Sunday, January 30, 2011

How the domain of math application restricts which part of math will be used/developed

Even World # 1 (my previous post, How to Easy Understand the Role of Mathematics in Physics, Economics, and Other Fields ,) has to have its own axiomatic systems and what you can do inside the world. Obviously, this set of axioms in World # 1 will dictate which axiomatics realizations we will have in the World # 2 (example, World # 1 are vectors and vector operations, and World # 2 is math, or World # 1 is physics, and World # 2 is mathematics). Confusingly, for instance, sometimes, the sub domain of mathematics of interest is called vector's mathematics, but there is no such thing. There is only a part of math constructs that are triggered or motivated by vectors. Vectors itself ARE NOT mathematics. They are spatial, arbitrary constructs, concerning length, direction, that happen to interest us. They also came out of the blue.

The confusion arises with rapid and continuous jumps between World # 1 and World # 2, so one can get a sense that math is so intrinsically connected with World # 1 that without it a specific formula would not exist. That's not true. Formula would exist in mathematics in any case, but what the numbers represent, i.e. what they are counts of, must be explained in World # 1.

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Thursday, January 27, 2011

Where all those number series in math are coming from

When I was an engineering student, I was wondering where all those number series are coming from. You know, those for which you have to prove the convergence, for instance.

I always thought there's some special reason, something behind the scenes that mathematicians used to construct those series. But, it turns out it may not be so. Or, whatever reason they had, it may not be a mathematical reason. The answer is, anyone can declare a series! By ZFC axioms, you are ALLOWED to form a series without any proof, almost directly from fundamental axioms of mathematics! You don't need to wait for high school education to do that. Once you are familiar with fractions, go ahead, create, construct, your own number series. And right there ask your teacher to introduce you to that delta-espilon definition of limit! No proof needed for constructing a series! How so? Because, essentially, fundamental axioms in math tell you that you can create numbers at free will, you can define sequences as you wish. So, you CAN define, say number 2, then multiply by itself, 2, 3, 4, 5,.. times and each time you decide to divide 1 by that and then sum them. Like here, 1/2 + 1/(2^2) + 1/(2^3) + 1/(2^4) + ..... Notice how arbitrary you came up with choosing the number and defining the sequence of operations. And you can define these sequences as much as you want. There is no other way! Nothing else behind the scene.

Now, proving whether it converges or not is a different story. However, look that YOU can define WHAT needs to be proven! Why would you define any series? Mathematician will say because you can! Do you really want to? You may not want to! That's the state of affairs too. Now, once you define it, math does not care whether you liked it or not, were you motivated or not, or, whether the series came form some physical or other process measurements, or from you just playing with numbers,. The series, as it is, can come into existence just because you can create them. You are allowed, again, by fundamental axioms in math to do that. And there can be many, many series, but again, they are all arbitrarily created. Try it by yourself! Just create a series, a sequence and try to find the sum. That's all what's to it.

You can read more:

Wednesday, January 26, 2011

Imagine mind as a painting

Imagine mind as a painting. Neurologists and biochemist are analysing how the paint pigment adhere to canvas fiber, they are interested in the canvas thread texture and material, and paint properties. Psychologists and cognitive scientists are focusing what is painted and the interpretation of the messages conveyed by the images on the painting.

Tuesday, January 25, 2011

Some parts of mathematics relationships are developed by quantifying completely non mathematical relationships

Some parts of mathematics are developed by quantifying completely non mathematical relationships, specifying what to count, why to count, way of counting, and sequence of specific calculations. Perhaps, without that "outside mathematics" relationships certain parts of math would not be developed. Example is calculus. It is fascinating, to me, how completely arbitrary, even subjective, sometimes very vague descriptions can lead into specific mathematical operations and calculation sequences. Example can be valuation of financial instruments, Black-Scholes equation and calculus. However, all mathematics can be developed without any outside input. We are used to think that, perhaps, physics, for instance, developed mathematics. But, it is not so. It may only motivate to select certain sets, numbers and sequence of calculations on them. These sequences can be developed within math too. Outside world may motivate us what to count, when, and in what order. Fair to say, it can lead to selection of certain mathematical mechanisms. But, again, those mechanisms can be constructed within math too, independently from any outside interference.

