Abstraction
Could computers understand and represent irrational numbers without knowing their exact values? Humans can. The measurement of generic intelligence is critical to further development of artificial intelligence (AI). However, the existing methods: Turing Test and its variants cannot really measure intrinsic intelligence capabilities. Based on the studies of knowledge development, several essential design goals for intelligence measurement are identified. A new method: Gu Test is proposed, to distinguish strong AI and regular machines, and meet some of these design goals. Further improvement could be done in future.
1. The Measurement of Generic Intelligence
Measurement is so important in sciences and technologies. Just as clocks are necessary to advanced studies of motion and speed, centrifugal governors are critical to make steam engines usable.
The measurement of generic intelligence is also important to artificial intelligence (AI). However, the existing measuring methods, such as Turing Test and its variants, are mainly behavior-based, knowledge-based, or task-based, etc., which cannot really measure intrinsic intelligence capabilities.
A new way of measurement : Gu Test, is proposed, to distinguish strong AI from regular machines.
2. Turing Test and Chinese Room concern
Alan Turing described an imitation game in his paper Computing Machinery and Intelligence [1], which tests whether a human could distinguish a computer from another human only via communication without seeing each other.
It is a black box test, purely based on behavior. To pass this kind of tests, computers only need to imitate humans.
So John Seale raised a Chinese Room issue [2], i.e., computers could pass this test by symbolic processing without really understanding the meanings of these symbols.
Also, Turing Test uses interrogation to test, so it only can test those human characteristics which already be understood well by humans and can be expressed in communication. Humans still have very limited understanding of life, psychology, and intelligence. So those intrinsic intelligence abilities which humans do not understand well yet could not be tested only via interrogation.
3. Variants of Turing Test
There are several variants of Turing Test which aim at improving on it.
One is Feigenbaum test. According to Edward Feigenbaum, "Human intelligence is very multidimensional", "computational linguists have developed superb models for the processing of human language grammars. Where they have lagged is in the 'understand' part", "For an artifact, a computational intelligence, to be able to behave with high levels of performance on complex intellectual tasks, perhaps surpassing human level, it must have extensive knowledge of the domain." [3].
There are two issues in this test. One is current computers only can store and process knowledge in some data forms. If some knowledge cannot be represented as data, then the "understanding part" of such knowledge would not be solved. The other issue is whether extensive knowledge is necessary to test strong AI, since individual humans may not have very extensive knowledge in certain domains.
Feigenbaum test is actually a very good method to measure expert systems. However, to measure generic AI or test strong AI, the essentials of "understanding part" need be identified. Whether they could be transformed into data and data processing should be parts of testing. Gu Test actually tries to solve these issues.
Another variant is Shane Legg and Marcus Hutter's solution [4], which is actually agent-based. In their framework, an agent sends its actions to the environment and received observations and rewards from it. If their framework is used to test strong AI, then it assumes that all the interactions between humans and their environment could be modeled by actions/observations/rewards. This assumption has not been tested.
Humans can play some roles of agents, but they are not just agents. Humans could make paradigm evolution, which usually means gain deeper observations, take better actions, and gain more rewards than what already in any definitions.
Actions/observations/rewards could be defined for specific tasks and agents. But how could these be defined for paradigm evolution, and for the whole life of humans ? Humans could make exceptions and innovations. Without these, there would be no Euclid, Galileo and Columbus, etc.
There are some essential parts of intrinsic intelligence which are not in their framework. Much more research could be done to further study the difference between humans and agents.
Even if Turing Test is enhanced with vision and manipulation ability, or with methods like statistical inference, etc., it still does not test the difference between knowledge and data, between simulated behavior and intrinsic intelligence, etc.
4. The Design Goals for Generic Intelligence Measurement
Based on the analysis done in previous sections, some design goals are proposed here:
1) Resolve Chinese Room issue, i.e., to test the real understanding, not just behavior imitating or symbolic processing.
2) Not just rely on interrogation. Find some ways to test those intrinsic intelligence abilities which have not been understood and expressed well.
3) Test those concepts, knowledge and intelligence which cannot be represented as data yet.
4) Involve as less domain knowledge as possible, since regular humans may not have much knowledge in specific domains.
5) Include those intrinsic capabilities commonly necessary in many domains, with which humans can develop intelligence in many domains.
6) Include a sequence of leveled tests, since humans are able to make continuous progresses in intelligence.
7) Include a framework to test structured intelligence and be able to make paradigm evolution, since humans can develop sophisticated knowledge structures and make paradigm evolution.
5. Gu Test
Based on these design goals, Gu Test is proposed. It should include a sequence of test levels, and be able to test structured intelligence and make paradigm shift.
