Tuesday, March 29, 2011

One Insight into Mathematics, Axioms, Logic, and Their Relations to Other Disciplines

Mathematics is, in the sense of methodology, like any other scientific discipline, including law, economy, psychology, biology, physics, chemistry. Or, more precisely, all other disciplines should be very similar to mathematics, if they are to discover new truths and solutions. I will demonstrate what are the two major similarities and one major difference. What differs mathematics from ALL of these disciplines is that mathematics deals exclusively with numbers, counts. It is, sometimes, hard to imagine that mathematics is independent discipline, given how much we, as students, and later in career, are fed (and fed up!) with numerous examples, starting with apples, pears, meters, acceleration, force, atomic mass, light wavelength etc.. Perhaps surprisingly to many of us, mathematics is an INDEPENDENT discipline and can be developed completely outside any example, i.e. examples (physical processes, decision generated numbers, measures) are not required for development and research in mathematics. Again, mathematics deals with counts, numbers exclusively, and that's it.

Now, similarities.

First major similarity, and the reason why mathematics is called one of the most precise sciences, is that it uses strong logic methodology. But note, this logic is used as a tool of thinking to solve problems in math, and develop mathematics. Logic is a separate discipline that can, and should, UNIVERSALLY be applied to any other scientific discipline.

Second major similarity is that mathematicians succeeded to define initial truths in mathematics, truths to start with, starting postulates or AXIOMS. But note, while axiom might be considered a mathematical term, it's meaning can be applied to ANY other scientific discipline or any other creative direction of thinking. Axioms are everywhere, we use it every day, but we do not call them axioms. Usually, these are initial assumptions in our directional thinking to solve a problem or to come up with a creative answer for something. The same truths are present in law, biology, physics, of course, the way they are discovered are different for each discipline. But, this is how it should be done. Define initial assumptions, make sure they are correct and start develop the system you are interested in.

Now, note how mathematics used LOGIC and AXIOMATIC approach to deal with counts! It is not that counts triggered development of logic and axioms. It's vice versa. Logic and axioms were there before, and are used to develop and enhance mathematics and are and should be used to enhance and develop other scientific and other creative human disciplines.

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    Saturday, March 12, 2011

    My Notes from Twitter on Mathematics and Applied Mathematics

    You can read more "How math can be applied to so many different fields and how we can use math in real life".

    Calculation in most cases, can be easy or straightforward. But what to calculate, which relationships to quantify is a matter of imagination

    Caffeine induced mathematical thinking..

    How come math and politics can have anything in common? What is it? Answer: our moral CPU examines math results then make political decisions

    From logic, p AND q => s. The "p" is your complex math calculation, while "q" can be anything from real world. Try it, see what "s" can be.

    You may ask why you need to calculate anything. Well some significant political decision may depend on that very calculation.

    Stochastic function is continuous but nowhere differentiable. Integrate that!! (hint, Ito's lemma :-)

    Integration should be introduced with a stochastic function, showing students limit in probability first, and then limit in Riemann sense.

    Forget about polynomials! You can create ANY type of functions in math, you are allowed (by ZFC axioms). If they interest you, even better.

    Polynomials are created from this air (from ZFC axioms). You are not required nor it's possible within math, to justify their introduction

    In math, you are free to create any function! No proof required! Then, since you created it, you may well be interested in their properties.

    When You Can Start Developing New Concepts Airplane, my blog http://tiny.cc/m5s1l

    Directional thinking supported by related systems of axioms.

    Combustion is, in its essence, an ELECTRICAL reaction between the fuel and oxygen molecules. From my blog http://tiny.cc/fboud

    The moment you find when a stack of bricks becomes architecture, you will find out when numbers and operations in math become beauty.

    Proof in math is like a proof in any other field. Logic of proof is same. The only difference is math has well defined starting assumptions.

    Word "IF" is a Magical Wand of Math. Using "IF" you can generate vast quantity of numbers, functions, patterns with no reference to reality.