Now, here is one interesting thing. Many of us are trying to find the relation between the real world and mathematics. How we can apply mathematics to real world? We may have a tendency to think that math is linked, that it is dependent on the real world. But it is not always so. Math can not distinguish, can not tell between real and fictional world or concepts. For example, if you have two dragons, and each dragon ate 53 kg of coal, how much coal in total they ate? The answer is 2 x 53 = 106 kg of coal. Here we used mathematics in a fairy tale situation. Moreover, we have used fictional, even untruthful concepts to do calculation. The dragons have nothing to do with physics or with real world! Especially coal eating. Yet, mathematics can not see that. It can not tell if the relationship between numbers you provide to it is fictional or real. Math can not see why you do or ask for calculation. It is you that specify what and why has to be counted, calculated and those requests you plug into math.

You may ask now, how math can be developed independently. Here is one illustration. Let's say a person writes number 7. Then he writes number 25. You don't know whether he counted anything or he just plays with numbers. He may add these two numbers, subtract. Whatever he wants. Let's say he multiplies them. So, 7 x 25 = 175. Looking at this mathematics operations you can not tell what is counted, if it is counted at all. It can be just play with numbers. Yet, it can be 7 train cars with 25 tones of grain on each, or 7 dragons with 25 kilograms of coal they ate, or 7 boxes with 25 CDs in each. The common calculation for both is 7 x 25 = 175. You can reuse this general mathematical result in all those cases. But, again, you can still independently play with numbers, as we have just seen, without keeping track what is counted. You may be interested to work with numbers only. Meaning, you can, just say now, 8 x 30 = 240. Then say that 240 > 175. These investigations of numbers properties constitutes mathematics as independent discipline.


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

Axiomatics Systems Do Not Need to be Part of Mathematics Only - Examples in Music, Electric Power System, Finance

Axiomatic systems do not have to be related only to mathematical field. Note that logic can be applied to many fields. Logic deals with truth values of the statements without asking where the statements and relationship are coming from. Similar to math which does not ask where the counts, i.e. numbers are coming from. If you have a conceptual system, with objects, relationships between objects and truth statements within it, call it World # 2, try to find the definition of the fundamental concepts from which all other objects and concepts definitions are derived and obtained. Go backwards in this definition search until you come up with, possibly, few concepts that can not be defined by anything within your system, your World # 2. These are fundamental objects, fundamental assumptions, truths, and they may form the axioms of the system. Now, these axiomatic objects can be clearly defined in other Worlds, outside your system (which in turn can have their own logic, axiomatic systems). For example, number of cars is clearly defined in traffic analysis, yet, that number of cars itself is undefined concept in math, since there are no cars in mathematics. Only a number. The inventions are usually created in the World #1 and relationships between objects are transferred into World # 2, i.e. your system.

We can learn one important thing from mathematics (in addition to discoveries that certain calculation applied in real world bring to us). Math has axioms and theorems. But note these are just exotic names for truth statements. Truth statements within math. Axioms are just starting truths that we have to have in any system. Theorems are "derived" truths within the system, in this case math. Moreover, theorems can be described by other parts of the math, and if you go backwards more and more you will hit set of axioms which can not be described by anything else within math but math is built from them. So, bear in mind, axioms and theorems are just exotic names for truth statements. Theorems can be proved from axioms and other theorems, while axioms are accepted without any proof WITHIN MATH.

