Monday, April 30, 2012

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

Edison on the role of theory in his inventions

“I can always hire mathematicians but they can’t hire me.” - Thomas Edison.

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

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":

  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,

Sunday, April 29, 2012

One useful exercise to reduce student's frustration with mathematics

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.

Tuesday, April 24, 2012

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

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 ]

Sunday, April 22, 2012

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

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,  ]

Friday, April 13, 2012

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

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 ]