Mathematics and Real World Applications Links. World #1 and World #2 Approach

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In mathematics, it is all about numbers, sets of numbers, and the sequences of operations on them.

Axioms how to link real world scenarios to mathematical axioms and theorems, have to be defined. The closer you are to the point of complete axiomatization of the real world domains you want to apply math at, the better. Mathematics will reward you with meaningful results. Quantitative aspects of real world axioms, theorems and laws are usually theorems within mathematics.

How that can help you to learn mathematics and the domains where the mathematics is apply to, whatever that means? The answers is that you have to separate the two systems of axioms. The real world system, let's call it World #1 and the pure mathematical system (with its own axioms!), let's call it World #2. Note that there are World #1 axioms (ideally), World #2 axioms and axioms how to link the premises, and eventually theorems, from these two worlds.

If you quantify objects and their relationships in the World #1, the quantities, and their relations, you obtained enter mathematics as initial or boundary conditions, or simply as numbers, set of numbers, pairs of numbers, as the starting points or starting premises. Note that these numerical starting points can be obtained inside math as well, without any extraneous motivation, i.e. within the realm of pure math only, sometimes directly from fundamental axioms.

The interpretation of numerical results is, apparently, up to you. It is you who will keep track of what is counted and why. The logic why you would do certain mathematical operations on the specified numbers, if coming from World #1, has to be firm and, ideally, has to be derived from an firm axiomatic system. Of course, math doesn't care if you axiomatized your real world domain or not. Mathematics will, without asking any questions, follow your instructions for calculations, and give you back results. Interpretation and usage of those results will depend on the correctness of your real world domain assumptions, accuracy of its logic, and  completeness and correctness of the real world domain axioms.

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How Free or Constrained We Are in Applying Mathematics to Real and Fictional Worlds

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There are several ways to “apply” mathematics, or more importantly, to obtain numbers and work with them. Here they are:

1. If you have 3 apples and you say that each one costs $2, how much money you will earn by selling all of them?
2. Measure the distance.
3. Physics laws, initial conditions, results of formula calculations.
4. Harry Potter or Hunger Games story.
5. Mathematical axioms.

The first example arbitrary associates a number with an apple. No measurements or physical law is required. Economical exchange and the quantity to exchange are solely based on human values. The selection of the price is usually how trader perceives the value, and it can be subjective, yet that subjectivity is the only way to go when agreeing on an exchange price of goods.

The second example is selective counting. The same way we define apples and want to count apples (and no other things), we decide we want to count how many of some unit length are in the given distance. We have in advance a unit length, say inches, or meters, and then a distance we want to measure. Note here that measurement is not a part of mathematics. Precision of a measurement is also outside mathematics. It is a method in the realm of physical world, how to count something, in this case length or distance. Measurement implies only that we agreed what and how to count, how to obtain numbers that will enter the numerical world of mathematics, often as pure starting points. Let’s say, we have 1m as a unit, and the length between two tables in a coffee shop. After the measurements we found that the distance between the tables is 2.3m

Let’s compare first and second example. First one has arbitrary numbers put together and multiplication selected as math operation due to need to sell the apples. Hence, math will see: 3, 2, multiply. 3 x 2 = 6. In the second example math will see: 1, and the count 2.3. That’s it. The difference in these two examples is that in the second one you are constrained by the physical distance you want to measure. You also specified the unit of length, 1m. Once these two things are specified, the measurement is not arbitrary. But, note, technically, it was arbitrary which units of length you have selected, and, in a sense, it is arbitrary which distance you want to measure. However, once this is established, selecting numbers is not arbitrary any more, it actually depends on the length and measurement unit.

The third example is a firm physics law. A physics law specifies what needs to be counted and then, very important, the relations between these counts. Are they are to be added, divided, multiplied, etc.. Note how you, in a physics formula, you still deal with counts, but you keep track aside what are those counts of. Now, in physics law, we have even less arbitrary things. It is not arbitrary anymore what needs to be counted (time, force, mass, energy, distance) but also the mathematical relations are firmly established (addition, division, multiplication etc). Interesting things is, mathematics, again, will see these quantities as given as starting point only. Specifying formula is extraneous to math.