Now, how we can use this elegant mathematical approach to other systems? And , I don't mean mathematical calculations, but more the way of thinking. What we have started with? What we have gained? Here are the answers. Let's talk about the concept of a number. I have posted many articles about it, and I will summarize here briefly the most essential conclusions, that may be already known to you anyway. Number is a common property, common count for all objects (counted objects) that has that very same number of elements. Hence, when we say 5, it can be pure number 5, but also can be 5 apples, 5 cars, 5 rockets, 5 CDs, etc. Then, when we discovered that we can investigate properties of numbers only, regardless where they come from, like their division, which number is smaller or bigger, various sets of numbers. In all these cases, dealing with "pure" numbers we don't need to reference any real world objects particular number can represent. Yet, obviously, when we get, in our investigation of these our numbers, some result, it can be applied to millions of objects that particular number can represent a count of.

Once we realized that the these abstracted numbers can be separate concepts and completely independent from any real world examples they may be generated from, we have started building a mathematical system. The important thing here is that we no longer needed apples, pears, cars, etc to deal with counts. We deal with numbers only. Like, 5 + 3 = 8, 6 + 9 = 15. And, this counting is true no matter what objects we apply to! It's universal result. Given this we can build a system of truth statements about numbers. Hence, mathematicians built axioms and started developing theorems from them.

And this is where this kind of mathematical thinking is and can be applied to any scientific or even art discipline. You can abstract some concepts from some real world domains. These concepts can be common for all the objects in this real world domain. Once these common concepts or objects or properties are established, you can put them in a separate discipline! And, along the way, discover and postulate truths about that new system. And, we are not talking here about numbers, but to any other system. So, the real world domain is World # 1, and the world of abstracted concepts , with just created axiomatic system specially for them, is World # 2. Now, World # 2 can be fully independent from World # 1, since it works with abstracted concepts from World # 1 anyway. And, many the results obtained by the developments in World # 2 can be applied to World # 1 given that we know, now, the specific objects in World # 1 that can be uniquely assigned to the abstracted concepts in World # 2.

One example is Euclidean Geometry. Line, point, triangle etc, are abstractions from any straight line or triangle from real world, like in geodesic measurements, land surveying, or even in architecture. But, Euclid abstracted these concepts and built axiomatic system for them.

Music is another example. Music piece has emotional, aesthetic value. But, system of notes has its own properties and hence can be consider an axiomatic system, i.e. World # 2, while meaning of the piece its aesthetic, emotional value is World # 1. Notice how World # 1, in music, dictates what will be written in World # 2, i.e. notational system.

Yet another example is with an Electric Power System, which I can call a multi layered example.  Let's consider a design of an extension of an existing power transmission line with a few generators and transformers added as well. The design requirements will be World # 1, the first layer,  and will consist of power requirements, load requirements, power angle restrictions, active/reactive energy requirements, power wheeling,  line capacity, transmission line corridors, economic constrains, environmental decisions. When all this is done, then the electrical circuitry can be designed. This is World # 2, the second layer. We will add electric generators, connect them with transformers, add transmission lines. This system, World # 2, has its own axioms, and they are related to strict definitions of electrical components we are going to use and the ways they are allowed to be connected (generators, transformers, switches, relay protection, reactive energy compensation, etc). Now, there is a third layer! It's mathematics used in electrical power system calculations. Mathematics is here World # 3. As we have seen, it has its own mathematical axioms, and calculations and results from such axiomatized mathematical system are readily used in power system. Note that math still doesn't care where the numbers are coming from. Math doesn't care if the numbers are coming from electrical circuitry, thermodynamics, or heat exchange measurements. Math will accept initial, boundary conditions, initial numbers, do the calculations and give the result back to the system that initiated the calculations. For instance, same PDEs can be used in electromagnetics, thermodynamics, aerodynamics, and even finance (Black-Scholes equation). It is us who have to keep track which units we are using, what is measured and counted, so we can assign obtained numbers to the correct objects, when we get results back from mathematics. So, in the case of electrical power systems calculations, we will use math universal results from, say, calculus, linear algebra, real and complex analysis, but we will also keep track what those numbers represent, namely, kW or MW, GW, A (amperes), V (volts), J (Joules), seconds, meters, kWh, etc, as we would keep track what is counted (using exactly the same mathematics equation) in finance, say (interest rate, strike price, option value, volatility, pay-off, etc) .

