An Essay on Human Intelligence

Norimasa Kobayashi
1996

Contents:

  1. Verbalization of Intelligence
  2. Information Science (Computer Science) and Verbalized Intelligence
  3. About Verbalized Intelligence
  4. Science
  5. The Physical World (The Universe)
  6. Science, Metaphysics and Technology
  7. Value
  8. Verbal Communication

1. Verbalization of Intelligence

First of all, what does it means 'to think'? Thinking is not limited to making or interpreting verbal expressions. For example, in 'The Emperor's New Mind' by Roger Penrose, there is a quotation of a letter Hadarmard (a French mathematician) received from Einstein.

The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements of thought are certain signs and more or less clear images which can be 'voluntarily' reproduced and combined . . . The above mentioned elements are, in my case, of visual and some muscular type. Conventional words or other signs have to be sought for laboriously only in a second stage, when the mentioned associative play is sufficiently established and can be reproduced at will.

Here, we can see that scientific (or mathematical) ideas that are usually thought as most logical are indeed very intuitive when they come into the minds of scientists (or mathematicians) for the first time.

There are more fundamental examples. Making up zig-saw puzzles and solving elementary geometrical problems are also nonverbal thinking. Professional chess players can draw chess board in their mind, and can think of good moves in a flash without reasoning (often those moves derived by theoretical thoughts are not so brilliant). Of course after the inspiration follows the verification of the goodness of the move but many of the inspirations are indeed correct. This intuitive activity of discovering a good move can also be considered as thinking.

Verbalization of a procedure and the procedure itself belong to completely different levels. Computers often find it very difficult to achieve tasks that are most easily done by human beings. On the other hand, we often find it too abstract to grasp the idea of an algorithm that is easily run by computers within a second. Think of adding two numbers in a decimal system. Computers do it according to a properly written program (and they cannot do it without a program) whereas elementary school pupils learn how to do it by heart through many exercises and examples; you can hardly expect them to understand the algorithm if it is told verbally to them (and therefore it is not told verbally). Moreover, unless one knows information science to some extent or he is the kind of a person who can bear relatively logical discussions, it is difficult for an ordinary adult to clearly and accurately tell verbally what procedures there are in add two numbers (I remember struggling to understand a program for adding two numbers written in Pascal a bit). However, no doubt every adult knows how to add two numbers by heart.

With respect to 'procedures', conferring the examples above, whatever sophisticated decisions made by an artisan can be considered as a form of thought, at least in naive terms. A chef's judging how much salt he should add to a dish is an intuitive thought in this respect. However, if we extend the concept of thinking to this extent, there arises some difficulty.
Let us take the example of chess again. In game theory, it is proved that in whatever game like chess, or zero-sum-finite-2 players game, the player having the first move is known to win in principle if he makes 'correct' moves. Now, even professional chess players neither know the best algorithm nor can think of correct moves by intuition (that is why chess still stays as one of the most interesting games of the world. If it is trivial which of the players is to win, there is no interest of playing the game.). Even when professional chess players do not make correct moves, we naturally describe the activity of their deciding a move as thinking. Moreover, we are no doubt thinking when we are playing chess as well (of course most of the moves we make are stupid at the standard of professionals). What if a player is a beginner? What about the players who play so fast as if they are not thinking at all (of course professional players can beat us easily even if they think only for seconds for each move)?

Generalizing the above situation, can we say any mental activity is thinking? Apparently, with respect to chess, the decision making of moves done by professional players seems to be 'more like' thinking than what is done by ordinary players. Is there any good criterion for 'the likeliness of thinking of a mental activity'? Is it possible to quantify that? After a little discussion, we have come back to almost the original question. It is too difficult to discuss this answer straight forward. I would like to start the discussion from the most orthodox type of thinking. That is verbalized intelligence.

2. Information Science (Computer Science) and Verbalized Intelligence

In this century, human beings have acquired a cogent way to investigate the verbalized intelligence -- information science. The difference between information science and philosophy may be that the former is scientific while latter is not (what it means to be scientific is itself very much controversial but I will treat this later). Let me revise some of the concepts of information science which I consider are relevant here.

Language

In the formal language theory, a 'language' is composed as follows:

I felt this definition was fascinating when I first heard of it. Of course, if you think of natural languages that we use everyday, this definition does not seem sufficient at all. However, I think it is the necessary condition; all languages actually fulfill this feature. The important point of this definition is that it insists that all the words are composed of finite elements. Even for those languages which have no writing, all the words can be parsed into finite number of syllables. This is the principle of language if it is a tool for communication. We believe that our memory is finite (which means we store all the information digitally) and therefore we can only store finite number of words in our brain. This definition also implies the creativity of language; without adding new symbols, we can define as many words as we want.

Information

In information theory, the amount of information is defined as the logarithm of the ratio of number of alternatives before and after the information is obtained. However, the probability that a particular alternative occurs must be known in advance in order to quantify the amount of information. In fact as far as I know, the discussion of information theory is only applied to the field of (tele)communication. I have tried once to quantify the amount of information obtained when we get a functional relationship between two continuous quantities and so on, however I came to know that this trial is not adequate.