Let’s look at  Newton formula F = ma. Virtually, no numbers are given compared to apples and price. What is given then? You are told that if you count mass and count acceleration of a body, then multiply these counts, you will get the quantity of force that is acting on the body. So, where is the freedom here, and where is the law, or constrain? You are completely free to select, arbitrary if you wish, completely up to you, a mass of a body, and acceleration. Example is, you arbitrary chose a car to drive from a dealer’s parking lot, and arbitrary accelerate when on the road, to test it. Of course, when you see other drivers driving their cars, you will have to measure their mass and measure acceleration, i.e. not arbitrary any more, it’s given by other’s driver’s arbitrary selection to you. The formula now tells you that it is the multiplication you have to perform on these two numbers to obtain the force on the car. That’s the value of the formula. A genius is required to select what to count and then to establish, discover, the relationships between these counts. Of course, the very first thing is to want to count something, as oppose to look for some other things in order to explain certain behaviour.

The fourth example, a Harry Potter story, signifies the fact that mathematics can not distinguish real from fictional world. Yes, math can be applied to real life and quantitative relations within physical world are important. But, math deals with numbers you supply to it, and with numbers only. It can not distinguish where these numbers are coming from. It is you who use the math and keep track where the numbers are coming from. Have you really counted, measured something, or just say you think that the number should be like that, math doesn’t care. If Harry Potter flies on his broom with the speed of 5 m/s, what is the distance he will advance after 7 seconds? The result is 5 x 7 = 35. He will fly over the distance of 35m. Note how math did not really care how you specified the numbers. Harry Potter’s broom or a rocket, or from the fictional world of Hunger Games, math does not know where the starting numbers and operations are coming from.

The fifth example tells you that, for math, it is sufficient, just to say, hey, here is the number 5, here is the number 7, do the multiplication and give me the result back. This is axiomatic approach and it is called pure math. Axioms of mathematics, more or less, tell you that the counts and operations are already available, you can pick them and define any sequence of operations on them. This is the fifth way you can obtain and play with numbers. No rockets, no apples, no currency, no physics laws, no length measurements are required to deal with numbers and hence to develop mathematics. Counts are there and you deal with them. One of the values of pure mathematics is that counts, numbers themselves and relations between numbers and sets of numbers, have some interesting properties, and results of that investigation can be used when you obtain numbers by any of the previous four ways, because the results will be applicable in each of them. Like, even if you don’t know what is counted, you will know that 3 + 5 = 8, in pure counts, pure numbers. It is a generally applicable result. For math, only the numbers you provide to it exists. You say here is the number 3, here is the number 5, add them. If this comes from any of the previous four examples, it is you, and not math, who will have to keep track what is counted and why you have chosen addition and not, say, division.

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[ Harry Potter, Hunger Games, applied math, applied mathematics, math and real life, real world math, examples of natural numbers,  counting, number concept,  ]

One More Example to Show What a Number Is and the Search for Truth in Various Disciplines

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Here is one more interesting and quite good example to define a number, to show, in essence, what a number is.Let's say you have three apples on the table. Let's do the following

• Do as many steps as you have apples on the table.
• Count or wait as many seconds as you have apples on the table.
• Count is many pencils as you have apples on the table.
• Do as many push ups as you have apples on the table.

You see, in all these examples, the count is the same, obtained by counting apples. It's number 3, count of 3. In each of those examples you can say that you matched all those objects with apples, in one to one fashion, to make sure there is the same number of each. By this matching, you can determine that the set of objects has the same number of elements as the set of the apples on the table.  Of course, you almost unconsciously used pure numbers, 1, 2, 3 to count other objects.You can see the universality of the concept of a count, number. Same count 3, number 3 is used to count truly any kind of objects.