Notice here that you can not use musical notes to develop electrical power system project, nor you can use generator ratings to compose a song. These are two different systems, but we have used common approach of thinking to axiomatize each and discover what is true within each of these systems. However, both of these systems can have moral, even emotional, aesthetics value, in our human experience system. But, music is specifically created to appeal to these human experience values, while electric power system is designed to transfer electrical energy over distance, which may have aesthetics value too, but it was not the primary purpose of that engineering project.

You also may notice that even World # 1 has maybe some predecessor world, and that, although World # 1 deals with axiomatized World # 2, the World # 1 itself can have axioms of its own, and their realizations can be dictated from some preceding world! And, that's true. It can be said that all these worlds may be nested going backwards, but there may be interconnections between them too.

So, you may ask, what is the secret of creative thinking. I would say, explore more and more worlds, try to see their interconnections, but also try to axiomatize each world, because you need to know what is true in each.

As for even deeper question, where any of the ideas come from, the very conception of a thought, the answer is in biochemical and energetic processes within brain, neurotransmitters, but also in very action of ADP and ATP that releases  energy and make the thought, emotional, or other action possible. A thought is a biochemical and energetic state in our brain, driven by biochemical processes. then, you may ask, where ATP and ADP got their energy stored, so it can be used in our organism for thinking or other activities? It came from plants, which in turn got it from Sun via photosynthesis. So, essentially, we use solar energy to think.

[ axiom, theorem, axioms, education, innovations, inventions, math, mathematics, philosophy, physics, solar energy, solar power, finance, axioms in finance, electric power systems, power systems, ] 

Monday, January 24, 2011

When You Can Start Developing New Concepts Airplane

New concept airplanes can be designed almost immiediatelly after a student undestands the importance of airfoil design and how the lift is created. Then, you need to recognize that the wing is a major device to create lift, and that engines are there to give a propulsion. So, start with various wing shapes, experiment with different engine number and locations on the wing, fuselage, and locate a small portion on the wing for a pilot cabin, taking care about visibility options for a pilot.

Thursday, January 20, 2011

How to Easy Understand the Role of Mathematics in Physics, Economics, and Other Fields

Where the ideas in mathematics come from? Where the inventions come from? What is the reason mathematics can be “applied” in so many fields?

Mathematics does not care where the numbers come from. Fundamental axioms of mathematics tell you that number exists a priori, it’s given, and that you don't need to prove the existence of a number in mathematics. Also, roughly, Zermelo-Fraenkel (ZFC) axioms, in essence, are saying that you are allowed to do the basic operations on counts, numbers. The only reason why these statements are called axioms is that the concepts of set and operations on a set can not be defined with anything INSIDE mathematics. All other concepts in mathematics are defined using sets, but the opposite is not true.

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

Again, math doesn't care where the numbers come from, while you DO CARE very much, if you are coming from physics, economics, applied statistics, and other fields with math in it! Look at the measurements. You may measure the distance, time, volume, number of cars, apples, atoms, number of CDs in your collection, you can count number of songs in your iTunes collection, number of dollars in your bank account, the price when you purchase something. What do you count, and why, i.e. any motivation why would you be engaged in counting, defines the area, the domain of business, science, that needs mathematical analysis in it. For instance, in physics, you count physical properties and you discover, in addition to physical relations between the objects, the quantitative relations between them. But, all those counts and math relations are abstracted, isolated from their source once they enter mathematical world. You have to keep by yourself, the track what you have been counting and what sequence of operations you have been performing on those counts.