However, currently only a first test step is suggested, to meet the goals from 1) to 5). The work to meet the design goals 6) and 7) will be left to future researches.
The first test step of Gu Test is : to test whether testees could understand irrational numbers without knowing their exact values. It is a white box test. Average humans with certain basic education can. Current computers most likely cannot.
It tests the real understanding; It does not rely on interrogation, but tests some intrinsic ability; Humans can pass this test, but they probably do not know why they have this ability yet; It tests some concepts and knowledge which cannot be represented as data; Irrational number is a primitive concept developed in Pythagoras' age, who is a poineer in philosophy and mathematics; The concept is necessary to so many domains, but involves very little domain-specific knowledge.
6. Future Research
Much more work need be done to extend Gu Test to meet the design goals 6) and 7). To really understand the essentials of intelligence, people have to study the history of knowledge development, philosophy, mathematics, sciences, etc.
References
[1] Turing, A. M., 1950, "Computing machinery and intelligence". Mind 59, 433–460.
[2] Searle, John. R., 1980, "Minds, brains, and programs". Behavioral and Brain Sciences 3 (3): 417-457.
[3] Feigenbaum, Edward A., 2003, "Some challenges and grand challenges for computational intelligence". Journal of the ACM 50 (1): 32–40.
[4] Legg, S. & Hutter, M., 2006, "A Formal Measure of Machine Intelligence”, Proc. 15th Annual Machine Learning Conference of Belgium and The Netherlands, pp.73-80.
Wednesday, December 28, 2011
Tuesday, December 13, 2011
Different Approaches For Knowledge System Development (version 2)
1) Introduction
Could computers measure and represent irrational numbers ? This question relates to some essentials of human intelligence. To understand this question, people have to retrace the whole history of philosophy and mathematics, back to Pythagoras age.
Humans with certain education could understand irrational numbers without knowing their exact values. Turing Test is not a good measurement for intelligence. The author of this article constructs Gu Test, which could test intelligence levels better.
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 artificial intellignece and education, etc. So although Dmitri Ivanovich Mendeleev and Albert Einstein are extremely important scientists, they are not discussed here since they essentially followed the classic scientific approach: Galileo-Newton approach.
It 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
The difficulties in artificial intelligence (AI) researches challenge the limits of classic scientific approaches. It would be helpful to review the history how people used various ways to develop knowledge.
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 and other countries, etc.
In theoretic research of AI, there is a very important issue: the difference between Euclid's constructive approach and many of modern mathematics. Say, could people construct irrational numbers on computers ?
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 strength and limitations of the first three ancient approaches, the medieval approaches, then Galileo-Newton approach. Several non-classic approaches and theoretic issues important to AI will also be discussed. Gu Test is proposed to measure intelligence levels based on these studies.
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 develop a coherence approach, but had significant influences on Pythagoras.
The first philosopher should be Pythagoras, who built a coherent systematic view. He formed a school of scholars to study philosophy, mathematics, music, etc.
Pythagorean are famous for Pythagorean Theorem. They are pioneers in mathematics, a systematic study. 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 an sensible world. Constructivism is important. But how to construct ?
This has a consequence for AI: how could people construct irrational numbers on computers, or could computers distinguish irrational numbers from rational numbers?
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 by many modern mathematicians. Measurement is again associated to nonlinear, chaos or deterministic uncertainty, etc. So people better think of it 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 to find the problems. When people discuss their arguments explicitly, they could find the problems and understand them better.
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 ignorance is not beautiful, not even pleasant, but an extremely critical step to make further progresses. 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 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, Ibn al-Haytham, and Galileo approaches later.
Sophism was an enemy to Socrates, and is also an enemy to future sciences. It does not provide a coherent approach.
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.
His syllogism is an unsuccessful summarization and simulation of the methods in mathematical proofs. Aristotle did not know how the logic really works in mathematics. So he missed a very important factor: how to make the premises in syllogism valid and concrete, i.e. the first principle.
Since Aristotle did not know how to apply logic and reasoning correctly, he did not know how to build coherent theories. His book The Physics brought little values to physics, but many misleadings. He just put togather whatever he knew or believed into huge collections without paying attention to coherence.
Aristotle was actually a naturalist. He made some contributions to zoological taxonomy based on observation. He is not the first one using observation. And he does not have a coherent approach.
5) 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. This becomes a problem in modern computer sciences: how could a computer represent irrational numbers and distinguish them from rational numbers ?
Measurability is still a critical problem in modern physics and AI. So, this usage 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 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 guarantees 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 is the beauty. Euclid's geometry is a good example of correct logic.
There are still limitations of Euclid's approach. It 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.
6) 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 very limited. 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 using 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.