You can deal separately with pure number 3, without linkage to any of the objects it can represent the count of. That's pure math. Once you start keeping track what you count, applied math kicks in.

Of course, you are always (as in any scientific discipline) interested to find the truth. Here, you may want to be interested to find truths about numbers. That's where logic enters, with its initial assumptions, axioms, theorems, proofs. You, essentially, always want to prove what is true in math. Mathematicians are after the proofs about counts. Mathematicians are after what is true about numbers, counts and their relations. Lawyers are after the proofs what is true with regards to law, moral, what is right or wrong, and with regards to other human values. Physicists are after the truths in physical world, where various forces, energy, motions are central focus in their investigation. Story writers and movie makers are after the true emotions and true moral messages their work will convey and show, even with fictitious plots, i.e. no matter whether the story is fictional or not, the message about human values, be it emotional, moral, must be real and true, and this message will be true if the story line is logically consistent with the story's framework, no matter how fictional that framework may be.

Now, back to the first example, with apples, steps, seconds, pencils, pushups. Mathematics, while apparently common to all those cases, can not define the actual concepts it has counted. What differentiate an apple from a pushup, and a pushup from a pencil, and a pencil from a second is not part of mathematics, and mathematics is, more or less, not part at all of that analysis and those very important relationships. Moreover, it is these non mathematical relationships that define the various disciplines and it is these non mathematical relationships that very often dictate the direction of mathematical development. These relationships dictate what, when, where, and why will be counted, measured, if required at all.

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Interrelations Between Deductive Systems and Inventive, Innovative Thinking

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Looking at one deductive system, call it A, in the context of other deductive systems, can show how these extraneous systems motivates development of the system A. By being "in context" I mean that theorems in one system, which has presence in the logical, conceptual surrounding of the system A, hence providing context for it, are axioms or starting propositions for theorems in the system A. The systems need not to be mathematical only. What is important is that the systems are based on deductive reasoning, and that they are axiomatized as much as  possible. I allow inductive reasoning, and definitely intuition as a method of discovery, but eventually, these both approaches will be morphed into a deductive structure and method. I wouldn't even differentiate inductive reasoning from deductive, but rather call it "dynamic deduction" or "deduction with self error correction".

This kind of deductive systems linking, where contextual nesting and inclusion can go infinitely (i.e. any system that provides context for system A can itself has its own context, etc), is a core of inventive, innovative thinking.

About Number Definition, Pure, and Applied Mathematics

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If we agree that math is about counts, and counts only (as it is, since numbers, counts come from the cardinality of sets and set theory) then geometry doesn’t belong to mathematics. It is, by some authors (mentioned in "What is Mathematics Really", R. Hersh), considered impolite to have any geometric drawing in a mathematical text. Geometry has link to mathematics as the morning purchase of vegetables on the local market has. The geometry is only more convenient (perhaps!) in representing numbers and their relationships. Geometrical figures do only one thing to mathematics – by measuring the distances, angles, etc. we generate numbers, and sets of numbers. None of ZFC axioms refer to anything geometrical in the same way that ZFC axioms do not refer to the bunch of carrots at the local produce markets.

Geometry can help to visualize certain mathematical relationships and results. But, the link between pure numbers and sets to the geometry is in essence arbitrary. Geometric interpretation of mathematical results are neither mandatory nor necessary.

For mathematics, it is completely arbitrary what or who generates numbers. The process of numbers selection, generation, numerical operations can be scientific, guessing, or a product of any dogmatic philosophy. Math couldn’t care less. As for geometry, the reason it has a strong presence in mathematics is just because of some of its practical applications. The reason why we can abstract real world into points, lines, planes, spheres is extraneous to mathematics. For whatever reason a line is drawn, and for that matter, what that line represents abstraction of, is not a part of mathematics. From math point of view we draw lines to generate numbers by measuring the lines’ lengths. Measuring process (with instruments, visually, or in any other way) again, is not part of mathematics. Math will see only the number you obtained.