The numbers, as math is concerned, are created from thin air inside mathematics, in other words, in math, you are allowed, without any additional proof, to declare existence of any number. Fundamental axioms of mathematics allow you to do that. Don't forget, when you measure distance and you say that distance between two cities is 12 km, it is not math. Only number 12 is math. All other stuff is YOU keeping track of WHAT and WHY you have been counting. Think about it! Then, let’s say you add a distance to another city, say, 5km. The total distance, if you travel, is 17km. It is 12km + 5km. But in order to obtain 17 you do not need kilometers! You need only existence of 12 and the existence of 5 and the desire or decision to add, sum them. So, 12 plus 5 gives 17. Now, you go back to your reasons why you have been counting, what you have been counting. It was kilometers, km, and you associate that description (which is not math!) to the count of 17. Note how math is isolated, how numbers are independent from what is counted. You can count cars too. First you counted 12 cars then 5 cars. Total cars counted is 17 cars. But look, math again does not care where the numbers 12 and 5 came from. For math it was enough to start with number 12, no matter where it came from, and get number 5, and, again no matter where it came from, to add them together to get count of 17.

Hence, math is only a tool for your physical, economical or other areas for which you may be interested to do quantitative analysis. There is World # 1 where specific, non mathematical, relationships exist between objects and entities you want to count. For instance, in physics, YOU decide you want to measure distance and at the same time you decide to measure time as well. In another example, car speed depends on the driver decision, and the relationship between the driver and the acceleration pedal is non mathematical. You only may supply to mathematics the counts of two objects, namely 15m and, say 3 sec, to calculate speed. YOU also, and not math, decide that you may be interested in their division, and you tell math, I want to divide these two numbers. Math will do that for you. Hence 15/3 = 5. But it is you that keep tracks what has been counted and divided. And YOU can add a label, in small letters, beside the number, to remind you what you have just counted, what you have measured, what you have quantified. It will look like this, 15m divided by 3 sec gives 5 meters for each second. Shorthand notation will be 15m/3sec = 5 m/sec.

Counts need not to be produced by physical process or by physical laws to enter mathematics. They can be produced by any rule. If that rule makes some sense to you, it can be a law. In economics, when you exchange goods, or buy something, it is the market that determines the exchange price. Say one apple for two pears. Or, one DVD for $12. How much five DVDs will cost you? It will be 5 * 12 = $60. But look, in order to perform calculation, math did not care where the numbers 12 and 5 came from. Math only needed 5, 12, and multiplication request to do the job.

Let's call the world of math the World # 2. Other worlds, like, physics, economics, engineering, that can produce or generate counts, we will call them World #1. Rules within World # 1 will generate counts, sequence of required math operations, rules that will label sets, and label counts. Independence of math is quite clear. Looking at 12 + 5 = 17 you can not guess did it come from physics, from measuring distance, or from counting number of cars, or from counting how many DVDs you bought this month (12) and next month (5). Math has no idea, nor you will know if you just look at the numbers, where the 12 + 5 = 17 comes from. That is because in math you are allowed to enter any number, as a starting point, and do any math operations on them. That's math by its definition.

I want to show that there are two distinct set of rules, system of rules when you want to use math. First set of rules are the rules within World # 1, like in physics, economics, statistical biology, engineering, chemistry, trading. These rules in most cases have nothing to do with math. In economics the count of how much items need to be produced come from the consumers subjective choice, or from supply or demand. One feels he will buy today three apples, and tomorrow five, perhaps. So, these rules, i.e. why to buy so many apples, are not mathematical, but once the number is defined it can enter the math world, World # 2 and you can do calculations with it. Also, relationships within physics between say charges, distance and force, like in Coulomb's Law for electrostatic force, are not mathematical. They describe the objects that interacts in certain way between each other, in this case producing force. Now, once you supply counts for charge, once you quantify charge, and distance, you can calculate the force, but note that math does not know what you have measured, it just adds or multiply counts as you requested. You have to keep track what physical law you have been using.