Naturally information theory does not deal with semantics. All the inputs are treated merely as sequences of symbols and the alternatives only mean the variation of the input sequences

Logic

The most fundamental analysis for the classical question 'What sort of statements are meaningful?' is propositional logic as far as I know. A statement is a type of a sentence which 'tries to' insist something (any statement has a subject and a predicate). The most fundamental thesis was that a meaningful statement is either true or false (even this rather modest thesis is known not to be 'true' today). The task of symbolic logic has been to treat the truth of statements provided that it actually has a definite truth value (such a statement is called a proposition). Therefore it ignores all the meaning of a sentence other than its truth value.

The most primitive type of inference is thus discussed within the field of propositional logic. Propositional logic analyses the truth value of compound propositions which consist of simple propositions and logical connectives (negation and conjunctions). Simple propositions each have a definite truth value. Logical connectives correspond to functions from T to T (or T × T → T) such that T = {0, 1} . Thus the truth value of any compound proposition is determined. However the scope of propositional logic is rather narrow because it does not touch the inner structure of a simple proposition.

Predicate logic deals with the content of even a simple proposition. Predicate logic no doubt has come out of the abstraction of mathematical inferences. In fact, here appears quantifier (universal quantifier ∀and existential quantifier ∃) and function which are both essentially defined on sets. The semantics of 1st-order predicate logic can be treated precisely on Herbrand space.

Inferences might have been originally used for purpose of inferring nontrivial statements but it turned out that even intuitively true statements have logical relationships, the most fundamental of which is consistency. Therefore the desire arose to organize true propositions into an axiomatic system. The most classical axiomatic system I know is that of Euclidian geometry. When propositions are organized into an axiomatic system, truth of all the propositions can be reduced into truth of axioms. Then after nearly 2 millennia Hilbert revealed that the inferences themselves mean something regardless of the truth value of axioms. At this stage, axioms mean no more than the finite relationships among undefined terms. Thus there is no point of discussing the truth of geometry. That is why non-Euclidian geometry can also be 'constructed'.

Now, think of a statement "5 + 7 = 12" that Kant mentioned in The Criticism of Pure Reason. From the view point of mathematics, it is true as a statement of the theory of natural numbers. More concretely, on the set of natural numbers, an operation of addition is defined and 5, 7, 12 are also defined by the successor function from the number 1 (or 0) and the axioms of equal sign are given.

df successor function -- function s:N → N
s(n) = the natural number next to n

Thus by definition, the statement "5 + 7 =12" is true. However, if the theory of natural numbers has no truth value, is this statement really true? What does this sort of absurd discussion have with the fact that "5 + 7 = 12" is intuitively true? I would like to talk about this problem later on.

By the way I would like to add some comments on predicate logic. I think it is only because quantifier is the powerful tool for analyzing the inference of non-mathematical propositions as well (ordinary propositions such as 'A person is a living organism.'), that the research of predicate logic was segregated from mathematics normally (in fact originally, Russell and Whitehead had the plan to establish a logical system which can derive all mathematical concepts; today after David Hilbert, we know that the axioms of mathematics are independent of axioms of logic. It is not possible to compose the set of natural numbers from only the axioms of what we can call 'axioms of predicate logic'. However, my insistence is that there is no clear semantic distinction between the axioms of logic and that of mathematics. At least we often say that the boundary of mathematics is not clear; for instance, whether the game theory, statistics, graph theory and so on are fields of mathematics or not does not seem to have a conventional conclusion.). Certainly mathematics is a logical system and it is natural that we feel that its form is idealistic. However it is not trivial at all, that all the sensible ordinary statements can be described as a formula of predicate logic. In fact we know that there is even propositions which cannot be interpreted within predicate logic. For instance, a statement 'John believes in God.' seems to have a definite truth value. Either John believes in God or he doesn't. However once you try to abstract the structure 'Believe(John, God)', you realize that Believe(*, *) is not a predicate. Believe(a, b) determines the truth value independent of the truth value of 'b', which does not make sense in compositional semantics (such statements are treated in modal logic). For another example, a proposition 'John walks slowly.' is not within 1st-order predicate logic. 'slowly' attributes 'walk', which is not allowed (there is no predicate for predicates in 1st-order logic). Thus we know that the situation is not very simple.

Fuzzy Theory

Fuzzy sets and fuzzy logic may be one of the first trials to treat the statements that do not have the truth value of 0 or 1. A fuzzy set corresponds to a word which has 1-dimensional meaning. There is a quantity which corresponds to a universal set of a fuzzy concept. For instance, the concept 'old' is a membership function from age (the set of natural numbers) to truth value space (here the set of truth value is the closed interval [0,1] instead of {0, 1}). The fuzzy set corresponding to the concept of 'old' is a subset (in sense of fuzzy set) defined by the membership function. Fuzzy logic treats ordinary inferences that involve fuzzy statements

We have to notice however that fuzzy logic is not necessarily better than classical logic. It is certainly the generalization of classical logic. Fuzzy logic involves classical logic as its limitation of a membership function to a step function (characteristic function). This relationship is similar to that of relativistic mechanics and classical mechanics in that the latter is the former's limitation of the speed of light to infinity. We can notice that classical logic is much simpler than fuzzy logic. In many cases, we are fulfilled with classical logic because we can obtain sufficiently accurate conclusions by approximating the fuzzy set into a crisp set and applying classical logic for the inference.

