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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