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 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. Probably he is a very important scholar misunderstood by many modern people.
Only after their efforts, big scientific progresses became possible.
7) Galileo-Newton Approach
Although Nicolaus Copernicus started the new age of sciences, he (and Johannes Kepler) still followed Euclid-Archimedes approach, the geometry view.
It is Galileo Galilei who formed many substantial understanding of the physical world and triggered a scientific revolution with his comprehensive and systematic thinking. Without such a way to think, people cannot build a new world theory from individual isolated conclusions. So his approach is different from Ibn al-Haytham's method. Isaac Newton developed his theories based on Galileo's work. This classic scientific approach is named as Galileo-Newton approach.
There are still limitations in this approach. It does not work well in AI, psychology, economics, etc. Those fields relate to humans. Measurability is still a big concern. There are even problems in Newton's four rules of reasoning stated in his Mathematical Principles of Natural Philosophy.
Actually, many important knowledge systems were not built with this approach.
8) Non-Classic Approaches
Although as great academic contributors, Charles Darwin, Adam Smith, and Sigmund Freud built their theories not with Galileo-Newton approach. These three theories are in the descending order of closeness to sciences. No surprise, their theories relate to humans.
Epicurus, Arthur Schopenhauer, Friedrich Nietzsche illustrated many important ideas, also related to human natures. Their ways are different from classic approaches, either.
9) Inadequate Summarizing Efforts
Many people tried to summarize the knowledge systems, such as: 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.
10) Measurement and Constructivism
Turing Test is not a right way to test the intelligence of computers. It is superficial and too subjective. Instead, Gu Test is designed to better measure the intelligence levels of computers.
Gu Test contains various tests for different intelligence levels. One necessary, but not sufficient test for strong AI is : could computers understand the real meaning of irrational numbers, measure their values, and represent them in their memory ?
Humans with certain education could understand what irrational numbers are and what PI is although they do not know the exact values of these numbers. Some experts could do certain accurate measurement for them.
So Gu Test could test some essentials of intelligence, and avoid the so-called Chinese Room issue.
In the physical world, measuring does not mean knowing the exact value. This is a difference between mathematics and physics. That is why "after Galileo founded Physics, Euclid retired from scientists, and became a mathematician only".
11) Gödel Theorems and the Limitations of Mathematics and Positivism
Gödel theorems illustrate the intrinsic problems in mathematics systems beyond certain complexity. There are foundational crisis in mathematics as in http://en.wikipedia.org/wiki/Foundations_of_mathematics#Foundational_crisis.
More important is, mathematics cannot explain all the potentials in reality.
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 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.
12) Main Contributions
The main contributions of this article are:
1) Identify an important intelligence phenomena: humans with certain education could understand irrational numbers without knowing their exact values.
2) Construct a Gu Test to better measure intelligence levels based on 1).
3) Show Turing Test is not a good measurement for Strong AI based on 2).
4) Clarify Galileo-Newton approach is different from Ibn al-Haytham's method.
5) Claim there are limitations and problems in Newton's four rules of reasoning stated in his Mathematical Principles of Natural Philosophy.
Please correct me if I am wrong.
13) Future Researches
To be explored and explained in future.
Could computers measure and represent irrational numbers ? This question relates to some essentials of human intelligence. To understand this question, people have to retrace the whole history of philosophy and mathematics, back to Pythagoras age.
Humans with certain education could understand irrational numbers without knowing their exact values. Turing Test is not a good measurement for intelligence. The author of this article constructs Gu Test, which could test intelligence levels better.
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 artificial intellignece and education, etc. So although Dmitri Ivanovich Mendeleev and Albert Einstein are extremely important scientists, they are not discussed here since they essentially followed the classic scientific approach: Galileo-Newton approach.
It 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
The difficulties in artificial intelligence (AI) researches challenge the limits of classic scientific approaches. It would be helpful to review the history how people used various ways to develop knowledge.
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 and other countries, etc.
In theoretic research of AI, there is a very important issue: the difference between Euclid's constructive approach and many of modern mathematics. Say, could people construct irrational numbers on computers ?
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 strength and limitations of the first three ancient approaches, the medieval approaches, then Galileo-Newton approach. Several non-classic approaches and theoretic issues important to AI will also be discussed. Gu Test is proposed to measure intelligence levels based on these studies.
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 develop a coherence approach, but had significant influences on Pythagoras.
The first philosopher should be Pythagoras, who built a coherent systematic view. He formed a school of scholars to study philosophy, mathematics, music, etc.
Pythagorean are famous for Pythagorean Theorem. They are pioneers in mathematics, a systematic study. 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 an sensible world. Constructivism is important. But how to construct ?
