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.

That famous Cauchy definition of the limit, and another view that explains the elusive concept.

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If you can come arbitrarily close to a value, in the limit process, then that value is the limit. Of course, there has been always a question "but there is still that small error there, no matter how many elements we add, and no matter how close we are". It is true if you do not let n -> infinity. If n-> infinity then error goes to zero. But there is another nice thing about it. The "arbitrary close" statement guarantees that WHEN n -> to infinity that value will be the limit. It does not say it is the limit if you have finite number of values, no matter how big that number is. It says that, essentially, the fact that you can come "arbitrarily close", i.e. "close as much as you want" to that value, in that, and only in that case, it guarantees that, when n-> infinity, that value is the limit. No other statement will guarantee that. No other statement will guarantee that anything similar will happen when n-> infinity. That's the statement you want.

[ applied math, definition of limit, limit in mathematics, concept of a limit, limit, limiting value, integral, differential, ]

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

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You may want to check the list of references.

Axioms and Theorems in Relation to the Mathematical Models of Real World Processes

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A mathematical model of a real world process is a set of numerical, quantitative premises driven and postulated by that real world environment, by its rules and by its logical systems extraneous to mathematics. Yet, these premises can be also derived directly from mathematical axioms. Moreover, while the premises are motivated by the real world processes and scenarios, the proofs of theorems, theorems built on these premises, are done and can be done only within the world of pure mathematics, using pure mathematical terms, concepts, axioms, and already proven theorems.

When illustrating to students applied math, it should be shown which premises are introduced from, and by the field, of mathematics application, and, as the second step, how these premises can also be defined from the inside of pure mathematics, without any influence of, or reference to the real world process or environment. Then, it has to be shown that the proofs of the theorems that use those premises are completely within mathematics, i.e. no real world concepts are part of the proof.

Successful assumptions will give predictable consequences. Axiomatizing that set of assumptions should ensure no contradictions in consequences. Usually, we are after a certain type, a particular set of consequences. We either know them, or investigate them, or we want to achieve them. Hence dynamics in our world of assumptions.

Theoretical, pure logic doesn't care what are your actual assumptions. It just assume that something is true or false and go form there. Sure, results in that domain are very valuable. But, we are after the particular things and statements we assume or want to know if they are true or false. Not in general, but in particular domain. Any scientific field  can be an example. Logic cannot tell us what are we going to chose and then assume its truth value. Usually it is the set of consequences we are after that will motivate the selection of initial assumptions. Then logic will help during the tests if there are any contradictions. If you are interested in specific consequences, in particular effects, investigate what causes those effects. When you have enough information about causes, make every attempt to axiomatize them. And, again, as mentioned, axiomatizing that set of causes should ensure no contradictions in consequences.

Axiomatizing the set of causes should ensure no contradictions in effects (consequences). When tackling the topic of applied mathematics, it should be explained how the mathematical proofs contain no concepts or objects from the real world areas to which mathematics is applied to. That very explanation will shed light on the realtionship between mathematical axioms, theorems and the logical structures in the field of mathematical application (physics, engineering, chemistry, physiology, economics, trading, finance, commerce...).

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.

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.

[ mathematics, real life math applications, learning math, teaching math, math applications, real world math applications ]

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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[ mathematics, math, math tutoring, philosophy, cognitive, cognition, learning math, learning mathematics, number, count, number definition ]

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

[ number concept, concept of a number, number, count, numbers, counts, integers, rational numbers, concept, math, mathematics ]

Mathematical Intuition (Poincaré, Polya, Dewey), Reuben Hersh, University of New Mexico. Link to the paper.

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Found an interesting paper on mathematics and intuition. Here is the summary. 

Mathematical Intuition (Poincaré, Polya, Dewey)

Reuben Hersh
University of New Mexico

http://explainingmath.files.wordpress.com/2011/07/mathematical-intuition-hersh.pdf

Summary: Practical calculation of the limit of a sequence often violates the definition of convergence to a limit as taught in calculus. Together with examples from Euler, Polya and Poincare, this fact shows that in mathematics, as in science and in everyday life, we are often obligated to use knowledge that is derived, not rigorously or deductively, but simply by making the best use of available information — plausible reasoning. The “philosophy of mathematical practice” fits into the general framework of “warranted assertibility,” the pragmatist view of the logic of inquiry developed by John Dewey.

The Concept of a Mathematical Function

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The concept of a mathematical function should not be first introduced as a formula, but as an arbitrary (ordered) pairs of numbers. Pupils are conditioned to think of a function as a continuous line or a firmula. Later there are issues with statistics when function is shown to be function a set of distinct dots, representing ordered pairs of numbers, i.e. there is no formula at all. Moreover, in probability and statistics numbers appear to be showing at random! 

The rule how you pair one number with another can be a formula, but also can be a completely random event. Math function is, first and foremost about pairing two numbers (or more in multivariable functions). Students should be aware that they can pair random chosen numbers, they do not need to calculate second number from first. The rule can be linked input or output, but that restricts the function in the way that you have to know input to get the other paired number, the output. Because, function can have a pairing rule "pick first number, then, ask another person to pick another number without looking at the first number, then pair two numbers". Rule is one thing. Paired numbers are another. I want to emphasize that function need not to be defined in a restrictive way by using words "inputs" and “outputs", which is more related to computer science. You do not have to know input to get output, in a function. Both elements of the ordered pair of an function can be completely random and independent from each other. Function is first and foremost a pair of ordered numbers. My examples show why the \"input\" \"output\" definition is restrictive and possibly misleading. In my view, the word pair, or more precisely definition "ordered pair" best describes the function. Then we can use word map, association of two numbers etc. Input and output really leads someone to think that there need to be formula or some dependency between output and input. But, it is not so. It can be, but that's too restrictive for function definition. As in my example, a function can be "pick an output that in no way depends on input". Or, pick one number, then cover it (hide it) then ask another person to pick another number. Pair these two numbers. Here, output in no way depends on input, yet this is a function.

 

[ math function, function, concept of a function, concept of function, mathematics, map, mapping, teaching math ]

Axioms and Propositions

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Axiom. Theorem. Starting point. Proposition. Premise. Assertion. Proof. Argument.
Before any argument, containing, say, two propositions, axioms for each premise should be stated first, so we know where the propositions are coming from and why we assume they are true.

It's not enough to say you based your decision on logic. Logic, but based on which set of axioms? Axioms of principles, values, feelings, or physics laws? Or, logic that uses axioms and premises on a hybrid axiomatic system, perhaps a combination of two or more mentioned? Perfect logical reasoning with wrong assumption is useless. That kind of logical reasoning is almost always worse than using intuition.



[ mathematics, math, applied mathematics, applied math, logic, mathematical logic, inventions, innovations,  ]

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.