1) Introduction
To really understand how humans develop intelligence and knowledge systems, people have to retrace the whole history of philosophy and mathematics back to ancient time, and study various kinds of methodologies for different purposes.
This article is not a complete review of philosophies and sciences. It only addresses the essentials and methodologies for knowledge development, for the interests of sciences, philosophy, education, and artificial intellignece, etc. So although Dmitri Mendeleev and Albert Einstein are extremely important scientists, they did not contribute significant different approaches. Essentially they followed Galileo-Newton's: the classic scientific approach.
This article does not try to cross the boundary between knowledge and religions. Only religious rites are mentioned.
By approach, it means coherent approach in this paper. Coherence does not guarantee correctness. However, incohenrence is always prone to problems and errors.
2) Various Approaches
Several approaches from ancient time are identified here. They are: Pythagoras-Plato approach; Socrates-Stoicism approach; Euclid-Archimedes approach; Yi-Jing approach from ancient China; approaches from ancient India; approaches from other countries, etc.
Some approaches from medieval age played important roles in sciences: they are Ibn al-Haytham's approach, Al-Biruni's approach, and Avicenna's approach, etc.
Galileo-Newton approach, the foundation of current sciences, evolved from Euclid-Archimedes and Ibn al-Haytham approaches. But there are differences between them. Some non-classic approaches from Charles Darwin, Adam Smith, Sigmund Freud, etc., are also different from Galileo-Newton approach.
The following sections will first discuss the strengths and limitations of the first three ancient approaches, then the medieval approaches, and Galileo-Newton approach; Several non-classic approaches and important theoretic issues will also be discussed.
The ancient approaches from Yi-Jing, India, and other countries, would be discussed in separate articles, if possible.
3) Pythagoras-Plato Approach
Thales is a pioneer in mathematical proof. Egyptians and Babylonians might know Thales Theorem before him, but he was likely the first one providing a valid proof for it.
Although Thales tried to explain natural phenomena not based on mythology, he is a Hylozoist who believed everything is alive, and there is no difference between the living and the dead. He did not build a coherence approach to develop knowledge, but had significant influences on Pythagoras.
Pythagoras probably was the first one who built coherent methodologies to develop knowledge and systematic views. He formed a school of scholars to study philosophy, mathematics, music, etc.
Pythagorean are famous for Pythagorean Theorem. They were pioneers in mathematics as systematic studies. They also proposed a non-geocentric model that the Earth runs around a central fire which suggests both the Sun and the Earth are not the center of universe. They might develop or formulate the idea the Earth is round.
Pythagoras also taught religious rites and practices in his school, so came his beliefs. He and Plato believed there be a perfect and persistent abstract world, and an imperfect and sensible world. They pursued the beauty of abstract perfection. Plato followed this philosophy and developed it into maturity.
Pythagorean made many early contributions to knowledge. They tried to construct complicated things with simpler ones. They believed whole numbers be simple and perfect, and tried to represent all numbers with quotient of two whole numbers. Here they faced the first 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: the way they construct or interpret mathematics may not fit into the reality. The beauty of mathematics may not be able to explain the imperfect and sensible world. Constructivism is important. But how to construct, and to what extent it works ?
Such an issue even has a consequence on modern sciences and technologies: how macro nonlinearity and micro quantum phenomena are related to irrational numbers ? How should irrational numbers be handled on computers ?
The issue of irrational numbers is actually related to measurement. Euclid described it in a better way as mentioned in Section 5, which is not understood properly by many modern mathematicians. Measurement is again associated to nonlinear, chaos, deterministic uncertainty, or computer modelling, etc. So people better think of this issue as a tip of an iceberg, rather than a solved problem which they could forget about it.
4) Socrates-Stoicism approach
Socrates led different beliefs. He did not take the beauty of abstraction as a doctrine. There is a Socratic method. He asked people to question each other. When people discuss their arguments explicitly, they could find the problems and understand them better. Socratic method is an important element of scientific approach.