Moreover, the reason why fuzzy logic did not use to be widely accepted in the western world for a while was that it was believed that continuous quantities should be treated within the field of mathematics using such concepts as functions. The thesis of fuzzy theory is that functional approach is not necessarily the best algorithm. Today, in the field of engineering, it is known that some of the systems are more easily controlled using fuzzy logic than describing the relationship of quantities as a function. Thus my impression is that it is not at all clear yet whether fuzziness is a very fundamental concept or merely a technologically useful idea.

3. About Verbalized Intelligence again

One of the important features of science is that it tries to limit its statements to reliable ones. It is modest in this respect thus keeps silence over what is not certain. On the other hand, I am not writing a book or what thus I would talk in much more naive words let alone the risk of ending up in a mere waste of time. Taking the results of information science in consideration, I would like to talk about verbalization again.

I think the functions of language today can be classified into three types.

Communication

If you see the primitive types of languages such as the ones used by animals, you can see that they are purely the tool for communication. The signs in the language suggest what is happening or what should be done. In this sense, the set of signs used by bees to communicate the direction of honey from their beehive is a kind of language.

Record

Until multimedia showed its development in recent years, written language was almost the only medium known for recording the information. Pictures were there as well, however due to the time and the effort required for input, it is natural that the amount of information stored in form of language has by far exceeded that of pictures.

Tool to Organize Thoughts

There are several kinds of sentences. Ordinary sentences that we read and speak are statements. However, setting aside exclamations which do not necessarily correspond to a set of words (in the sense of the previous section), imperatives should not be ignored. A computer program is an example of a set of imperatives. The important difference between statements and imperatives is that the latter can directly correspond to an action while the former essentially gives a relationship among concepts.

If you treat a language as purely a symbolic system, only 'relationship' in this respect can characterize the language. The third aspect of a language deals with the relationships themselves. Merely describing an object by a single word is not at the stage of verbal thinking. It might have much to do with pattern cognition or other types of perception but this kind of activity is not verbal, or not within a language in other words. A verbal thinking occurs when a language has sufficient number of terms, and the concepts corresponding to terms form complex relationships such as hierarchy (classification), feature (such as colour, size, eta). Let us assume that verbalization means to express a known subject as a set of relationships of terms in a language. Here we also assume that the terms used in the expression are all trivial to the subject (thinker) or directly correspond to an intuitive image.
I think this definition of verbalization is not unnatural and we actually often try it. The best example that I can think of is history. Past is just there and it does not change. Then, why do we try to make up a story to describe the past? Another example is science. Science has drastically changed its character in this century but roughly until the last century, the aim of science had been to investigate the static structure of nature. Application was not the major problem. If the nature does not change anyway in sense of scientific laws, why do we bother investigating the nature scientifically? I think the utility of verbalization can be summed up as follows:

Decomposition

As is already mentioned, verbalization translates the object into the relationships of more trivial terms. In other words, the object is treated as a system being described as a composition of elements each of which corresponds to a word. In computers, a compiler knows what to do when a supported command is given. A program is a set of commands, which decomposes the whole procedure into a set of simpler (trivial) procedures.

Abstraction of a Structure

A verbal expression sets a perspective on the object. The same object can be described in many ways. It is similar to the fact that a same person looks completely different in different photographs.

This is the essence of history. History is not the genuine record of facts (or primary source materials). Today we have video cameras. However most of personally recorded tapes have nothing to do with history though what they recorded certainly form a part of the past. History is a story (whether it is true or not!) and it involves a judgement of what is important and what is not. Thus it is natural that German word Geschchte means both story and history.

Some people might misunderstand that physics provides quite a universal perspective. However, they have to be aware that physical perspective is specific for acquiring physical answers. For instance, if you want to know the translational motion of a rocket in universe (for simplicity there is no force applied to the rocket), all you need to know are its mass, initial position and total momentum; and the output that you can get are only its position and momentum at any time. You have to notice that it does not deal with the information such as what is done inside the rocket, what colour the rocket is and so on. Moreover we have to see that this is the strength of classical mechanics. It suggests that if you only want to know the information with respect to the motion of an object, you only have to know a few parameters; whereas until we came to know classical mechanics, we had not known what was essential to control or estimate motion. In other words, the theory of mechanics enabled us to ignore unnecessary data by limiting the degree of freedom.

The idea of structuralism is even more fascinating. Think of a sentence 'Maybe, this phenomenon is caused by magnetic effect.'. In sense of logic, this statement has no meaning because it has no truth value. However we are stimulated in some way when we hear such a statement. The reason is that the statement provides a perspective. For instance suppose the phenomenon is a disease. Normally, we do not relate physical effects to human health. Hearing that the disease might be related to the physical environment, we have now obtained the idea of investigation which we might never come up with before we heard of it.

