Branch: Introduction to the Axiomatic Method

The central focus of BoP is mathematics, theoretical physics and computer scienses. Why this focus and what do these sciences have to do with the axiomatic method?

In order to answer this question, let us first ask and answer another question: Is mathematics a natural science or does it belong to humanities?

Most of the mathematical objects and disciplines deal, in fact, with natural phenomena. However, note that the objects, which mathematicians usually deal with, i.e. numbers, vectors, matrices, equations, functions, are purely abstract. They only exist within the human mind, they are a pure product of the human thought.

Therefore, an answer to the second question can be that mathematics neither belongs to natural sciences nor to humanities. In fact, it belongs to the third kind of sciences - the structural sciences. Structural sciences deal with patterns and structures. For instance, in mathematics,

• the disciplines of geometry study the patterns and structures of space,
• differential equations describe the patterns and structures of movement,
• probability theory and statistics deal with the patterns and structures of randomness,
• logic is the discipline studying the patterns and structures governing formal argumentation,
• topology fathoms the patterns and structures of position and location.

Like mathematics, also theoretical physics and computer sciences belong to structural sciences. And this is what BoP is dedicated to. The axiomatic method is a pillar of structural sciences. Therefore, structural sciences are predestined to demonstrate what axiomatic method is and how powerful and useful it is for these sciences.

What is the Axiomatic Method?

Unfortunately, the axiomatic method it is not broadly known. The method is very simple and it always follows these steps:

1. Assert the truth of a couple of statements (and call them axioms or postulates).
2. Add the postulates to a list (and call that list theory).
3. Given all statements in your theory, logically derive new statements which are true (and call them propositions or theorems).
5. Continue with step 2.

This method is as simple as - like a snowball - very powerful! It allows constructing complex theories from easy to understand basic axioms. The rules of logic ensure the truth of every single theorem in your theory.

Is the Axiomatic Method Important for Education?

Yet only at first sight does the axiomatic method inhere in structural sciences only. For this reason, unfortunately, it is not broadly known. There are many reasons, why the axiomatic method deserves the same significance in education as literacy does, including these:

1. It can be used to derive theories, which successfully describe the world around us. For instance, the Euclidean geometry, which is a theory based on only 5 axioms, is a theory enabling us to correctly measure every-day lengths, areas, and volumes.
2. The axiomatic method is just another word for analytical thinking. It is a "toolbox" for logical thinking, going far beyond structural sciences like mathematics. Problem-solving services of craftsmen, investigations, scientific works, audits, legal proceedings, etc. all implicitly require the capabilities of the axiomatic method.

Thus, the ability to use the axiomatic method and to think logically should be taught from the very beginning of every elementary school, going hand in hand with teaching the ability to read and write.

Criticism of the Axiomatic Method

Nevertheless, there are some substantial limits of the axiomatic method. One important limitation is its "bivalence" - it always depends on only two outcomes: either a theorem is true or it is false. Therefore, mathematics and every other theory about the world, which was derived using the axiomatic method, can only model the real world but never be the real world. A theory about the world cannot describe it completely. To be more concrete: Just think about statements like "sports is good for you" or "it is 10 o'clock" and try to assign them a value of being true or false.

Assigning the values true or false to each logical statement is known as the law of the excluded middle, which is an axiom itself and is used (either explicitly or implicitly) in almost all mathematical theories. It is a concept of "absolute truth", i.e. the assumption that statements can be either true or false, but never something in between. In fact, some famous mathematicians and philosophers used this the concept of "absolute truth" to define pure mathematics:

"Pure Mathematics is the class of all propositions of the form '$p\text{ implies }q$' where $p$ and $q$ are propositions containing one or more variables, the same in the two propositions, and neither $p$ nor $q$ contains any constants except logical constants [...]. In addition to these, mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth." (Bertrand Russell, from Principia Mathematica, 1903).

Like Russel, many mathematicians at the beginning of the 20th century believed in the concept of "absolute truth". The scientific progress made in the 19th century seemed to be unbroken. In his opening speech to the 1900 Congress in Paris, another famous mathematician, David Hilbert (1862 - 1943) said:

"Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?"

In his euphoria about the genius of the human mind, Hilbert proposed in the Parisian Congress, among others, to solve some still unsolved problems1. The second problem from his list of unsolved problems was to prove the consistency of the axioms of arithmetics (Peano axioms). As long as the consistency of these axioms remained unproven, one could not with certainty decide whether $1+1=2$ or $1+1\neq 2.$

Many mathematicians started desperately to solve the second Hilbert problem, but none succeeded in doing so. At the latest since the discoveries of Kurt Gödel (1906 - 1978), some decades after Hilbert, mathematicians finally lost their faith in the (wrong) concept of "absolute truth" and, consequently, the power of mathematical reasoning and the axiomatic method. Gödel shocked the wrong belief that the axiomatic method was capable to prove or disprove the completeness and the consistency of every given theorem or axiomatic system. These limitations became known as Incompleteness Theorems. They basically state that:

1. Given a theory derived using the axiomatic method, it is impossible by means of this theory to prove or disprove that it is free of contradictions (i.e. its consistency).
2. If we have a consistent theory derived from a given axiomatic system, there will always be statements formulated in this theory, which cannot be proved or disproved using this theory (i.e. the theory never becomes complete).

In other words, Gödel proved that using the axiomatic theory it is fundamentaly impossible to confirm that there are no contradictions in a theory. Even worse, it is also fundamentaly impossible to prove or disprove all statements which can be formulated inside the theory - the will always remain some statements left in the theory, which cannot be proven using the theory - leaving us in the uncertainty about their truth.

Despite these theoretical (and philosophical?) difficulties, BoP consequently uses the axiomatic method as the best approach known to this day to establish theories, which model the real world surrounding us.

Axioms: 1
Parts: 2 3 4
.s: 5

Thank you to the contributors under CC BY-SA 4.0!

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References

Bibliography

1. Govers, Timothy: "The Princeton Companion to Mathematics", Princeton University Press, 2008,
2. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011

Adapted from other CC BY-SA 4.0 Sources:

1. O’Connor, John J; Robertson, Edmund F: MacTutor History of Mathematics Archive

Footnotes

1. Which later became his famous twenty-three Millennium Problems, some of which are still unsolved today.