Socrates asked many questions, but did not give many answers. This might be a good attitude. Socrates taught by playing a role model. By admitting his ignorance, he suggested other people also to admit their ignorance.
Admitting their own ignorance is not beautiful, not even pleasant, but it is an extremely critical step to make further progress. However, this attitude is offensive to many people. Socrates was voted to death eventually, probabily under accumulated anger from others.
Then Socrates played a role model again by accepting the death to show the rule of laws.
Also, Aristotle believed it was Socrates who identified the method of definition and induction, etc., which are essential to sciences in future.
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, Ibn al-Haytham, Galileo-Newton, Darwin approaches, etc., later.
Sophism was an enemy to Socrates, and is also an enemy to future sciences. It does not provide a coherent approach, rather it tries to gain advantages by twisting the facts.
5) The Confusion of Aristotle's Philosophy
Aristotelianism is not coherent, too. Although Aristotle adored Socrates and claimed he was against sophism, his way is actually a mixture of Socrates, Plato and sophism without coherence. So he included assertions like a flying arrow is at rest in his book. He might enjoy such a twist, and cannot distinguish sophism from real profoundness.
His syllogism is an unsuccessful summarization and simulation of the methods in mathematical proofs. Aristotle tried to develop a perfect logic to guarantee the correctness of thinking. However, this only shows he did not really know how mathematics work. So he missed a very important factor: how to make the premises in syllogism valid and concrete, i.e. the first principle. Euclid provided a solution on this issue for some problems later.
Furthermore, he was not aware of that the logic based on ignorance or shallowness could suppress innovation and scientific progress. His assumed profoundness is a mental trap which hindered scientific progress for a very long time. His problems were illustrated by Galileo and Gödel later.
Since Aristotle did not know how to apply logic and reasoning correctly, he did not know how to build coherent theories. In his book The Physics he just put togather whatever he knew or believed into huge collections without paying attention to coherence, which brought very limited values to physics, but many misleadings.
Aristotle was actually a naturalist. He made some contributions to zoological taxonomy based on observation. He is not the first one using observation.
Aristotle's world view was invalided by Galileo in future. His logic was disproved by Godel later. His single logic would lead to social Darwinism, which actually takes opposite positions from real Darwinism on many issues, as mentioned in later sections.
His philosophy is to develop knowledge only good enough for real life, which might help people to absorb funds, build influences with assumed profoundness (declaring some simple logics based on ignorance or shallowness would be the only correct mindset, which is really fascinating), or even gain powers, etc.
So his philosophy is thought as one kind of realisms by many people.
6) Euclid-Archimedes Approach
Euclid is the first one who established a concrete systematic theory for a domain. He is more like a scientist, 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 word "measure", instead of a number divides another number. Many modern mathematicians think not being abstract enough is Euclid's limitation.
However, Euclid using measurement for division, implies the accurate value of irrational numbers cannot be measured by rational numbers. Measurability is still a critical problem in modern sciences and computer modelling. So, such an expression is not Euclid's limitation, but his insight. 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 many parts of modern mathematics.
He constructed the geometry system by some simple axioms, and derived other theorems from these axioms. Although this looks like Pythagorean's way to construct complicated numbers by simpler whole numbers, they are different.
Euclid guaranteed the correctness of derived theorems by making axioms simple and straightforward, thus self-evident. So Euclid solved the first principle issue in certain extent. Pythagorean did not show why whole numbers could be used to express all numbers, they just believed it be the beauty. Euclid's geometry is a good example of correct logic.
Euclid's approach mainly works in mathematics, and cannot find all theorems and laws in real needs. When Euclid worked on optics, his approach for first principle faced a problem: he did not have justified reasons to choose emission theory instead of intromission theory, although no justified reasons to make the opposite choice at that time, too. These limitations are partially solved by Ibn al-Haytham approach and Galileo-Newton approach, 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.