I often think that Ernst Mach was great in that he doubted the concept of absolute space and time. His discussion might not have brought about much results in terms of propositions, however the perspective he provided was certainly succeeded to Einstein who constructed relativity.
I think a considerable proportion of statements are of this kind (already this particular statement is meaningless in logic already because it involves the word 'think'). When we have a conversation with a person, and hear her or his opinions, we do not demand that they are true. We stand on a viewpoint of relativism in a way. We are stimulated by obtaining a new perspective and we are satisfied.

This type of acquiring a new perspective is in line with information theory in naive terms. Before the perspective was presented, there was infinite number of alternatives that can describe the object whereas after the presentation, we can limit the relevant description of the object to the presented perspective if it is relevant. When we encounter a type of problem which has been never experienced before, we first have to try to set a perspective, and this process is far from logical inferences.

Even more extreme is interrogatives. Questions have no truth value essentially however they have meaningful contents. Professors say that they are stimulated by good questions of students. Indeed, in science, it is often more difficult to set up a good question than to answer them. For instance, Einstein started doubting the concept of absolute time and set up the question 'What principle can explain Lorenzian transformation?'. If such a question was presented, I imagine it was not difficult for either Poincare or Lorenz himself to find the principle of constant speed of light as well. I think the greatness of Einstein lies in that he could set the question with respect to space and time clearly (however this is not the case with general relativity since I feel that general relativity is difficult enough to construct even after getting the idea of the principle of equivalence).

Clarity

By verbalizing an object we can often grasp the concept more clearly than just leaving it as it is. Think of procedures for example. Traditional artisans did not teach their skills to their disciples. Craftsmanship was to be conveyed by heart. In Japan, it is said even today that apprentices have to 'steal' the skills from their masters. On the other hand, nowadays in chain restaurants like Mc.Donald's for instance, the employees are taught what to do through a manual and majority of the procedures are exactly verbalized in it. It is much more efficient and accurate to communicate what should be done by words than by letting them copy what the seniors are doing.

However of course we should be aware that verbalization is abstraction thus

it is not always able to cover all the necessary components of the object. In such a case, a new term or a new language must be introduced, or you have to think of other ways to grasp the object (that is one of the reasons why it is often much better to imitate seniors than learning theories. In fact, when you learn your mother language, you do not learn a theory of how to pronounce but follow natives talking, and naturally that is the best way for it). Physicists had to invent calculus in order to describe change.

Wittgenstein thought in his early stage that all the 'meaningful' statements can be translated into those of a formal language such as predicate logic. However, it seems that verbalized intelligence is much broader.

4. Science

I think most people agree that science treats propositions. Scientific statements must be 'true'. Even the statistical statements are 'true' within the theory of statistics. However, what does it mean to be true? Logic treats the correspondence of a statement to truth value but it does not tell us what truth actually is at all. It is a working hypothesis of logic. Likewise, logic does not tell us why logic itself is effective.

Apparently, what truth means depends on the situation. Imagine a person's believing in a particular statement. So far as he does not try to convince others to accept the statement, it is his right to think so even if the statement seems false to others. We do not usually 'discuss' such a case. The truth that 'we' refer to has to be common among us. That is the most fundamental condition of any concept that (we think) we share (we call such concepts as objective). Thus, a single statement must have the same truth value to all of us. Fuzzy logic treats the subjectivity in that it allows different membership functions corresponding to different people's standard. However, if we would like to share the truth, the membership functions must be nearly the same.

Now, my feeling is that any set of propositions on which 'the researchers' can share a unique precise procedure of verifying their truth can be referred to as a science. Some of the procedures are common to all sciences:

I think the most fundamental condition that all sciences fulfill is logical consistency. Imagine a referee examination of a science journal. The standard a referee uses to judge that a paper is not false is, I think, consistency with the established facts, theories (this is quite important; usually in science, an idea is not admitted unless it is consistent with well-established previous researches; philosophers of science like to tell that relativity and quantum mechanics are incommensurable with Newtonian mechanics, but on the contrary, these two theories are constructed on the hypothesis that they must be identical with the Newtonian mechanics a the limit of c(speed of light) to infinity and h(Planck constant) to 0!) and consistency within the paper itself (of course a new work does not always demand complete truth. If a new perspective is indicated in the paper, it might be sufficient). Even for experimental papers, consistency is one of the most important standard for the judgement of falsity since it is unlikely that the referee makes check over the experiment itself. When the statistical processing of data and the reasoning seem adequate, the paper is considered as reliable. In fact it often happens that experimentalists fabricate data. For instance, when there is a reliable theory and it is expected that the result of supporting experiments will turn out to be positive, then if an experimentalist does apparently appropriate experiment, makes up a positive result and submit the paper, there is practically no way to check whether he really got the result or just made it up. Some experiments need so delicate techniques that no one but a particular researcher can get good result!

The second fundamental condition of all sciences is that its truth depends on experiments, measurements or observations. In philosophy of science, experiment and observation are strictly distinguished but let us not care about it at the moment for simplicity. An experiment is a set of procedures to verify a hypothesis by checking the corresponding phenomena in the 'physical world'. Experiments stand on working hypotheses. For instance, when you measure a distance with a ruler, you have to assume that the ruler is accurate. Think of the experiment by which chemists in 19th century discovered the law of conservation of mass. It must have used glass flask assuming that substances do not pass through glass. Today we might check that a container is well sealed up by using the law of conservation of mass! In philosophy of science, this situation is called 'the theory-laden observations'. Philosophers think that in order to interpret experimental results, theory is essential. This idea seems plausible but as far as I know, the techniques of experiments involve intuition. We can tell that an object is a rigid body without knowing the theory of mechanics. Just look at it. If it is 'solid', it can be treated as a rigid body! It is only the habit of philosophers that they want everything to be verbalized. I insist that the working hypotheses of experiments are basically intuitive and the intuition can be acquired only through training. What is important is that the working hypotheses are shared by the researchers, no matter whether they have theoretical background or not.

