Flow of Quantification Results - Pure and Applied Math

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When you specify how much of some objects you need or want to count, when you first have a pure number in mind and only after that you chose the objects to count, based on that number, you bridge the space and connect pure and applied math. The number you had in your mind belongs to pure math domain, while the quantifiable objects you decide to count, together with the chosen number, belong to the applied math domain.

Applying mathematics means, as per the illustration, filtering out the units, objects counted, and dealing with pure sets and counts, numbers only. With these numbers and quantitative relations you enter the world of pure math, obtain the results, by doing new calculations or by using already proven theorems, and then return back the result to the real world, reattaching the units on the way back.

But, it also means, that the logic, reasoning within pure mathematics, similar to the chain of political reasoning and decisions before a certain action is taken, is important when it is required to know exact quantity that will be used in the real world scenario. Accuracy of pure mathematical processing, calculations, proofs, theorem resuse, is a significant, a central factor to obtain a correct number and hence go ahead with some directive how much of some objects need to be counted. Of course, initial conditions, numbers entered the pure math mechanisms, are coming from quantification in the real world, and that link will be the major connection for units reattachment and decision what to count and at which magnitude.

Hence, the importance of theorems, theorem proofs, although they may seem too abstract and distant from real world application at the first sight, has central role in obtaining accurate results that will be used back in decision making in real world domain from which initial conditions originated.

Quantification result, or the obtained number, need not to be used in mathematical domain only or to quantify something else. It can be used, as a true or false fact, or as an information, as a part of any decision making process of any domain, in any hybrid axiomatic system, or any logical system as a part of logical expression and in any logical connective.

Set Theory, Units and Why We Can Multiply Apples and Oranges but We Cannot Add Them

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You have, probably, been told that you can not add apples and oranges. Why is that? But, you may realized or may have been taught that you can multiply them. How is that possible? And this state of affairs may be following you through your school, education, and even (non-mathematical!) career. Here is the explanation.

Let’s say there are some apples on the table and let’s say we want to count them. We decide we want to count apples. And there we go. Suppose there are eight apples on the table, and we correctly count them, thus obtaining count of eight. Eight apples. The most important thing here is what we decide to count. We decided to count apples. And nothing else. It is the apples count we are interested in and not any other objects. That’s our definition what belongs to our set, specific set that (qwe decided!) will contain apples only. Now, if we see pencils on the tables, pears, oranges, books, they don’t match our definition, they are not apples, and hence they will not be added to our “set”. That’s the reason we “cannot” add oranges and apples. It is our decision that we want to count apples only, and our decisions if more apples are put on the table we will add them.

If we decide that our set, the things we want to count, will have other objects and that we want to have a total number of objects we are interested in, then we have to specify that in our definition. We have to say, now, that we have decided to count, as members of our set, say apples, books, and oranges. We may not be interested at all how many of each are there, we just want their total number. IN this case, we clearly can add apples, books, and oranges together, because it is our definition of  what belongs to a set that determines elements and number of elements in that set. And, with this set definition, we clearly can add apples and oranges, and  for that matter any object we decide will belong to our set of interest.

Let’s see again in which scenario apples and oranges can be added again. Suppose that, on our table, we have 8 apples, 5 books, 7 oranges, and 3 pencils. And suppose  that we define the set as “count all fruits on the table”. IN that case we will not count books and pencils, but we will correctly add together apples and oranges, because they are fruits and that’s the definition of a  set membership. Hence, our set will have 8 fruits (apples) and 7 fruits (oranges), giving the sum of 15 fruits. 

The conclusion is that the set membership definition determines what will belong to a set, what kind of objects, and that this definition will determine which objects we can add together. Definition of the set membership is essential to determine which objects we can count together.

Ok, so, we clarified that, when the set definition says “count only apples” we can not add apples and oranges together. But, when you say “multiply apples and oranges” we can do that. Why?