The entries from Worlds # 1 to math world, World # 2 always look like an original invention, from math side, and it is true because math is not aware why you needed to, say, add or multiply two numbers!

There is a set of distinct, non mathematical rules present in each field, outside mathematics. These rules, in their own world can define the initial counts (sometimes called boundary or initial conditions) and define set of required math operations. Math world, in turn, accepts those request, does the calculation, with its own rules (theorems, lemmas etc) and gives back the result to the World # 1. World # 1 keeps track what is counted, measured, quantified, and assigns the math result to the appropriate object. And this is how you can use math in any discipline where you can and want to quantify objects and their relationships.


Here are more links you might like as well:
 

Tuesday, January 11, 2011

One Way How to Increase Motivation to Exercise

We do not like to do monotone things, in a gym, for one hour. We do not like to exercise because of that boredom. Lack of motivation has a strong presence in our plans to be fit.

In order to improve motivation I came up with one idea. What if the gym is organized as a series of rooms (in the same area). Each room has a different exercise equipment. You go from room to room and do not spend more than two, to three minutes in each room. You do as much as you can in each room and move on. You have different space each time you start the specific exercise and you KNOW you will not be on one machine for one hour! So, you will go through ten rooms say, and do ten different exercises in each room, in average two, three minutes spend in each. That's 20 to 30 min good workout, if not more! Not bad to do couple of times a week! It's quick, it's fast, there is a rapid change in exercises, and monotonicity and boredom are eliminated.

Some Thoughts How a Number In Math Can Be Obtained

A number, in math, can come from measuring, counting, observing, watching, listening, dreaming, intuition, guess, random selection, ...

Limit, in math, and hence derivative and integral, are defined by incorporating a "PICK" operation in the sequence of required operations.

Pick a wrong number, and no matter how accurate your addition, subtraction, multiplication, or division is, result may be useless.

Before you add subtract, multiply, or divide, you have to pick a number. PICKING a number is very important mathematics operation.

In addition to +, -, * and /, there is one more important operation in math: PICK a number! That is how limit in calculus is defined!

Fiction and Science Fiction Writing Styles

While in fiction story telling you don't need to prove the technical possibility of the devices that are part of your story, like ships, cars, liquids, weapons, etc, in science fiction writing you not only have to develop emotional, moral, human impact but also have to use technically accurate and technically possible solutions for those devices and machines. Unfortunately, the strong presence of technical detail, to prove the scientific background of futuristic devices and machines featuring in a science fiction story, takes away dramatic, human part of the story. It is, apparently, hard to achieve, at the same time, high dramatization in a SF work, while maintaining technical accuracy of inventions, technical devices, and machines in the story.

Here we see why the scientific papers are, essentially, very boring (scientist do not describe their possible excitement while making that particular scientific or engineering discovery, that's for later movie screenplay and fiction writers). The engineers and scientists insist on technical and scientific accuracy (which is, actually, the only way to communicate the findings), while fiction writers signify the importance to explore human experience, emotional, and moral value of the story events and social impact of technical discoveries, devices. Hence, you will never find a paper title classified as electrical engineering story, but only electrical engineering paper. However, once the technical paper fulfils its role to accurately describe the invention, discovery, construction, scientific finding, it can be judged or analyzed from the humanistic, moral, and emotional point of view. That's the exact point where the pure fiction or science fiction writing creativity takes place.

Here is my breakdown of the genres and its major requirements:

1. Technical, scientific paper – describes, from purely technical, scientific point of view a machine, invention, discovery, without going into any analysis what kind of impact it can have on a society from a moral, emotional, psychological, economic, political, legal or any other point of view.
2. Fiction – arbitrarily employs in the creative process any kind of machines, devices, fantastic inventions, discoveries (huge ships, light speed travel, extra strong weapons, supernatural powers) without taking care are they possible or not. The importance is put on humanistic, moral, emotional, social effects and consequences from dealing and using these machines.
3. Science Fiction – carefully employs and describes machines, inventions, discoveries that feature in the story, making attempts to logically predict the futuristic technical solutions, while at the same time, using fictional characters and dramatizing events, try to analyze the humanistic, moral, emotional, social effects and consequences of dealing with and using those machines.