This has a consequence for AI: how could people construct irrational numbers on computers, or could computers distinguish irrational numbers from rational numbers?
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 by many modern mathematicians. Measurement is again associated to nonlinear, chaos or deterministic uncertainty, etc. So people better think of it 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 to find the problems. When people discuss their arguments explicitly, they could find the problems and understand them better.
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 ignorance is not beautiful, not even pleasant, but an extremely critical step to make further progresses. 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 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, Ibn al-Haytham, and Galileo approaches later.
Sophism was an enemy to Socrates, and is also an enemy to future sciences. It does not provide a coherent approach.
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.
His syllogism is an unsuccessful summarization and simulation of the methods in mathematical proofs. Aristotle did not know how the logic really works in mathematics. So he missed a very important factor: how to make the premises in syllogism valid and concrete, i.e. the first principle.
Since Aristotle did not know how to apply logic and reasoning correctly, he did not know how to build coherent theories. His book The Physics brought little values to physics, but many misleadings. He just put togather whatever he knew or believed into huge collections without paying attention to coherence.
Aristotle was actually a naturalist. He made some contributions to zoological taxonomy based on observation. He is not the first one using observation. And he does not have a coherent approach.
5) 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. This becomes a problem in modern computer sciences: how could a computer represent irrational numbers and distinguish them from rational numbers ?
Measurability is still a critical problem in modern physics and AI. So, this usage 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 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 guarantees 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 is the beauty. Euclid's geometry is a good example of correct logic.
There are still limitations of Euclid's approach. It 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.
6) 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 very limited. 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 using 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.
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 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. Probably he is a very important scholar misunderstood by many modern people.
Only after their efforts, big scientific progresses became possible.
7) Galileo-Newton Approach
Although Nicolaus Copernicus started the new age of sciences, he (and Johannes Kepler) still followed Euclid-Archimedes approach, the geometry view.
It is Galileo Galilei who formed many substantial understanding of the physical world and triggered a scientific revolution with his comprehensive and systematic thinking. Without such a way to think, people cannot build a new world theory from individual isolated conclusions. So his approach is different from Ibn al-Haytham's method. Isaac Newton developed his theories based on Galileo's work. This classic scientific approach is named as Galileo-Newton approach.
There are still limitations in this approach. It does not work well in AI, psychology, economics, etc. Those fields relate to humans. Measurability is still a big concern. There are even problems in Newton's four rules of reasoning stated in his Mathematical Principles of Natural Philosophy.
Actually, many important knowledge systems were not built with this approach.
8) Non-Classic Approaches
Although as great academic contributors, Charles Darwin, Adam Smith, and Sigmund Freud built their theories not with Galileo-Newton approach. These three theories are in the descending order of closeness to sciences. No surprise, their theories relate to humans.
Epicurus, Arthur Schopenhauer, Friedrich Nietzsche illustrated many important ideas, also related to human natures. Their ways are different from classic approaches, either.
9) Inadequate Summarizing Efforts
Many people tried to summarize the knowledge systems, such as: 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.
10) Measurement and Constructivism
Turing Test is not a right way to test the intelligence of computers. It is superficial and too subjective. Instead, Gu Test is designed to better measure the intelligence levels of computers.
Gu Test contains various tests for different intelligence levels. One necessary, but not sufficient test for strong AI is : could computers understand the real meaning of irrational numbers, measure their values, and represent them in their memory ?
Humans with certain education could understand what irrational numbers are and what PI is although they do not know the exact values of these numbers. Some experts could do certain accurate measurement for them.
So Gu Test could test some essentials of intelligence, and avoid the so-called Chinese Room issue.
In the physical world, measuring does not mean knowing the exact value. This is a difference between mathematics and physics. That is why "after Galileo founded Physics, Euclid retired from scientists, and became a mathematician only".
11) Gödel Theorems and the Limitations of Mathematics and Positivism
Gödel theorems illustrate the intrinsic problems in mathematics systems beyond certain complexity. There are foundational crisis in mathematics as in http://en.wikipedia.org/wiki/Foundations_of_mathematics#Foundational_crisis.
More important is, mathematics cannot explain all the potentials in reality.
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 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.
12) Main Contributions
The main contributions of this article are:
1) Identify an important intelligence phenomena: humans with certain education could understand irrational numbers without knowing their exact values.
2) Construct a Gu Test to better measure intelligence levels based on 1).
3) Show Turing Test is not a good measurement for Strong AI based on 2).
4) Clarify Galileo-Newton approach is different from Ibn al-Haytham's method.
5) Claim there are limitations and problems in Newton's four rules of reasoning stated in his Mathematical Principles of Natural Philosophy.
Please correct me if I am wrong.
13) Future Researches
To be explored and explained in future.
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