The answer lies in the two step process we always do, but we may not be aware of that. And, in some language imprecision as well. You do not multiply apples and oranges, You multiply the numbers obtained by counting apples and oranges. Let’s suppose you want to multiply 5 apples with 3 oranges. But then, let’s, for a moment, focus only on 5 apples. Or, even there is a basket of apples, say around 30, beside the table. You can say “I want 5 apples on the table”. You take five apples from the basket and put them on the table. Now, you can say, I want 3 times 5 apples on the table. Then you  take, from the basket, groups of 5 apples, 3 times. You essentially took 3 x 5 = 15 apples from the basket. But, where that number 3 came from. Ok, you can say, and you will be right, it came from your head,  you just imagined number 3 and decided to count 3 x 5 = 15 apples from the basket. So, you have this, 3 x 5apples = 15apples. But, notice! While you arbitrarily imagined that number 3, it can also come from counting another objects! You can say, you have counted people in the room, there were 3 of them and each of them will have to have 5 apples. Hence, you obtained number 3, this time not from your head, but from real counting of the people in the room. And, again you will have 15 apples on the table, from the basket. We can write that as 3 people x 5 apples = 15 [ people x apples ] . The “unit” here is [people x apples ] and essentially it tells us HOW we have obtained numbers used in the multiplication! By these “units” we keep track what we have counted. So, it is not at all that we have “multiplied people and apples”, but that we have multiplied numbers obtained by counting people and apples. If we use oranges and apples, and say, I want to put 5 apples beside each of 3 oranges, how many apples I will need to take out of the basket, the answer will be 3 x 5 = 15 [oranges x apples ].

Only numbers can be multiplied, added, divided, subtracted. Objects, concepts, like apples, oranges, people, cars, pencils, books, can not be ‘multiplied”, they can be counted only. By counting them we obtain the numbers to work with. It is with these  numbers only that we do mathematical operations. You know that  4 + 3 = 7 no matter if we count apples, or oranges, or cars. It is an universal result. When you write down 4 + 3, a friend beside you doesn’t know are you counting in your head CDs, dollars, or apples, but he knows that the result will be 7 no matter what.

The Links Between Different Axiomatic Systems and Cross-Axiomatic Ideas Generation

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Combinatorial creativity needs to recognize the cross-axiomatic links between partially or fully axiomatized systems combined. Puncturing the axiomatic membranes around conceptually delineated ideas using cross-axiomatic probing needle is the key to creative thinking.

The systems I talk about here need not to be mathematical at all. Axioms and theorems can be a part of any system that uses logic.

Let's take a look at the drawing. Small red, blue, green squares are theorems. The diagram shows that each system, S1, S2, S3, S4, S5, is axiomatized. The system itself is developed from the set of axioms, as indicated by the rectangle where they reside. It can be seen how the theorems (small squares) are derived from the axioms and other theorems. Even a proof is indicated to show that connection.















































Now, notice how the theorems from system S1 influence the theorem conception, even definition, in the system S2. Example can be the processes in physics that motivated developments in mathematics, but there are many other examples between other fields as well. We can see that these small squares have two fold connections. Let's look at the system S2. One line comes from system S1, and another comes from S2's own axioms. That fact shows one of the most important thing, and it is that theorem in one system (S2) can be motivated by the other system, system extraneous to S2, in this case the system S1, but also that those theorems can be defined directly from the axioms in the system S2. The proof of these theorems, in system S2, can come later, and if needed, can be and must be constructed only from the axioms (and already proved theorems) in the system S2. That leads us to the second most important point. The theorems in one system, example here is S2, must an can be proved only using axioms and already proved theorems from the system S2, no matter how clear and inspiring motivation and illustration is coming from the system S1.

But, in order to advance in the creative work, we don't have always time to prove each and every step or a conclusion. Most often we would accept that the theorems, say in system S2, give real, true, correct consequences when the results are applied to the system S1. Example of this can be mathematics applied in physics or engineering. As Reuben Hersh wrote, "controlling a rocket trip to the moon is not an exercise in mathematical rigor.". This connection and method to accept that a theorem is true in one system as long as it gives and has the correct and desired, true consequences in another, linked system, that uses the theorems, is essential for our uninhibited, creative, combinatorial thinking, the use of our intuition that proved so successful in many scientific discoveries and engineering inventions.

When we take a break of a tough problem at hand we have been trying to solve for hours, and start an undemanding task, not related to the problem we have been solving, we give our unconscious mind time to process and frequently find solution, while working in the background. We daydream with systems we just loosely axiomatize or don't axiomatize at all. We work with assumptions that we perceive or assume are correct and true, and we probe them with other systems to check whether the results are correct or even possible. That may be core of the creative thinking.