Archimedes, one of the greatest engineers, was highly influenced by Euclid.
7) Medieval approaches
Pythagoras is very insightful in mathematics, philosophy, music, religious practices, etc. Plato developed the philosophy in his style into maturity. Socrates and Stoicism knew the way to develop rationale and ethics. Euclid and Archimedes designed theoretic and real systems in rigorous forms.
They all made big contributions to knowledge development, and still have big influences so far, but also with problems. Their accomplishments are still in very limited extent. It is some medieval scholars who brought in some new factors critical to future sciences.
Ibn al-Haytham used some procedure to do scientific research: observe, form conjectures, Testing and/or criticism of a hypothesis with experimentation, etc. He used this procedure to prove intromission theory.
However, this procedure is not a complete scientific approach. He can only use it to reach individual results, still under Euclid's geometry view. Although he did some brilliant work in optics, due to this geometry view and his way of thinking, he cannot gain deep and comprehensive understanding of physics.
More important, Ibn al-Haytham did not know the problems of Aristotle's world views. So he did not know to get out of the mental trap created by Aristotle's philosophy, to establish new paradigns in sciences and develop systematic theories, which are critical to scientific revolutions. The tasks to change mentality were left to Copernicus, Galileo and Darwin, etc.
Al-Biruni put an emphasis on experimentation. He tried to conceptualize both systematic errors and observational biases, and used repeated experiments to control errors. He might also be a pioneer of comparative sociology.
Avicenna discussed the philosophy of science and summarized several methods to find a first principle. He developed a theory to distinguish the inclination to move and the force. He discussed the soul and the body, the perceptions, etc. Probably he is a very important scholar misunderstood by many modern people.
Only after their efforts, big scientific progresses became possible.
8) Galileo-Newton Approach
Nicolaus Copernicus constructed a systematic theory deviating from Aristotle's world views, thus starting the new age of sciences. However, he (and Johannes Kepler) still followed Euclid-Archimedes approach, a geometry view. He left these problems of world views and mental trap unexplained.
It is Galileo Galilei who formed much substantial understanding of the physical world and triggered a scientific revolution with his comprehensive and systematic thinking. With his remarkable work "Dialogue Concerning the Two Chief World Systems", Galileo combined Socratic Method and Ibn al-Haytham's method. More important, he showed the differences in world views and asked people to get out of the mental trap of Aristotle's philosophy. His systematic thinking and world view led to new scientific theories and establish new paradigms in physical sciences. These had not be done by Ibn al-Haytham's approach, who only could make individual isolated conclusions.
Isaac Newton developed his great theories based on Galileo's approach. So this classic scientific approach is named as Galileo-Newton approach. Although Newton did a great work, he did not explain why he could do these, i.e., summarize well what are the real differences between their approach and Ibn al-Haytham's approach, and what are the limitations of their approach. It is the responsibility of future philosophers to explain the mechanisms behind these.
Galileo-Newton approach mainly work in worlds without considering the effects of life. Thomas Robert Malthus and John Maynard Keynes followed this classific scientific approach and tried to apply it to life and human societies. They made some progresses, but very limited. And they missed something very important to human beings.
Actually this approach does not work well in artificial intelligence, psychology, economics, and other humanity and social sciences, etc. Those fields relate to humans. Measurability and modelling are still big concerns. People could look at Newton's four rules of reasoning stated in his Mathematical Principles of Natural Philosophy, to figure out what the problems really are:
"1.We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.
2.Therefore to the same natural effects we must, as far as possible, assign the same causes.
3.The qualities of bodies, which admit neither intension nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever.
4.In experimental philosophy we are to look upon propositions collected by general induction from phænomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions".
Actually, many important knowledge systems were not built with this approach.
9) Non-Classic Approaches
As great academic contributors, Charles Darwin, Adam Smith, and Sigmund Freud built their theories not with Galileo-Newton approach. These three are listed here in the descending order of closeness to sciences. No surprise, their theories relate to humans.
