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

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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...).

Contextual triviality. What Has to be Done When Someone Uses the Word "Logic".

You can download all the important posts as  PDF book "Unlocking the Secrets of Quantitative Thinking".

When someone uses the word "logic"he/she should immediately point out what is initially assumed to be true/false and what truth results are derived, i.e. what can be proved from those initial assumptions.Word logic is nice to use, it can be fancy, it can show you want to be precise in your communication or explanation. However, the axiomatic system should be known and understandable for all the parties that are part of the "logic" communication. Saying that something is logical doesn't mean it is obvious, and it doesn't mean it does not require a proof. If something appear to be "trivial", the logical context of axioms and premises should be clear to all participants who want to accept that "trivial" remark. Contextually "trivial" is OK.

The other day, while browsing mathematics section at Indigo bookstore, I have noticed a book "Logic for Mathematicians" by A. G. Hamilton, (Google books http://goo.gl/17kEj). I was pleased to see that the book reflects my view that logic is an independent discipline from mathematics and that mathematics is only one of the area of the application of logic. While Frege may have integrated both directions of thinking, I am glad that A. G. Hamilton "presented the subject matter without bias towards particular aspects, applications or developments, but an attempt has been made to place it in the context of mathematics and to emphasise the relevance of logic to the mathematician.".

To me, this is important because of my view (most of the posts in this blog) to differentiate clearly the worlds that define what is to be counted, measured etc, from the mathematical world that accepts pure numbers as starting points. Logic is used in both worlds.



[ logic, mathematical logic, math, math concepts, axioms, mathematics, teaching math, teaching mathematics, understanding mathematics, Frege, Hamilton, ]

Real World Example of Natural Numbers

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In real world application of math we have to keep track what we have counting, while pure math cares only about pure numbers.

This is an example of counting rounds in an UFC match (UFC - I do not approve nor like).

[ math, mathematics, math application, axioms, examples of natural numbers, natural numbers, axioms, sets, number theory, ]

Mathematical Proof for Enthusiasts - What It Is And What It is Not

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With the term mathematical proof we want to indicate a logical proof, i.e. proof using logical inferences, in the field of mathematics. So, it should really be “a proof in the field of mathematics”. Also, we have to assume, and be fully aware, that proof must be “logical” anyway. There are really no illogical proofs. Proof that appears to be obtained (whatever that means) by any other way, other than using rules of logic, is not a proof at all. Assumptions and axioms need no proof. They are starting points and their truth values are assumed right at the start. You have to start from somewhere. If they are wrong assumptions, axioms, the results will show to be wrong. So, you will have to go back and fix your fundamental axioms.
When you have first encountered a need or a task for a mathematical proof, you may have asked yourself "Why do I need to prove that, it's so obvious!?".

We used to think that we need to prove something if it is not clear enough or when there are opposite views on the subject we are debating. Sometimes, things are not so obvious, and again, we need to prove it to some party.

In order to prove something we have to have an agreement which things we consider to be true at the first place, i.e. what are our initial, starting assumptions. That’s where the “debate” most likely will kick in. In most cases, debate is related to an effort to establish some axioms, i.e. initial truths, and only after that some new logical conclusions, or proofs will and can be done.

The major component of a mathematical proof is the domain of mathematical analysis. This domain has to be well established field of mathematics, and mathematics only. The proof is still a demonstration that something is true, but it has to be true within the system of assumptions established in mathematics. The true statement, the proof, has to (logically) follow from already established truths. In other words, when using the phrase "Prove something in math..." it means "Show that it follows from the set of axioms and other theorems (already proved!) in the domain of math..". Which axioms and theorems you will start the proof with is a matter of art, intuition, trial and error, or even true genius. You can not use apples, meters, pears, feelings, emotions, experimental setup, physical measurements, to say that something is true in math, to prove a mathematical theorem, no matter how important or central role those real world objects pr processes had in motivating the development of that part of mathematics. In other words, you can not use real world examples, concepts, things, objects, real world scenarios that, possibly, motivated theorems’ development, in mathematical proofs. Of course, you can use them as some sort of intuitive guidelines which axioms, or theorems, you will use to start the construction of a proof. You can use your intuition, feeling, experience, even emotions, to select starting points of a proof, to chose initial axioms or theorems in the proof steps, which, when combined later, will make a proof. But, you can not say that, intuitively, you know the theorem is true, and use that statement about your intuition, as an argument in a proof. You have to use mathematical axioms, already proved mathematical theorems (and of course logic) to prove the new theorems.

The initial, starting assumptions in mathematics are called fundamental axioms. Then, theorems are proved using these axioms. More theorems are proved by using the axioms and already proven theorems. Usually, it is emphasized that you use logical thinking, logic, to prove theorems. But, that's not sufficient. You have to use logic to prove anything, but what is important in math is that you use logic on mathematical axioms, and not on some assumptions and facts outside mathematics. The focus of your logical steps and logic constructs in mathematical proofs is constrained (but not in any negative way) to mathematical (and not to the other fields’) axioms and theorems.

Feeling that something is "obvious" in mathematics can still be a useful feeling. It can guide you towards new theorems. But, those new theorems still have to be proved using mathematical concepts only, and that has to be done by avoiding the words "obvious" and "intuition"! Stating that something is obvious in a theorem is not a proof.However, using own intuition to construct a proof or to formulate theorem is definitely useful.

Again, proving means to show that the statement is true by demonstrating it follows, by logical rules, from established truths in mathematics, as oppose to established truths and facts in other domains to which mathematics may be applied to.

As another example, we may say, in mathematical analysis, that something is "visually" obvious. Here "visual" is not part of mathematics, and can not be used as a part of the proof, but it can play important role in guiding us what may be true, and how to construct the proof.

Each and every proof in math is a new, uncharted territory. If you like to be artistic, original, to explore unknown, to be creative, then try to construct math proofs.

No one can teach you, i.e. there is no ready to use formula to follow, how to do proofs in mathematics. Math proof is the place where you can show your true, original thoughts.

[ set, set theory, concept of a set, sets in mathematics, real world, applied math, applied mathematics, axioms, math education, math proof, mathematical axioms, mathematical proof, mathematical theorems, mathematics, theorems, tutoring ]

How Mathematics Can Be Used in Real Life

You can download all the important posts as  PDF book "Unlocking the Secrets of Quantitative Thinking".

How to distinguish between pure math and applied math? How math can be applied to many different fields? How we can use math in real life? Why math looks so complicated? Where all those volumes of math come from? Can I revisit my math from primary and secondary school and finally understand what is it about? Can I learn math now? How I can teach my kids or students effectively primary and secondary school math?

I will be trying to answer these questions in the next several posts.

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[ applied math, applied mathematics, axioms, math and real life, math education, math, mathematics, physics concepts, tutoring, calculus, school, education ]