Which fields to select and include into this game is the most challenging thing and usually amounts to an invention, innovation, new idea generation, and unexpected success in the obtained results. This combinatorial thinking is the core of the generation of new ideas.

The diagram B shows how the new, hybrid, cross-axiomatic systems are created, by our thinking, combinatorial process. Note how theorems from the S3 and S4, when combined can be a theorem in the system S5 and very often some of the axioms in the system S5.

As I have mentioned, we usually don't need, and do not axiomatize systems we work with during our intuitive, creative thinking, and even during the design. Axiomatization can come later. Axioms, and proofs constructed from them, can eliminate any contradictions within the system, that may creep from our, potentially incorrect assumptions since we may have been tricked by nice motivations and examples coming from other connected systems.

Here is another diagram, an example in, mostly, scientific fields. Picture shows that each field is encircled within its own system of axioms (discovered or not).

But, the links with other systems allow these other systems to "puncture" through the surrounding circle of axioms to get into the other fields, and motivate generation of theorems within it. Then, when, and if, a rigorous proof is needed, these theorems will be proved only with the axioms of that system (and not by theorems or axioms from the system that motivated them) and by already proved theorems.

You have to keep open mind and not to be bounded by only one system, even if it has firm axiomatic framework for itself. To be creative, you have to see how it relates to other systems. For instance mathematics. There's a rich world of ideas right behind math axioms. Axioms deny you access to them, yet it's from these ideas mathematics axioms came into being.

One may ask, how we can advance in discoveries, design, even everyday actions, without proving theorems of the systems we are working with on a daily basis.

We can do that because we work with hybrid systems, where an assumption is accepted as true if a combination of the component systems (that contain those assumptions) gives truthful, real, useful, and correct consequences in other system that uses them or is connected via some functionality to them. The consequences we can imagine that can happen, given premises we have at hand and we deal with, allow us to avoid the immediate proofs for these premises and play more freely, without inhibition, with combinations of different fields, ideas, systems.

Combinatorial creativity needs to recognize the cross-axiomatic links between partially or fully axiomatized systems combined. Puncturing the axiomatic membranes around conceptually delineated ideas using cross-axiomatic probing needle is the key to creative thinking.

To me, for instance, a deliberation is cross-axiomatic attempt to draw plausible conclusions from partially axiomatized systems at hand http://t.co/S3783HnQ

Mathematics, Intuition, Real World Mathematics Applications. Random Notes.

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Edison on the role of theory in his inventions

“I can always hire mathematicians but they can’t hire me.” - Thomas Edison. http://www.brainpickings.org/index.php/2012/03/28/the-idea-factory-bell-labs/


Economics and mathematics

John Maynard Keynes, a student of Alfred Marshall, makes a statement in
his General Theory of Employment, Interest and Money (GT) which reflects
Marshall’s statical method: "To large a proportion of recent ‘mathematical’
economics are mere concoctions, as imprecise as the initial assumptions they
rest on, which allow the author to lose sight of the complexities and
interdependencies of the real world in a maze of pretensions and unhelpful
symbols". 

About proof

The premise may be true because it gives true consequences. Let's prove it later.

Daydreaming is a comfortable investigation of a number of premises that can give true consequences without a need for premises' proofs.

If the consequence is true, for a certain premise, it is a nice sign to go ahead and prove the premise within its axiomatic system.

Even if pure math is clear if contradictions, it can not guarantee that non-axiomatized field of applied math will have meaningful results.

Proof, in any field, with possible exception of mathematics, is often consider convincing, even correct, if the consequences, effects under investigation seem to be reasonably predictable in the whole mash up of a number of non-axiomatized systems and their relationships, in the web of their cause, effect connections. But, that should not put mathematics in any special position, because the proofs in other fields, and the investigation methods, in the way they are, are the best what can be done at the time. If the cause is not axiomatized, the effect can be predicted in only of handful of special cases, which is what we have to deal most of the time anyway.

References for the book "Unlocking the Secrets of Quantitative Thinking":

You can download all the important posts as  PDF book "Unlocking the Secrets of Quantitative Thinking".