Fiction writers are not, usually, technically inclined, but they give the powerful human message in their work, and are possibly way better selling authors on the market. Science fiction writers, in most cases, lack the deep human message in their writings, but are good in presenting where the technology might go in the future.

Mathematics in Physics - Distance, Time, Speed

We are used to use mathematics with physical concepts quite automatically, without thinking what is going on in a formula.

Let’s clarify a bit what is going on under the imaginative hood in a physics formula.

Look at the speed. Everyone knows that speed is quotient of distance and time. You obtain distance, then divide it by time. But, let’s take a close look at these two physical concepts separately.

Distance. We have a physical feeling what a distance is. We can measure it. We can count steps, kilometres. Now look at our counts that comes from distance. Let’s say we counted 5 meters. No matter what we have counted we have a number 5. That’s the count. Can be a count of meters, or apples, or atoms. That’s how mathematics enters physics. However, once we are in physics, suddenly it matters WHAT is counted! While mathematics does NOT CARE when you add 3 + 2 to obtain 5 where did you get those 2 and 3, in physics it matters what you count. In physics it matters where the counts come from!

Let’s consider example of 5 baskets and each basket has 3 apples. How many apples are there? Of course you will multiply 5 baskets by 3 apples to get 15 apples. But, one may ask how I can multiply baskets and apples. These are two different things. The trick here is you first abstract, or, obtain separately, or extract, a COUNT of baskets, keeping in mind WHAT you counted! That’s the count (or number!) five. Then, you obtain a COUNT of apples, which is 3. These COUNTS are now separated from objects that generated the counts, form what was counted! And since they are PURE counts, we can do with them what we can do with counts (numbers) in math! We can add them, subtract them, multiply, or divide. In this case will multiply COUNTS (not apples, not baskets). Let’s do it. 5 x 3 = 15.

Once the multiplication is done, we have to go BACK to our world of baskets and apples, in the world that generated counts, in the world where, actually, matters what is counted. Note that transition. While in math world (let’s call it World #2) we deal only with counts when we return to the world of countable objects (apples, baskets, let’s call it World # 1) we do take care which counts belong to which object!! And that would be the major approach, trick to master the whole physics, or, any applied mathematics discipline (economics, finance, engineering, trading, biology). In all these disciplines you have a separate thinking, separate rules, logic, specific for that particular discipline, that defines some kind of relation between countable objects! The rules what and why is counted, in any of these disciplines, are quite alienated from mathematics. In most of the cases these rules have nothing to do with math! You do not need mathematics to come up with a concept of basket, apple, distance, time, atom, speed, force, cars, money, water, air. You do not need mathematics to say I will buy 12 litres of gas. And, say, later you decide to buy additional 5 litres of gas. Total would be 12 + 5 = 17. Mathematics enters only when you dealt with pure counts, and not WHY you decided to by first 12 then 5 litres! Math enters when you wanted to add these two numbers. For you, it matters that those numbers represent litres of gas, but from math point of view you could add the number of the buttons on your coat. These are all concepts from World#1 (physical world, economics world...) and each discipline has its own rules WHAT to count and WHY. Once you decide you want to quantify the relationships between objects in Labelling counts. In the World # 1 you will need only to take care, keep tab, what is counted and when, and then, somehow LABEL, those counts, most likely with an object name what is counted. For example, you can label count 3, in the basket-apples example, with apples to get 3(apples). In this case “apples” is the label WHAT is counted. Then, you can label number 5 with 5(baskets).

