axiom in another. Which systems to link in this way is in the center of innovative thinking.
[ applied math, applied mathematics, math applications, poker, basketball, financial math, poker and math, math concepts, financial mathematics ]
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