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, ]

You can read more from my blog entries:

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, ]

You can read more from my blog entries:

- Real world examples for rational numbers, for kids
- Math and its relationship with real world
- How math can be applied to so many different fields?
- Where the graphs in mathematics and physics come from?
- Tweets about math, physics, and how to approach math calculations
- One insight about mathematical axioms, logic and their relation to other disciplines
- How the ideas are born and notes on creative thinking
- Mathematics Axiomatic Frontier
- Why math can be an independent discipline?
- More on creative thinking, math, innovations, physics, emotions
- Comparison between making movies and math and physics
- More tweets about math
- Domains of math applications and math development
- Where all those number series in math are coming from?
- How to understand the role of math in economics, physics, engineering, and in other fields

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