    Can you have debates, opinions in Engineering? Yes, but not on Physics Laws. You can debate initial and boundary conditions!

    p AND q => s. Your "p" is your strong will, motivation, decision. You decide what is "q" from your world of ideas. Then "s" will follow :-)

    Math should help you think more freely. Math shouldn't prohibit all your other thoughts and thinking in order to correctly add two numbers.

    When writing a fiction story, remember, human values are at stake and not physics laws. Work the story events backwards.

    Formula for good acting. Get into your character weeks before the scenes. Then, your response to the scene events will be almost unconscious

    Formula for good acting. Always gesture first, even for a second, then say a line. It will look like you are fully in character..

    Good acting can make any difficult, artificial part of movie plot work.

    I would let students write an essay why they don't like math, and then work with him or her on case by case basis.

    Women Mathematicians. Danica McKellar, Hollywood star (IMDB: http://tiny.cc/lwhrh) and published math author (J. Phys.) http://tiny.cc/cwrk8

    Where math comes from and origins of fundamental math concepts for better understanding mathematics.Blog http://tiny.cc/lyah8

    The confusion in applying math is that in math you have only one number 2, while in field of app. you have to keep track where 2s come from.

    Track to creativity. Chose world of interest. Axiomatize it. Chose world where you interpret results. Axiomatize it. Play with both worlds.

    When my world of axioms clashes with your world of axioms, fallacies will fly like sparks!

    Social values, moral, ethics and engineering brain circuitry - my Blog, http://tiny.cc/ckbdq

    The first course in calculus should include Ito's Lemma and Ito's calculus, to show other kind of limits (limit in probability)..

    Finding patterns within facts in any subject of study, even very boring ones, can make that subject way more interesting.

    Comedians not only discover what properties are the same between two objects or concepts, but they also select those which are funny!

    Mathematics doesn't need to know nor does it care what are you quantifying! Language describing way of quantification doesn't belong to math

    Quantification is broader term than measurement. You can quantify by simply assigning a number, sometimes even without counting.

    Never trust the other side of a logical connective. :-)

    Can you learn to play chess? No. You can only learn rules. How to play it will be up to you.

    I think logically. It's just that my assumptions are wrong.

    Think logically! Don't let the emotions, feelings, sex, intuition, love interfere with your decision...whom you are going to marry.

    The difficulty of being creative is in the fact that you don't know what should be on the other side of a logical connective.

    My article that will change how math is thought :) http://tiny.cc/bvn7i

    My blog for anyone who is interested what mathematics is all about and how to use math any time you want, http://tiny.cc/0nl49

    How the new ideas are born by creatively combining different directions of thinking and axiomatic systems. http://tiny.cc/1ur1q

    When a school claims "we'll teach you C++ so you can be a programmer" is the same as "we'll teach you English so you can be Shakespeare".

    A picture is worth a thousand words - mathematics and disciplines it is applied to - http://deconstruktion.blogspot.com/

    Illustration of the relationship between mathematics and disciplines it is applied to. http://tiny.cc/lbdfm #math #mathematics

    Insomnia, red wine, Family Guy, iPod, Kutta-Joukowski Theorem, Panel Numerical method, aerodynamics, aviation history...those are my nights.

    More Math Insights. Mathematics Axiomatic Frontier and How Outer Worlds Puncture it to Generate Numbers. http://tiny.cc/lbdfm

    Where the Graphs in Math and Physics Come From. Why do we use graphs to represent functions? http://tiny.cc/coiie #math #physics

    Automobile Engine Principles, from fuel chemical bonds energy to crankshaft and camshafts, now with illustration video! http://tiny.cc/38l7i

    Easy to grasp, explained and illustrated by me, principles of internal combustion engines, http://tiny.cc/s97w5 #physics #cars

    How emotions enter math: "Boeing wasn’t happy with the way the Air Force calculated the higher fuel costs of the A330," http://tiny.cc/fl4vx

    While number 5 is only 5 in math, it really matters what, when, and where you counted, in physics. Was it meters, seconds, mass, speed..?

    Our emotions are like a guitar, like strings, waiting inside us to be plucked and played into a melody not yet heard..

    To be a good writer you have to know the matters very well...

