Tuesday, May 29, 2012

The Links Between Different Axiomatic Systems and Cross-Axiomatic Ideas Generation

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

Monday, May 7, 2012

Unlocked secrets of quantitative thinking in the palm of your hand

My updated booklet, 157 pages, hard copy, "Unlocking the secrets of Quantitative Thinking". You can also download the book in PDF file format from the right pane.

http://explainingmath.files.wordpress.com/2011/07/unlocking-the-secrets-of-quantitative-thinking-applied-and-pure-mathematics-april-26-2012-e.pdf


You may want to check the list of references.

Thursday, May 3, 2012

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

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...).