The idea of 'research program' by a Polish philosopher of science Lakatos seems to describe the above situation satisfactorily. According to 'research program', a scientific theory consists of hypotheses (or propositions) that vary in importance. The hard core consists of a set of principles or assumptions that are not to be tested by experiments. Rather, scientists try to make up the models with which they can apply the principles. For instance, in Newtonian mechanics, the principles are the equations of motion (or the principle of least action), the law of gravitation, Euclidian space and independent time, and the atomic principle. What researchers try to do under the research program of Newtonian mechanics is to establish models, or more broadly interpretations, such as rigid body, fluid, gas and statistical ensemble so that they can apply the principles to the reality. What are checked by experiments are the models. In naive terms,

reality = scientific principles (hard core) + models (protect belt) + individual features (initial conditions, noise, ...)

Individual features of the subject are initial conditions, noise and other differences from the applied model, and so on. The important insistence of the idea of research program is that a single experiment can not function as a counter evidence (refute) to the principles of a theory. It is the procedures of the experiment or the model introduced to interpret the result that are doubted. In this respect, the truth of a theory cannot be verified by experiments.

The most famous example I know that supports the idea of research program is the estimation of the existence of Neptune. When astronomers calculated the orbits of the readily known planets (that were Mercury up to Uranus at the age) accurately using perturbation, they found out that the difference of the observed value from the theoretical value was larger than the observation error. They were so confident with their observation and calculation (the accuracy of astronomical perturbation is often surprising) that they had to challenge the inverse problem of estimating the factor that caused the significant difference. The point is that they did not doubt the effectiveness of the above stated principles of Newtonian mechanics but they tried to look for the source of force that was not known at the stage. The hypothesis they made that there must be another planet somewhere that influences the gravitational field of the solar system seems to be quite natural. An important feature of this example is that when the difference in calculated value and the observed value was found out, the principles of Newtonian mechanics were not doubted, but it was used aggressively to search an unknown factor.

Engineering may be a better example. When experimental data and estimated values show discrepancy, I expect that engineers rarely doubt the effectiveness of scientific theories they apply but they try to improve the model they constructed.

As is already mentioned, it is almost unlikely to refute a theory by a single experiment. Then how can we judge whether a research program is effective or not? Lakatosh called the research program which succeeds in estimation and explanation successively 'progressive', and the one which fails 'regressive'. When two research programs treat the same subject, a 'regressive' one is replaced by a 'progressive' one. For instance, the ether theory of Lorenz is logically equivalent with the special relativity except the law of E = m c2 but the latter is more progressive than the former. However, if you compare special relativity with non-relativistic mechanics, you know that in most cases we encounter where we can ignore the speed of light, the latter is more effective than the former.

Now, do we demand only truth to science? The answer is no. Science is attractive for the researchers and has some utility even for outsiders so that not a large proportion of people deny to be a sponsor (governmental expenditure spent for science) of science as a whole. Then what is the value of science? Is progressiveness (or productivity) only the value of science? I think in naive terms technological application and universality are also as important.

I think it is needless to discuss the former. There is no doubt that sciences have contributed to construct the contemporary civilization. Thus let me talk about the latter. Science is essentially different from mere database or personal intelligence such as his experiences. Indeed, I think this is one of the criteria that distinguishes science from other intellectual activities. Science is the organization of 'facts' (or hypotheses). With such theoretical tools as logic and maths, sciences have succeeded to organize the relevant intelligence into a theory that can be used to estimate unknown and new phenomena. It is similar to the generativity of languages that enables us to describe new situation using the existent terms.

Philosophy likes to discuss the universality, thus physics has always been treated as the prototype of all sciences. Actually, the most important theme of physics has been the unification until recently, and no doubt still is one of the most important. Most physicists had believed as though principles were important and the application of principles were no more than a matter of calculation. The principles of physics are the elementary particles and the interactions among them. Their insistence was that only because human beings did not have enough power of computation, they could not apply the principles sufficiently to understand the nature. Nature was simply too complex to understand, and the complexity they thought were matter of individual properties thus not the subject of physics. Some physicists still seem to feel that principles are at least more important than application, and probably therefore there are still so many particle physicists today.