References for the book "Unlocking the Secrets of Quantitative Thinking":

1. “Probability: Elements of the Mathematical Theory”, C. R. Heathcote.
2. “What Is Mathematics, Really?”, R. Hersh.
3. “Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being”, G. Lakoff, R. Nunez.
4. “Introduction to Set Theory”, K. Hrbacek, T. Jech.
5. “Foundations of the Theory of Probability”, A. N. Kolmogorov.
6. “Stochastic Differential Equations: An Introduction with Applications”, B. K. Oksendal.
7. “Essays on the Theory of Numbers”, R. Dedekind.
8. “Fundamentals of Mathematics, Volume I”, S. H. Gould.
9. “The Electrical Engineering Handbook”, R. C. Dorf.
10. “The Algorithm Design Manual”, S. S. Skiena.
11. “The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number”, G. Frege, J. L. Austin.
12. "Introductory Applied Quantum and Statistical Mechanics", P. L. Hagelstein, S. D. Senturia, T. P. Orlando.
13. "Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables", M. Abramowitz, I. A. Stegun.
14. "The Feynman Lectures on Physics, Volumes I, II, III", Richard Feynman.
15. "Electromagnetics", J. D. Kraus.
16. "Textual Strategies", J. V. Harari.
17. "Fundamentals of Aerodynamics", J. D. Anderson Jr..
18. NACA Airfoil Sections, Report No. 460.
19. "The Limits of Interpretation", Umberto Eco.
20. "The Open Work", Umberto Eco.
21. "A Theory of Semiotics", Umberto Eco.
22. "Metaphors We Live By", G. Lakoff, M. Johnson.
23. "Pi: A Source Book", L. Berggren, J. Borwein, P. Borwein.
24. "Option, Futures, and Other Derivatives", John Hull. 
25. "Energy Risk", D. Pilipovic. 
26. "Commodities and Commodity Derivatives", H. Geman.
27. "The Mathematics of Financial Derivatives", P. Wilmott, S. Howison, J. Dewynne.
28. "Components of Nodal Prices in Electric Power Systems", L. Chen, H. Suzuki, T. Wachi, Y. Shimura.
29. "Film Form: Essays in Film Theory", S. Eisenstein.
30. "Introduction to Topology", T. W. Gamelin, R. E. Greene.
31. "Quantum Mechanics for Applied Physics and Engineering", A. T. Fromhold, Jr..
32. "Complex Variables and Laplace Transform for Engineers", W. R. LePage.
33. "Partial Differential Equations of Mathematical Physics and Integral Equations", R. B. Guenther, J. W. Lee.
34. "Point and Line to Plane", W. Kandinsky.
35. "Bauhaus", F. Whitford.
36. "The Power of the Center: A Study of Composition in the Visual Arts", R. Arnheim.
37. "Visual Forces, An Introduction to Design", B. Martinez, J. Block. 
38. "Why great ideas come when you aren’t trying", Matt Kaplan, Benjamin Baird and Jonathan Schooler,

One useful exercise to reduce student's frustration with mathematics

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One useful exercise for kids learning math. Let them decide what to do with numbers.
Let them select the number by themselves from these two groups of numbers, as well as operations on them.

Then, another exercise can be to ask student to measure, count something and find the number that represent that count. Adding another object to that count, and find the matching operation on the picture.

These exercises will show kids that mathematics can be dealt independently of real world objects. It will also show that it is us who can chose the numbers and sequences of operations on them, or we can obtain numbers by measurements. It should be shown to student that math does not know how we obtained the numbers. For math it is important only with which numbers it is dealing with. It is us who keep track, aside from that diagram, what is counted and why. That can be called applied math.

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...]

An Example How to Learn Probability and Statistics Using World 1 and World 2 Approach

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This is an example, actually guidelines, how to learn mathematics, specifically material in the book "Business Statistics" by D. Downing and J. Clark.

Please read my previous posts about separating mathematical world, World # 2 from non mathematical axioms and logic, called World #1. Here World # 1 is motivation to develop probability and statistics material. World # 2 is pure mathematics.

It is sufficient to recognize premises in World # 2 motivated by World # 1. Note that mathematics has well established set of axioms, and that these premises can be developed without any mentioning of real world examples related to the statistical analysis. Again, please read my previous post or my book.