    You probably wondered what 2.5% meant in my previous post. That's my point! Mathematician sees just that, 2.5%. No other meaning is required

    2.5%

    The best starting point to learn about combustion is energy calculation of oxygen splitting fuel molecule forming accelerating CO2 and H2O.

    Computers work with two bits, 0, 1. Humans work with interpretation of bits. Consequently, a single bit can have billions of interpretations

    A very decision whether you will add or subtract two numbers has nothing to do with math. It can come from non mathematical thinking.

    Field of math application is to math is as a driver is to his car. To physicist it matters where he drives, while mathematician doesn't care

    Applied math - clash and harmony between two or more axiomatic systems, one of which is in mathematics.

    If math is derived from 9 axioms where teachers found all those apples, pears, giving, getting, having, and claiming these words are math.

    Worse thing than making a bad decision is when others know you decided wrong yet they take advantage of it watching you sit and do nothing.

    Mathematics is never used as a motivational method for a student to do something else. But student has to find another motivation to do math

    Issue how math is thought is that teachers mix nonmathematical reasoning with math statements derived directly from fundamental axioms.

    In order to explain to student that 2 + 3 = 5, at one point it should be signified that no examples are necessary to state the addition.

    Kids should be told the fact that "Mary gave Pitter two apples" has nothing to do with math. It's sociology, psychology, economy.

    For each real world example in math students should be shown that formula can be gotten from axioms.

    Where numbers come from? How about from love? http://yarzabek.wikispaces.com/, "Let me Count the Ways I Love You"...

    New fields with applied mathematics. Quantitative physiology, quantitative finance, quantitative biology, quantitative sailing :-) etc ....

    My goal here is not to talk about stochastic PDEs, integrals, other differential equations. I want to show where they are coming from.

    Mathematics and other fields and why math can be independent discipline http://tiny.cc/gwdgy

    In order to make a winning invention you have to discover the right system, figure out its axioms, and generate the winning premises.

    Logic deals with truth values, but, determining what is true, in the first place, has, in many cases, nothing to do with logic.

    Instead of addition, subtraction, etc...I would introduce excercises with operation "pick a number" in the first five lectures in math.

    You can not "apply" math in some field, simply because, in math, you do not know what are you counting. You can only "utilize" math.

    No matter what direction of thinking you are taking (logical, practical, irrational) emotions are there to stop or encourage you to do that.

    Wednesday, March 2, 2011

    How the new ideas are born. Creatively combining different directions of thinking and axiomatic systems.

    Since math has initial axioms and sets, and membership to a set, that are a-priori accepted as undefined concepts in mathematics, it means that reasons why they are generated and hence the counts and sets cardinalities, are outside of mathematics, and those reasons have to be remembered and manipulated separately. This means there is no other way to remember sets, other than by “brute force”, i.e. remember them as they are. If the different sets are coming, or are generated, from physics, economics, trading, quantitative finance, stochastic stock price movement, engineering, counting apples, etc, knowing where and why they are defined, is not a part of mathematics. That has to be clearly distinguished during any initial mathematics lecture. Of course, any logic, any language that describes relationships of concepts within a certain discipline, say in physics, that can help remembering with which sets we are dealing with, is welcomed. However, also, the same logic has a meaning in the field of its conception, namely, physics, engineering, economics, etc. But, it has to be remembered, although it looks like numbers and counts are defined by physics concepts, they are not. They are only used to quantify stuff in Physics, and physics definition of counts, numbers and their relationships are not necessary to define them in mathematics. These counts, functions, can be defined directly in math, essentially from mathematics fundamental axioms, independently from any field in which math is “applied”.

    I can conclude that axioms, for any system, are barriers, but also a connection boundaries between any two worlds that are axiomatically defined. I will keep the examples of Physics, as World # 1, with its own axioms and Laws, and Mathematics, as World # 2, with its own axioms. So, if these systems are separated and are fully defined with their own axioms, how then, can be any connection between them, if axioms are fully sufficient to define everything in each system (like it is said for mathematics axioms)? The answer is you can form a new axiomatic systems by combining premises or even axioms from the previous two with logical connectives, “AND”, “OR”, implications etc…Remember those p, q, r, s from pure logic course? Those p, q can come from Physics and Math respectively! And a new logical statement, i.e. statement that has a truth value, true or false, can be made. Thus, we are forming a Hybrid Axiomatic System from Physics and Math which we are accustomized to see or it’s better known as a Textbook in Physics, with descriptions of phenomena that can make an impression to the reader that those Math formulae can not exist without Physics. But, they can.