However, it is known today that the situation is not so simple. First of all, even if we assume the existence of a unified theory and that it were an axiomatic system on 1st-order predicate logic, Goedel's incompleteness theorem and the related works seem to show that proofs of propositions cannot be always automatic. Secondly, Lakatos' idea seems to indicate that the construction of models (application of principles) and the principles themselves are independent. What many physicists seem to have felt were that models were necessary only for practical use and it was possible to correspond the system to elementary particles uniquely thus laws of physics would determine the estimation uniquely. Let us again assume that this were possible. It is known that in many of nonlinear systems, an infinitely small difference in initial conditions evolves into infinitely large difference, often known as a butterfly effect. This shows that in principle, it is impossible to estimate the future accurately only by physics. Moreover, even if we were able to know the hamiltonian, we have to keep in mind that the number of particles that compose ordinary systems is the order of mols (or 10 power of 23 ). If we consider all the interactions among all the particles, the amount of computation shows combinatory expansion, which is known as NP(non-polynomial)�@problem in computation theory. Therefore it is impossible (not unlikely!) that in the future computers will solve all sorts of Schroedinger equations. For instance, analytical solution for the simplest model of spin-spin interaction called Hubbard model is not yet found. Thus there have been several trials of numerical computation of the correspondent Schroedinger equation. The world record of the number of particles used for the computation (of course the computation was done by one of the fastest super computer of the time) is about 20, as far as I know!

Today, most of the application of contemporary physics is within the field of material science. The theoretical aspect of material science is basically the application of quantum mechanics. Looking for new models and composing the hamiltonian of the models are the major work. Also often, the method of solving the relevant Schroedinger equation itself analytically and the approximation are also important topics. Especially, the method of approximation has been one of the most important topics of mathematical physics. Experimentally, often a guideline of looking for a new phenomenon has little relationship with quantum mechanics. For instance, in the field of surface science, discovering a new orderly structure on a particular surface is treated as a new work. However, it is seldom that an adequate model which can explain the formation of the structure satisfactorily is soon submitted. Moreover, it is almost impossible to estimate theoretically what sort of structures appear on a given surface before it is observed.

By the way, I would like to stress that I intentionally discussed that the truth of science is determined by 'researchers' because I consider it need not be 'sufficiently rational being' or 'anyone'. For instance, I think physics is difficult enough not to allow a considerable proportion of people to understand it. Even I who specialized in physics in University of Tokyo do not even have a will to understand particle physics. I have had some friends in the physics department but at the stage of undergraduate, at least most of them only seemed to have reached the level of slightly understanding particle physics but far from being able to criticize the content of it. Thus I say that the truth of science is verified by the researchers and normally outsiders have nothing to do with it. Then how can outsiders know that a particular field of science is true or not? The apparent conclusion is that there is no way for the judgement! All outsiders can do about science is to learn the field to his ability and, with regard to propositions beyond their learning, merely to believe what the experts say. The judgement of the truth of science thus basically stands on the trust over scientists.

Scientists are often referred to care only peer review. They do not care whether outsiders approve of their work or not unless they are government officers or corporate directors who have the power over the budget of researches. How can we deal with the gap between scientists and outsiders? Any science makes the discussion open to everyone; which means anyone with the will to learn the science is welcome. This openness may be almost the only reason why outsiders can trust the scientists. In that respect, I feel that over-specialization has brought about the uncertainty in truth of science.

5. The Physical World (The Universe)

The existence of the unique physical world is almost trivial to most of us unless one is an extreme idealist, insane or a religious cult. However, I feel the concept of unique physical world is rather artificial. At least, the questions I came to notice are:

Let me first indicate my intuitive idea about the position of an arbitrary subject within the physical world in the following diagram.

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The diagram stands on the viewpoint of materialism. A subject in the diagram only functions as an observed existence. A person in materialistic view is nothing but an objective structure in the physical world. Empirically, it seems relevant that even the consciousness of people is within the physical world in that the state of consciousness at least partly depends upon the environment to which the subject belongs.

Observation OB is like a mapping from the physical world to the sensitive organs SO if we ignore the resolution of the organs. Environment E is a subset of the physical world and can be naively defined as E=OB-1(SO). Cognition transforms the stimuli to the sensitive organs into the image I within the consciousness. I is a set of data on short-term memory. At this stage, it is known in cognitive science that the data relating to grasp a certain pattern is more likely be cognized than others, and this seems apriori. For instance, when you see a persons face, you notice the positions of eyes, nose and so on and you can easily identify who is who. Moreover, when you see a picture of a person you can easily tell that it actually is a picture of a person. A dog seems to be able to do the former but not the latter. It might be difficult for a computer to identify who is who by face. This shows there is a selection of data before the stage of consciousness. If we roughly accept this view, we know that all we have in our minds with respect to what we 'feel' is the image I.

For simplicity, let us limit the sensitive organs to only sight which is apparently most related with science. The visual image we get when we keep our eyes open is like a movie. Retina is like a video tape that can be overwritten. This image is the 'reality' for the subject, not the physical world (this is my demand!). All we have in our minds that we can consider similar to the physical world are the succession of visual images at best (when we assume that our memory is good). All the descriptions with respect to the reality must be constructed by analyzing the data visually seen. Thus the very concept of the physical world is the 'structural' extension of the environment (like the analytical connection of a real function into a complex function). In other words, we assume the structure of the physical world to explain features of the environment. Let us come back to the diagram. The environment is different from the content of a movie in that it is partially interactive with the subject. The subject can partly change his environment by acting to it (A). Here, I mean act by a very broad sense. Looking in a different direction is an act as well. We empirically know that much of the features of the environment are determined independent of you. In that sense, of course the environment is different from a story you make up in your mind; you have a complete control over the content of the latter but not the former. I call such features of the environment physical features. These physical features are the structures enabling the extension from the environment to the physical world (physicists like to speak of this fact that there are structures found in the environment as 'there is order in the world'. It is suggestive that a structure is an order.). Thus the concept of the physical world is artificial in that it is a model to explain what is going on in partially uncontrolled environment.