Hence, it is sufficient, and necessary, to learn these premises. Note that they do not require proof, or more precisely, many of them follow direct from basic math axioms. Then, learn real world explanations that can motivate their selection and introduction. Clearly separating these two worlds you will be able to firmly understand mathematical treatment of business statistics. At any point you should be able to define the premises and show the separation boundary between pure mathematics and business field (hint: they even use different vocabulary). To help you further, no business term can ever be used to prove any mathematical theorem mentioned in this book no matter how business situation "motivates" mathematical concept introduction.

It is interesting that math students are taught how real examples motivate math new concepts and new directions of math development, but then it's not emphasized how no real world object or concept can be used in any mathematical proof. 

A Guide to Interdisciplinary Innovative Thinking

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Application of one scientific field (or any system that has a logical structure) in the other, usually means that, when both systems are axiomatized, the link via logical connectives between the two fields' postulates, hence creating new premises in the new, hybrid system, will mean that postulates in one system (field!) will dictate selection of premises in the other. Some particular combination of these can lead to an invention.

Sometimes, the key to an invention is a selection of two (or more) systems and linking them together. Linking primarily means links via logical connectives, i.e. selecting premises, forming theorems.Sometimes, we already know the fields (systems!) but we need to find winning connection between the two (or more) of them. Where intuition fits in? It fits in selecting appropriate fields and selecting correct and useful links between them. Don't forget, an axiom is not provable within the system it defines, i.e. within the system it is developed from them. Choosing right premises and choosing to search for axioms is usually inspired by the linkage to the world outside the one that we look to find the axioms for.

Here are some examples. Each field, or system, is assumed to have its own set of axioms, postulates, and theorems, whatever that means in that system. As you will see, the system does not have to be mathematics. Note the selection and links.


Music -> Emotions.

Instrument -> Music -> Emotions

Physics -> Mathematics -> Human Language

Engineering Design Requirements -> Physics -> Mathematics -> Human Language.

Emotions, morals -> Paints, canvas -> Painting

Electric Power Systems -> Economic dispatch

For readers' exercise, try to define axioms in each of the systems and illustrate how the postulates, theorems in one system dictates premises in the other.

The power of a good question is that it can point to the areas of knowledge you need to familiarize yourself with. It can also initiate effective knowledge filtering and selection of the right facts that will be connected in a new, original way, to answer your question.

You probably got your engineering degree for knowing how to solve differential equations, not how to select useful and innovative initial and boundary conditions.

A mathematician and an artist. An accomplished NASA and IBM statistician and scientist talks about his sculptures. http://www.bbc.co.uk/news/magazine-16976145

A conception of an idea - axioms and brain

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An idea is conceived in your mind. But that's the different question than axioms in mathematics. How an idea came into an existence at the first place is a question for biochemistry, energy paths, oxygen driven, in our brain's biochemical processes. But, when we talk mathematics, we use our oxygen driven conceptual mechanism to limit our ideas that can be generated from math axioms only. Note that first axioms must be conceived, then thoughts from axioms. They are all "puff" generated from energetic processes in our brain.

We can think, that's apparently given. How the idea is created in our head, or, even worse what is it, is not a part of mathematical study. We can say that an idea is a state of our mind, molecular, energetic, a dynamic state of biochemical processes, that keeps the idea present in our brains, purely on an energetic level.

We can conceive an idea or a thought, that can be called a postulate, and then use logical thinking to derive theorems from the postulate or axioms. You have to be sure that your next mathematical thought is originated in axioms and that it can be derived, proved by them.

Thinking freely, without axiomatic boundaries, is also an attractive scenario. Free train of thoughts can give initial and starting conditions, initial premises in, most likely, any axiomatic system.

An Example of a Novel Way to Understand Math in Real World - Financial Mathematics

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Probably every 7^th grader will be able to do the following mathematical tasks.

Let ‘s assume we are given the following numbers

            1,000,000        1,435,900        1,500,000        1,400,000       

Pair the numbers 1,000,000 and 1,435,900, like this:

           

            (1,000,000 ,  1,435,900)          Pairing done.

Subtract  1.435.900 from 1,500,000 and show the result:

            1,500,000 – 1.435,900 = 64, 100         Subtraction done.

           

            64,100                                                 Result shown.

Subtract 1,435,900 from 1,400,000 and show the result:

            1,400,000 – 1,435,900 = - 35,900      Subtraction done.