For example, when we write 2 x 3 = 6 (without any explanation) will the reader know where 2 and 3 came from? Of course not. It can be from 2 baskets, each one having 3 apples. Or, it can be from 2 cars, where each car has 3 passengers. Why do you need a rectangle with sides 2 and 3 to explain you this mathematical result? You don’t need it.

As much as apples, cars, are not part of mathematics, in the same way is not rectangle or any other geometrical figure. Geometry is perhaps interesting because it selects, generates certain sets of numbers that are of interest in everyday applications, like lines, squares, rectangles, circles. It is quantification of these figures and their measures that matter to mathematics, and not figures themselves. The thought  process that takes place in defining a circle as an ideal abstraction of all real world attempts to make a circle (as well as a straight line abstraction of all straight directions) is a nice thing to think about, but that’s not part of mathematics. Once you “idealize” circle, math cares only about the numbers you provide by measuring them.

Simply put, no geometric figure should be considered an element or part of pure mathematics because none of the theorems in math are proven using them. If seemingly geometry terms are used in proofs or appear to be a focus of study, like trigonometry or differential geometry, it is because the axioms of geometry are part of it, but, they are not part of mathematics. Mixing ZFC axioms and geometry axioms is like mixing ZFC axioms and axioms of any other system, including "marbles used in counting", carrots methods of purchase, quantitative finance rules, etc..
 
"The formalist makes a distinction between geometry as a deductive structure
and geometry as a descriptive science. Only the first is mathematical. The use of
pictures or diagrams or mental imagery is nonmathematical. In principle, they
are unnecessary. He may even regard them as inappropriate in a mathematics
text or a mathematics class."  ("What is Mathematics Really" Rueben Hersh)

Of course, it doesn't mean you should not use them to better communicate your ideas, investigate new directions in math or other sciences, or visualize a bit more difficult concepts in mathematics.But, you have to clearly differentiate between mathematics and these non mathematical objects and concepts.

Mathematics is defined not by objects it counts, nor by reasons or logic why those objects are counted, but with concepts used to define mathematics axioms and to define proofs of mathematical theorems.

[ to be continued...]

From Reuben Hersh's book "What is Mathematics Really"

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From Reuben Hersh's book "What is Mathematics Really".

Any proof has a starting point. So a mathematician must start with some
undefined terms, and some unproved statements. These are "assumptions" or
"axioms." In geometry we have undefined terms "point" and "line" and the
axiom "Through any two distinct points passes exactly one straight line." The
formalist points out that the logical import of this statement doesn't depend on
the mental picture we associate with it. Nothing keeps us from using other
words—"Any two distinct bleeps ook exactly one bloop." If we give interpretations
to the terms bleep, ook, and bloop, or the terms point, pass, and line, the
axioms may become true or false. To pure mathematics, any such interpretation
is irrelevant. It's concerned only with logical deductions from them.
Results deduced in this way are called theorems. You can't say a theorem is
true, any more than you can say an axiom is true. As a statement in pure mathematics,
it's neither true nor false, since it talks about undefined terms. All mathematics
can say is whether the theorem follows logically from the axioms.
Mathematical theorems have no content; they're not about anything. On the
other hand, they're absolutely free of doubt or error, because a rigorous proof
has no gaps or loopholes.

The Definition of Number

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After a number of years dealing with mathematics in your primary, secondary school, there still may be a question what the number is. Moreover, unless you are a professional mathematician, with PhD in your resume, I can safely assume that your frustration and fear of mathematics is still present.

What is a number? Seemingly popular approach I am using here does not reduce the strength and significant clarity of the definition. Bear with me, and listen carefully :-) You may find out many interesting things!

Here is the clearest approach to defining number.   

Number is a count.