Let me give some examples:

the concept of time

Kant discussed that the concept of time (and space) is apriori to us. However, in physics, the concept of change and the concept of time is not identical, and Kant's statement seems to be more adequate when we replace 'time' with 'change'. Time is a physical quantity and is a real number. It is determined by comparing change of the phenomenon with a state of a clock. A clock is considered to function 'periodically'. The periodicity of a clock can be judged by an experiment. If you already have a 'correct (periodic)' clock, then you can check the correctness of another clock by comparing the two clocks. However, in this scheme there is at least one type of clock whose correctness must be determined independent of other clocks. The periodicity of this particular clock can only be inductively 'assumed' by ascertaining that by using this clock to measure time, the physical world can be orderly explained. Thus the concept of physical time is far from that being felt.

the atomic hypothesis

We have never seen an atom directly, however we believe that all matters consist of atoms.

interpolation

Suppose you see a person (let me call the person seen in this occasion as A) and see him again in another occasion (B). You have no doubt that 'two people A and B' you saw are identical. You interpolate his continuous existence during the time between the two occasions that you met him. This continuity is not always trivial for an infant, thus it cannot identify anyone but its mother who is always with him. Naturally, if you do not have enough memory, you simply do not remember A when you see B.

dark matter

I do not know how much the hypothesis is effective but in the contemporary astronomy, the existence of dark matter is often discussed. We cannot observe dark matter with light which is the most fundamental medium of observation in physics. However, still it does not necessarily mean that the concept of dark matter is not physical. The reason is that it interacts with what can be observed through gravity.

the figure of the earth

If you only demand ordinary information necessary in your daily life, there is no problem in thinking that the ground is flat. Indeed, many people are expected to have believed that the figure of the earth (or the ground on which we live) is flat before Magelan traveled around the world. My insistence is that since we cannot 'see' the whole earth directly today as well, we need to think that the earth is spherical only when it is necessary in our lives.

the concept of 'the world'

Today, we the people of developed nations use the term 'the world' without any doubt meaning the society (assuming there is one) of whole human beings. Through mass media and telecommunications, we can know what is going on far away on the earth. Moreover, an educated adult is expected to follow 'the world news'. Imagine 200 years ago. Most people did not have any means to know what was going on far away in real time. The image they had over the term 'the world' must have been very different from what we have today. However, I would like to emphasize that 'the world' is only a small part of most people's lives still. As I said, the only way through which we access the world is mass media or computer network usually. It is far from 'feeling' it. For instance, it is empirically true that people are easily disguised by the content of broadcasts. We can hardly imagine that someone else can disguise what we directly feel (although it is possible to some extent).

At this stage, I think I succeeded to convince you that the physical world is not equivalent to the visual image you have in your mind. It is rather like a map in naive terms.
Now let us move onto the second question. What do we share in the physical world? Certainly, the images we get from the physical world are different. However, we can see the same 'thing'. How do we know that we are actually looking at an identical object? It is only through communication and the assumption that the pattern cognition is common to all of us. We have to be aware that the correspondence of a concept to a phenomenon in the image in your mind is intuitive. For instance, when you learn your mother tongue, there is really no way to define the terms that describe the environment.

Let me give an example. Think of a sentence 'The flower is red.'. The image of 'Red' for me might be quite different from that of yours. You might like red while I may not. However, there must be at least two things common:

Science is often referred to as the study of relationships among the concepts corresponding to the physical world. On the other hand it is only through intuition or pattern cognition how we correspond scientific concepts to phenomena in the physical world. That is indeed one of the difficulties of science which we do not see in mathematics.

Let me sum up my view point. First of all, I do not doubt the existence of the unique physical world basically common to all conscious beings including such beings as dogs. It is the working hypothesis of all activities of mine with respect to communication. If we admit the empirical fact that surprisingly many features of what we respectively feel can be shared through communication, it seems natural for us to assume the existence of the unique physical world. Here I would like to stress that the 'assumption' of existence and the existence are equivalent. When virtually everyone can accept that a thing exists, then there is really no point of objecting to the existence unless you have a strong reason for insisting it is better not to think that way.

However, at the same time, I also have to emphasize that the assumption of the existence of the physical world is only through the fact that we share much of our experiential world. We have to be aware that what we can share naturally have contents. We imagine the unique structure that makes these contents common to all of us. In that sense, even if we imagine the existence of the physical world, the concrete image we can have over it is no more than the logical extension (extrapolation) of the images we have. Therefore in this respect, there is in principle no 'unique' treatment of the physical world. To illustrate this view of rather constructivism better, I would like to make a short quotation from Glaserfeld[1987].

6. Science and Metaphysics, Technology

Having chatted over the physical world within which science judges the truth of its statements, I would like to discuss further delicate problems with regard to the boundary of science and other fields of intelligence.