    Now, let’s leave Physics for now, and consider more real life examples, say, a price of a mobile phone. Look how we have two concepts floating completely separately in our minds. A mobile phone and, say, price of it. At this point we don’t know what the price is. It can be any number. Mobile phone is in non mathematical world, price as well, but a number (for the price) is in mathematical world. Note how arbitrary that number can be. The relationship between two worlds will be “price of”. On one side of this relationship is “mobile phone’, and on the other is “number”, a count (of related currency). So, the associated pair can be shown as (mobile phone, 3.75). But note how mathematics keeps its independence for whatever is on the left side of the number. It can be mobile phone, it can be a pen (price of pen), it can be an apple (apple, 3.75), or DVD like this (DVD, 3.75). This independence of mathematics from what is counted is allowing mathematics to be a separate discipline, with its own logic and axioms.



    Inception of new ideas and inventions come from finding, differentiating clearly one or more axiomatic systems that must have some kind of relationships, and then combine premises, assertions, theorems from these systems, using, generally, logical connectives. For instance music. One system is that we can move fingers in many ways. Second system? We don’t know yet. We can use hands for many things. We have to be after specific, particular consequences.Choosing right second system is an art. So, we chose guitar. Now we have two systems. We can play a guitar. But, what is melody? Now, we introduce third system, not a neighbour who hates music, but, maybe more perceptive person, or our own perceptions for beautiful melody. We have now three systems, and through trial and error, say, we can compose a winning song.

    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.

    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 relationship between mathematical axioms, theorems and the logical structures in the field of mathematical application (physics, engineering, chemistry, physiology, economics, trading, finance, commerce.

    Here is the illustration of nested axiomatic systems that can lead to an invention.



    If you still ask where the ideas are ultimately born, i.e. where that first layer of axioms come from, it will be in biochemical neural paths configurations whose electrical activity is triggered using oxygen and ATP as energy currency to initiate new states in our minds that are called ideas.

    [ inventions, invention, creative thinking, evolutionary psychology,  applied math, applied mathematics, axioms, creativity, education, genius, ideas, mathematics, physics, ]

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    Tuesday, March 1, 2011

    Mathematics Axiomatic Frontier and How Outer Worlds Puncture it to Generate Numbers

    Mathematics Axiomatic Frontier and How Outer Worlds Puncture it to Generate Numbers.



    Any number within Mathematical World (bounded by ZFC Axiomatic Frontier, including Axiom of Choice :-) is seen within Math World as generated directly from the Fundamental Axioms. But, applied mathematicians and other people using math within their disciplines (Civil Engineers, Traders, Physicist, as shown in this illustration) have to keep tab what they have just counted, i.e. what a particular number is associated with, and where it comes from. And that, the language, the particular discipline's language and logic, as it is illustrated, is outside mathematics, and that fact should be signified to students when introducing them to math. There has to be a clear distinction between a particular discipline discourse, discipline's specific logic, language of explanation, definitions, and pure math, math that can be traced, essentially, to the fundamental ZFC axioms.

    Each thin arrow that goes from "Outer" world, world where the math is "applied", can generate a number, inside Math world, can generate a relationship, formula, even differential and integral equation, right there. When the calculation is done, within Math, the result will be returned via thick arrow to the discipline where the "request" for calculation came from.





    The drawing shows one more property. Math is not "applied" to the fields, but, it's rather used as a direction of thinking in order to solve some particular problem. The world "applied" probably came in use because the mathematical constructs, and math development in general, can be done without any of the Outer World "bubbles", disciplines (Economics, Physics, Civil Engineering, Electrical Engineering), and then, when there is a need, that math results can be matched with some requests from the Outer World. While certain domains of math development have been motivated by physical processes, that math can be developed completely from mathematical axioms, without analyzing any physical process.

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