Tuesday, January 25, 2011

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.

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