I will repeat again, number is a count. The purity and significance of this definition can not be emphasized more. While it is simple, it conveys many more important messages than other definitions and approaches you may have read about before. One of the most important message, in my view, of this definition is that it implicitly specifies what you can do with counts. Knowing what you can do with counts, you actually filter out all non mathematical concepts that may be mixed during "bad" mathematical lectures over the years. Also, thinking of numbers as counts, you define what pure mathematics is about! And that can help you answering the questions how math can be applied (about what "applied" means we will see later) in so many different fields, and what differentiate pure and applied math.

What you can do with counts is what mathematics is all about! So, what can you do with counts? You can add them, subtract them, divide, multiply. You can, then, do any number of these operations in any sequence you want. No apples, pears needed to do that! Count 5 is a universal count. It can come from counting apples, pears, cars, atoms, money, steps, seconds. That number 5, count 5 is a universal thing for all of them. While you can eat 5 apples, drive 5 cars, wait 5 seconds, with count 5 you can not do that. But you can add another number 5 to it! or deduct count of 3 from it. Or do any other "counting" operation! Note very important thing -> how you call these counts, i.e. are they integers, positive, negative, odd, even, rational, are just labels we attach and associate to the concept of a count! There is only count we are dealing it all the time.

Looking at count 5 only, i.e. number 5 only, you can not tell whether it came from apples, pears, or counting seconds. So, how you will differentiate count 5 of apples and count 5 of seconds if count, number is actually so universal concept? There is no other way than to keep track by yourself what you have counted. Technically, you will write a small letter beside the number, beside the count to remind you what it is a count of! Or you can remember that in your mind. Whatever works for you.

So, whenever you read about those exotic mathematical concepts, like matrices, determinants, integrals, equations, algebra, arithmetic, you will know one thing - it is all and only about pure counts we have just talked about. There is nothing else there. For instance, a matrix is a set of counts arranged in rectangular fashion on page. But, you do not need even that rectangle. You can just imagine in your mind the same set of counts and differentiate between them in any way you want. It just happened that it was convenient to write those numbers in a rectangular grid on paper! It was just convenience.

The logic, the reasoning in the world of counted objects is separate from the logic that deals only with counts. You can investigate properties of counts only, completely independent from the real world objects they might represent count of. This is the topic of mathematics. To find what is true about counts i.e. numbers. Hence the proof. But note here, we are really interested in counts' characteristics, no matter which objects have been counted! What these characteristics can be? We can have odd numbers, or even! We can have prime counts. Some counts can be divided by others, while some not. But, the numbers are numbers, it is us who give them names to keep track of some of their properties or just we want to deal with some numbers while leaving other numbers alone!. Naming numbers is not a mathematical operation. It just help us describe, label numbers, counts, we want to deal with.

We can compare the way we can deal with numbers to the way a sculptor deals with clay. It is only the clay that he works with and nothing else. Clay! But, what clay represents when it is shaped, what sculpture represents is not about clay! The motivation how the sculptor will twist, press, mold, shape clay is outside clay's world. Same in math! The reason why we add, subtract, divide, or even select numbers to deal with, frequently are outside mathematics! The motivation can come from us buying CDs or from an economist measuring supply and demand, or police measuring speed of the car.

Now, back to sculpture again. The sculpture can represent anything. The similar thing is with numbers. In mathematics we are dealing with numbers only, the same way sculptor deals only with clay! But, if we want to interpret math results and use math in some other fields then we will have to keep track what we have counted, measured, keep track which objects are numbers, counts associated with! That would be called applied math. And, again, the numbers can represent count of many, many different things.

In developing and understanding a subject, axioms come late. Then in the formal presentations, they come early. - Rueben Hersh. #math

The view that mathematics is in essence derivations from axioms is backward. In fact, it's wrong. - Rueben Hersh http://books.google.ca/books?id=R-qgdx2A5b0C&printsec=frontcover&dq=what+is+mathematics+really&hl=en&sa=X&ei=6O9BT5RIqdfRAcyqteMH&ved=0CDsQ6AEwAA#v=onepage&q=what%20is%20mathematics%20really&f=false

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