In theoretic research of artificial intelligence, there is a very important issue: the difference between Euclid's constructive approach and many modern mathematician's approaches. Please note the word 'constructive' here.
For those who are only interested in technologies and engineering, they better know, one of the greatest engineers, Archimedes, was highly influenced by Euclid.
Unfortunately, most of current students spend much time in learning geometry, but they know nothing about the philosophy and methodology established by Euclid. So they could apply mathematics in a wrong way later, such as in economics which is almost dominated by mathematical solutions now.
So comes this study. Several approaches for knowledge system development are identified here. They are: Pythagoras-Plato approach; Socrates-Stoicism approach; Euclid-Archimedes approaches; Yi-Jing approach from ancient China; approaches from ancient India and others, etc.
Galileo-Newton approach, the foundation of current sciences, evolved from Euclid-Archimedes approach. But the two are also different somehow.
The first three approaches are discussed here. The others would be covered if time and resources are available in future.
Please note only the differences and problems are discussed here, with no mention who are correct and who are wrong. However, Aristotle's approach is not listed here, because it might be wrong, not just with some problems. I discussed The Aristotle's Approach in a different article, since it is more controversial.
1) Pythagoras-Plato Approach
The first philosopher should be Pythagoras, because he called himself so. He is famous for the Pythagorean theorem, of the length of three sides of right triangle. Here mathematics and philosophy started togather.
Those who think philosophy is almost dead in modern time, may be wrong. A long list of philosophies in mathematics is attached at the end of this article. Mathematics and philosophy are highly related to each other, from the beginning to now.
Pythagoras also taught religious rites and practices in his school, so came his beliefs. He and Plato believed there is a perfect and persistent abstract world, and an imperfect and sensible world. They persued the beauty of abstract perfection.
Pythagorean made many early contributions to knowledge. They tried to represent complicated things with simpler ones. They also gave mathematics their own interpretation. They believed whole numbers are simple and perfect, and tried to represent all numbers with quotient of two whole numbers. Here they faced a mathematical crisis: some, actually most of the numbers, cannot be expressed in such a way. They called these irrational numbers.
This discloses the problems of Pythagoras-Plato approach: could the beauty of mathematics be used to express rationale and fairness, etc., as done by Pythagorean ?
Please note irrational numbers are actually related to nonlinear and chaos or deterministic uncertainty. So although people accept irrational numbers now, is this mathematical crisis really solved ? As said, the studies here is not about history, but about the development of knowledge systems, artificial intelligence, and educational psychology, etc.
2) Socrates-Stoicism approach
Socrates led different beliefs. There is a Socratic method. He asked people to question each other to find the problems. When people discuss explicitly their arguments, they could find the problems and answers easily.
Socrates asked many questions, but did not give many answers. This actually might be a good attitude. Since he asked all people including himself to admit they are igorant.
Admitting their ignorance is an extremely critical step to make future progresses. However, this attitude are offensive to many people, and Socrates was voted to death eventually.
Also, "Aristotle attributed to Socrates the discovery of the method of definition and induction, which he regarded as the essence of the scientific method." (from wikipedia).
Socrates promoted rationale and ethics, which was followed by Cynicism and Stoicism. However, the strange behaviors of Cynicism actually showed the frustrations faced by this approach: they did not find effective ways to discover much more knowledge. This task would be achieved by Euclid-Archimedes approach and Galileo-Newton approach later.
3) Euclid-Archimedes Approaches
Euclid is the first one who established a concrete systematic theory for a domain. He is more like a scientist, rather than a pure mathematician.
He did not concern much of the beauty of abstraction. In the proof of the number of prime numbers, he used the language 'measure', instead of if a number is dividable by another number. Many modern mathematicians think not being abstract enough is a limitation of Euclid.
Please note measurability is still a critical problem in modern physics and artificial intelligence. So, this is actually Euclid's strength. He caught the essentials of the problems.
He usually constructed solutions rather than just proving the existence of unknown solutions. His system is incredible concrete after more than two thousand years, much more concrete than much of modern mathematics.
He constructed the whole system by some simple axioms, and derived other theorems from these axioms. Although this looks like Pythagorean's representing complicated numbers by simpler whole numbers, they are different.
Euclid guarrentees the correctness of derived theorems by makng axioms simple and straghtforward, thus self-evident. Pythagorean did not show why whole numbers could be used to express all numbers, they just believed it is the beauty. Euclid's works are good examples of logic application.
There is still a limitation of Euclid's approach. His system mainly works in geometry, and cannot find all theorems and laws. This limitation is partially solved by Galileo-Newton approach, please note only 'partially'.
Euclid was truly thought as a scientist in early days. Only after Galileo founded Physics, Euclid retired from scientists, and became a mathematician only.
Galileo-Newton approach is still limited in its application domains. It does not work well in artificial intelligence, psychology, economics, etc. Measurability is still a big concerns.
Pythagoras is very insightful in mathematics, philosophy, religious practices and beliefs. Plato developed the philosophy in his style to maturity. Socrates and Stoicism knew the way to develop rationale and ethics. Euclid and Archimedes designed theoretic and real systems in rigorous ways. They all made big contributions to the development of knowledge systems.
They still have big influences so far, but also with their problems. Even Galileo-Newton approach did not solve most of the problems. The problems already solved are only a small portion.
From irrational numbers, to the modern foundational crisis in mathematics, there are some essential problems not solved yet. People only tried to avoid them. New approaches and philosophies are needed. Usually, establishment of a new domain means a new type of philosophy.
For the foundational crisis in mathematics, please see:
http://en.wikipedia.org/wiki/Foundations_of_mathematics#Foundational_crisis
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The philosophies in mathematics:
(From http://en.wikipedia.org/wiki/Philosophy_of_mathematics)
Mathematical realism
Platonism
Full-blooded Platonism
Empiricism
Mathematical Monism
Logicism
Formalism
Conventionalism
Psychologism
Intuitionism
Constructivism
Finitism
Structuralism
Embodied mind theories
New Empiricism
Aristotelian realism
Fictionalism
Social constructivism or social realism
Quasi-empiricism
Popper's "two senses" theory
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