In physics, it is custom not to use parenthesis but just to put a letter beside the number to label what is counted. Hence, for meters it will be 5m, for time it will be 5sec, etc. And it is up to you what are you going to do with these counts! You can divide them, add, multiply etc. What are you going to do, and why, has nothing to do with math! It’s your intuition or physical knowledge, or inspiration! Math is like an obedient servant who will do what you told him to do! Add 3 + 2! Sure, 3 + 2 = 5. But, this “servant” does not know what you have counted, what you have added and, moreover, what would be the reason for addition!

Let’s look at our speed example! You measured distance, and obtained, say 10 meters. Then, simultaneously, you measured, counted time, and obtained, say, 5 sec. Now, inside the World#1, i.e. the world of distance, time, our physical real world around us, only WE can decide what makes sense to do with measurements and counts. We can easily say let’s multiply 10 by 5 and obtain 50. But, that really does not help us if we want to get SPEED. To get speed, we need to divide 10 by 5 to get how many meters per second we have been traveling! So, let’s do it. Dividing it we obtain 10 / 5 = 2. Now, what is two? Is it distance? No! Is it time? No! Ok, you can say it is speed, but we do not know that yet. We dealt so far with distance and time. You can say that 2 we have obtained is actually 2 meters per second, and name it as speed. Look how we have dragged the description what is counted to the new physical concept, speed! And, moreover, we can quantify speed using the counts of distance and time. Don’t forget, number two is still only a number, but, you drag the label with it, the description what you counted or even more how you obtained it. It is 2 meters per second or 2 m/s. That unit designation [m/s] i.e. [meters/second] tells you the information form World # 1. It tells you that you have counted, first, distance, in meters, than you divided that COUNT with the count obtained from counting time. What you did is you have divided COUNT of meters by COUNT of seconds, and not METERS by SECONDS.

So, we do have these straightforward steps, COUNT of meters / COUNT of seconds to somewhat abbreviated notation, COUNT(m) / COUNT(sec) to very abbreviated notation 10m / 5sec. You can see that units in physics are nothing more than labels to COUNTABLE objects that we have a special interest or motivation to count!

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Thursday, January 6, 2011

Fiction Stories, Fairy Tales, and Physics

Why fairy tales, fiction stories, and movies can be accepted so easy and with so much pleasure, yet they may have a complete disregard of physics laws? We are thought physics in school and yet that knowledge seems to be useless for many of us. Way more success and, if you wish, commercial success, can be achieved with writing completely fictional stories without any obligation to follow laws of physics in them. Why is that and how is that possible?

The answer is humans have a strong emotional faculty in their brains. Emotions, moral, then ethics is always used as a final judgment about everything that is thrown at a man. Even the newest technological inventions, never seen before, will be immediately judged by a person for its emotional, moral, ethical, human value. Yet, the reverse is not true. You can not use emotions and morals to construct, say, bridge, or electrical motor, or internal combustion engine.

With fantastic, fictional stories, as long as you present a story that has emotional, moral, ethical value in it, it does not matter whether the things inside stories are technically feasible or possible. As long as you trigger emotions, ask moral, ethical questions, explore what men and women value in their human experience, it does not matter how big is the space ship, is it possible to travel faster than light, or are there are Dragons or not. Pick something that will stir emotions, something that will show which values are in stake and don't worry if physics laws are followed or not. The story is not about a physics laws, it is about human values. The story is about which emotional, moral, sociological, personal values are at stake when you construct an engine, say, by specifying nicely boundary and initial conditions and then solve numerically (or in closed form) related partial differential equation (PDE). Engineers and physicist will do the PDE solution, but writers and philosophers will tell you what is nice or not to do with your device. However, engineers can also write a story if they shift their creative focus from specifying boundary conditions, solving numerically PDEs to answering the questions how their device will be used, and what value to humanity, society, individuals will the device, or PDE solution, have from emotional, moral, ethical point of view.