Some people classified Charles Darwin as a follower of Francis Bacon's methodology. It is not true. Darwin did much more than observation, induction, etc., empiricism methods, to construct his great theories for life and human history. He is the greatest historian in history. He established a paradigm to combine sciences, life, and humanity, etc.
Although Alfred Russel Wallace was thought as a co-founder of evolution theory by many people, I only discussed Charles Darwin here due to pure academic reasons.
It is also the responsibility of future philosophers to illustrate the differences between Darwin's approach and Ibn al-Haytham's approach, and the differences between Darwin's approach and Galileo-Newton approach.
People also should pay attention to the differences between Darwinism and Social Darwinism. They actually took opposite positions on many issues. Darwinism cherishes diversities, and does not agree with the single logic of Aristotelianism.
Epicurus, Arthur Schopenhauer, Friedrich Nietzsche also illustrated many important understandings, related to human natures. Their ways are different from classic approaches, too.
10) The Progresses in Natural Sciences So Far
After Galileo-Newton approach was established, many theories in natural sciences were proposed such as periodic table, genetics, relativity theories, quantum theories, etc. They mainly followed Galileo-Newton approach. Relativity theories are just refinements to Newton dynamics, just as Kepler refined Copernicus' circle orbits with ellipse orbits. Today people know planets do not run in ellipse orbits, too. So people do not know whether relativity theories are the final theories.
There are good examples for work cross different approaches, such as the synthesis of evolution and genetics theories in life sciences. But people do not have good theories in psychology and intelligence yet, and do not know whether Galileo-Newton approach would work well in these fields. Even in biology, most mechanisms can not be explained yet.
The progresses mentioned above mainly happened before World War II. After World War II, there are genome, and the Standard Model of particle physics. The significance of these two progresses could be compared with periodic table. However, since periodic table is the first of such type of models, proposed much earlier, so genome and the Standard Model of particle physics are less important than periodic table, and far less important than Euclid, Copernicus, Newton, and Darwin's theories.
11) Inadequate Summarizing Efforts
Many people tried to summarize the knowledge systems, such as: Francis Bacon, René Descartes, David Hume, Immanuel Kant, etc.
Georg Wilhelm Friedrich Hegel, Karl Marx, Bertrand Russell even made more ambitious efforts.
Just they all missed some or many important aspects.
12) Godel Theorems and the Limitations of Mathematics and Positivism
Godel incompleteness theorems illustrate the intrinsic problems in mathematics systems beyond certain complexity. Actually any systems including Peano Arithmetric cannot be both complete and consistent. So the main goals of Hilbert's program cannot be achieved. This is the foundational crisis in mathematics.
Actually Godel theorems show Aristotle's philosophy is wrong. There is no perfect logic in this world. This happend almost three hundreds years later, after Galileo showed Aristotle's world views are wrong. Darwinsim actually does not agree with the single logic of Aristotelianism, too.
Positivism is an ideal goal for Galileo-Newton approach. However, as said, Charles Darwin, Adam Smith, Sigmund Freud and many others built their theories and systems not strictly with this approach.
In the whole knowledge system, positivism, just like perfect logic, is more like a Utopia, rather than a reality. Even so, as one of important doctrines in sciences, positivism should not be ignored, especially in the conclusion stage.
13) Future Researches
Some of the challenges for future are identified here:
1. How could people establish new paradigms and develop better world views ? These are the mechanims missed in Ibn al-Haytham's approach. Galileo, Newton, Darwin did these well. But they did not summarize well how they achieved these. Francis Bacon, René Descartes, David Hume, Immanuel Kant, etc., did not summarize these well, too.
2. Identify the limitations in Galileo-Newton approach.
3. Identify the limitations in Darwin's approach.
4. Develop an approach for future researches on life, psychology, and intelligence, etc.
It is up to future philosophers to figure these out and illustrate the mechanisms behind these.
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