            -35,900                                                Result shown.

Would you be surprised that this is mathematics that is thought in undergraduate studies in quantitative finance? Now, of course, that’s not the whole story, because there are other more exotic parts of mathematics that are taught as well, even within the same course. Those other parts are probability, statistics, and stochastic calculus, for instance. Yet, it’s amazing that this simple calculations are found in very important illustrations of some central financial concepts.

So, where is the trouble? Why not teach 7^th grader quantitative finance and financial mathematics, since the student already has required mathematical knowledge? As a matter of fact, it is possible. Why it is not done is a different story. A few wrong turns in math education and you are lost in numerical labyrinth for the rest of your career. I want to rectify that.

Here is the actual explanation where these numbers come from. The following is the excerpt from an excellent, extraordinary book (“Options, Futures, and Other Derivatives”, 5^th edition, John C. Hull)  in which the definition of a forward contract and certain trading technique associated with it are defined.

While mathematicians are satisfied with the starting, almost innocent phrase ”Let’s assume", "we are given", or "suppose”, and start writing numbers on paper a-priori, without any explanation, directly from fundamental axioms (be it ZFC or Peanno), a financial specialist must deal with, has to strictly define logic and provide reasoning where and why these numbers came into consideration. It is not enough to say that “we assume”, or, “the numbers are given”. Why are they given? Who gave them? Where from?

What a mess of descriptions. For instance, payoff  from forward contract. It is only one of many non mathematical concepts, concepts extraneous to mathematical world, concepts never used in the proofs of mathematical theorems. The others are buy, sell, outcome, position, forward contract, trade, bank, treasury! One has to know these definitions, their relationships, logic that applies to them way before even considering to enter number handling generated by these concepts. Postponing the introduction of the role of proof in mathematics teachers may further blur the boundary between pure and applied math.

Payoff appears to be the word of the day! It specifies a numerical procedure to be performed on selected numbers. Also, legal terms are thrown into the payoff and forward contract definition mix as well, like corporations is legally bind, it's obliged to keep its part of the contract, be it buy, or sell the asset. Obliged is an additional property to buy GBP 1,000,000 for $1,435,900. Note how these properties are added as flavours to the numbers 1,000,000 and 1,435,9000. Math here is very simple. two numbers are given! That's it! But, what is behind the definition of "given" is very important in finance.  From mathematical point of view ( i.e from setology point of view) the numbers' existence is guaranteed by fundamental axioms. No explanation necessary. All those classification, including different currency, who owns the currency, who buys and who sells, is outside math! So, these two numbers are linked together through some kind of specific financial logic, which has its own vocabulary and conceptual relationships. Mathematically, it's sufficient to pair these two numbers, like this:

(1 000 000,  1 435 900 )

What is attached to this pair of numbers is the reasoning why we paired them,  and the financial definitions extraneous to mathematical world. It's the world of financial concepts and relationships, and they do not belong to math.

As we continue reading about this forward contract trade, we come to the concept of "spot exchange rate". This is another "number generator" or "number picker". Now, we have a triplet of selected numbers, with the exchange rate definition in the background:

(1 000 000, 1 435 900, 1 500 000)

From math point of view, the number 1,500,000 is added arbitrary, i.e. math does not see the reason where and why this number is coming from. It's given. We suppose it. It's assumed is there. How we can ignore the fact of spot price presence, exchange rate, etc? Because in order to subtract these numbers, the mentioned definitions are irrelevant for the subtraction. Hence, the words "It's given..." immediately isolate pure math operations and numbers from the set descriptions(of sets they belong to) and from the objects definitions they represent count of. Notice how the words "Let's suppose" can blatantly keep you in dark about, sometimes, beautiful logic inisde the field math is applied to, and how it can suck out any pleasure in working with applied mathematics.

Then, there comes the question "how much is forward contract worth?". This particular question dictates which numbers will be picked and which math operations will be performed on them. 

Again, from mathematics point of view, it is specified, without any further explanation, which numbers are in game. For math, it is enough to use word “IF” and start generating numbers and their relationships. This “IF” implies usage of fundamental axioms. Hence, IF you have number 1,500,000 and IF you have number 1,435,000, deduct the second number from the first. That’s what matters to mathematics. To finance, the reasoning why you deduct second number from the first (and not vice versa).

Payoff = S_T – K

Who would expect that the number 1,435,000 will have the following description: it is a six months forward offer quote for USD-GDP currency exchange. Note how this definition has almost nothing mathematical in itself (except number six, for six months). Cardinalities of a set are not part of this definition. You have to know all this just to pick one number! And that knowledge matters. If your non mathematical logic is flawed, you will select a wrong starting number and, even if your subsequent mathematical operations are perfectly accurate, the result will make no sense within the applied field, because the initial number was wrong.

Educators can ask math students the following question: "Give me an example what number 1,435,900 can represent count of". And, student will start searching from his or her experiences what can have that count. it can be 1,435,900 apples, 1,435,900 pears, 1,435,900 oranges, 1,435,900 birds, 1,435,900 rockets, even 1,435,900 thoughts. But, would you expect from a student to give you the following interpretation: "Number 1,435,900 is the number of dollars a bank is offering in a 6-months forward contract, in currency exchange for 1,000,000 British pounds on August 16, 2001" ? Probably not. You don't expect student to know financial concepts at that early age. And, moreover, why would the financial field should be in focus for this example. So, what is the point of asking student to give you example of a number, unless you want to indicate that there have to exist World # 1, in this case financial world, that has its own set of rules, axioms, premises, definitions, whose logic will define to which set the number 1,435,900 belongs? This financially based description apparently can not be deduced by looking at the number only. And, that there is World # 2, world of pure math, which starts with "given" numbers, no matter what is the reasoning of obtaining that number. Distinguishing these two worlds is in the essence of understanding mathematics.

Look at the freedom of how the scenarios in this trade are created: if spot rate falls to 1,400,000. Note the "number picker"! It's "spot rate"! Look how the fundamental axiom that number exist, is disguised in this functional description why and what picks actual number. From math point of view, this is the same as "number is given", 'let's suppose", "let's assume".

The interplay between pure math and our numbers is further advanced with actual logic between numbers' sources. While the sources themselves do not belong to math world, the numbers picked by them do. Hence, the rules between these "sources" (which are extraneous to mathematics) indirectly influence which numbers will be picked and will enter the specified calculation. Also note that the actual math operation is motivated by the things and reasoning outside math. We want to do subtraction because we want to find the payoff! Payoff, as a concept, has nothing to do with math. If it does you will see theorems in mathematics proved using or referencing this concept. But there are no such theorems. So, we have payoff as:

Payoff = 1,400,00 - 1,435,900 = -35,900

So, here, you deal with specifications what each number represents, and financial logic and reasoning structures what to do, what sequence of mathematical operations to perform. Whether the result of this calculation is called "payoff" or "orange with freckles" or "space carrot", mathematics couldn't care less. Math sees only two numbers provided to it and math operation to do. It's up to you to keep track what is counted and why.

In financial application of mathematics, we want to generalize the math operations on specific numbers, using, apparently the English language and financial terms and definitions from the financial domain of counting.  Hence, we will say, the payoff from a long position in a forward contract on one unit of asset is:

S_T – K

Look how we have described the logic and requirements what to do with numbers. This reasoning is completely outside mathematics, and the sequence in subtraction matters to the financial domain. Math just see the numbers and subtraction. What these numbers represent, i.e. which set they belong to, is described in financial terms. We have S_T = spot price and K = forward contractually specified price. The pairing of these numbers, before even any mathematical operation is done on them (in this case it is subtraction), is specified by the definition of a forward contract, by existence and definition of market, concept of spot trading. This is what is required for you to know to pick right cardinalities at the first place, before doing any mathematical operations on them.

You have a number, say, 1,500,000. Pure number. Units are not yet assigned to it. The number will remain the same, but the definition what it represents will change in accordance to some domain rules. In finance, these rules are defined by "buy" "sell" "obliged" "exchange" "asset" "forward" "payoff". These definitions and rules change the ownership of that cardinality, that number. Note how number, and for that matter quantity of dollars it represents, remains unchanged. What is changed is who owns that amount of currency, and that's not a mathematical concept.

It is a "fierce pairing of numbers" and "fierce changing" of descriptions of sets to which those (same!) numbers belong to which is part of the forward definition and many aspects of trading. The similar conclusion applies to other domains of applied mathematics.

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