Mathematics

Is mathematics a science? It is usually not regarded as science since it is not limited to the description of the physical world. However, today, the boundary of mathematical sciences and mathematics is very vague. The reason has much to do with the development of computers. Computer simulation can 'realize' model in the physical world. Once a phenomenon exists in the physical world, it is a subject of science. In this respect, the simulated phenomena are all real. I think that more abundantly or universally the structure seems to exist in the physical world, more scientific the theory is. For instance, I think the theory of groups, Euclidian geometry, calculus and linear algebra are all scientific in this respect. It is ridiculus to insist that a single statement 'Space is described by Euclidian geometry in an inertial system.' is a scientific statement of mechanics, and regard the axiomatic system of the geometry itself as a pure logical system.

Technology

In science, 'something newism' is essential in researches. In this century particularly, the most important task of experimental sciences is to look for new phenomena. However, as Francis Bacon said, intelligence is power and looking for a new phenomenon mostly requires the control over the nature. It is not like an adventure looking for a treasure. A new phenomenon is not there. It has to be 'made'. The best example maybe the synthesis of a new compound. If the activity of making up things is within science, what is the distinction between science and engineering (technology)? Today, this boundary is also quite vague, but the only difference I can think of is that in science a new phenomenon is often unknown whereas in engineering a plan (or the target of researches) always exists before the realization of a product. Information science and computer science especially has both technological and scientific aspects.
There is another aspect of the physical world today which makes it difficult for us to distinguish science from technology. That is the fact that the world has been essentially changed by human beings. The universality of science had seemed to be able to deal with new situation until roughly the last century, but it turned out that different problems demand different theories, thus the technological metamorphoses of the physical world brought about not only new methods within the existing theories but also new sciences such as chaos, fractals, fuzzy, computer science and so on.

Metaphysics

Metaphysics deals with values which seem to be essentially personal thus is not science. However, if a value can be related to a state in the physical world, the theory which treats the value can be regarded as scientific in a way. For instance, theory of optimization might explain the process of evolution.

7. Value

The attitude that logical positivists took against metaphysics had been to reject its effectiveness by logical procedures. The outline of the procedures they took was roughly as follows:

  1. translate the insistence of metaphysics into logical formulae (or propositions) of symbolic logic and discuss the logical structure of the propositions
  2. if the logical properties or the axiomatic propositions seemed irrelevant, the insistence are rejected
  3. those insistence that cannot be translated into logical propositions are regarded not to have any content from the first place.

They had indeed tried to analyze metaphysics in this scheme (linguistic analysis), and the main stream of logical positivism reached the idea that only natural science was worth being discussed. Indeed, physics has already existed as (at least apparently) a highly logical system, and as was mentioned, there had once been the atmosphere of thinking that a unified theory would explain everything, thus it seems natural some philosophers thought that the research of science would be sufficient as intellectual activities.

I think that the above stated scenario is much better than merely trying to discuss without conventional arena. At least, I can say that it is ridiculous if a same thing is discussed in more than two completely different manners, which is often seen in metaphysics. Subjective concepts most clearly exemplified by value belongs to such a theme worthless of discussion. For instance, if one thinks A is better than B and another thinks the other way around, and both of them do not have a will to change this preferantial order, then it is no use for them to discuss which is better.

However, there is an essential problem in the scenario. As I have already stated in section 2, the idealistically logical symbolic languages, even if they exist, are not at all so simple as the known predicate logic that the logicalists must have thought of. Moreover, even if the idealistic languages are known, it is not at all clear who is qualified as a competent interpreter.

The attitude of logical positivists' having limited their scope of research to the field of natural science seems even more irrelevant. It is not at all clear that values are not shared by all human beings. Also, it is oversimplifying to think that subjective matters are not verbal problems of philosophy. One most typical example that I think is quite logical and at the same time has much to do with ethics is social choice theory. The impossible theorem by Arrow, which proved that democracy is impossible under any election system that fulfills apparently adequate conditions, clearly has much to say about what political system should be, although it is not a statement about natural science. The consistency of several values can also be treated logically as well.

To my impression, we might assume the existence of an idealistic world of values. Within that value space, we may discuss the consistency among several values, optimization and so on. The objectivity of such an idealistic world is not necessarily important. Any particular person can discuss the similar properties within his own subjective value space. It is empirically known that such methods as systems analysis can treat certain class of problems.

8. Verbal Communication

I have finally reached the worst result that I cannot see any clear conclusion of the essay I have written. However, to finish the essay, I would like to add some more comments on the communication in natural languages.

Though I have pointed out a possibility of treating values logically, the reality is not covered with idealistic languages. Our perception is only through our cognitive framework such as culture, religion and mother tongue. My department insists that natural language is an essential tool for communication because of this.

The dependency of contents on different languages should not be ignored. There are of course properties that cannot be translated. There are very many misunderstandings between different nations, derived from the fact that they respectively assumed the generality of their value though it is not always true that they share the value. However at the same time, it seems fruitful to think about the language-free properties of the intelligence of human race such as natural science, generative grammar and so on.

I think ordinary chatting at worst has an effect of exchanging information of the physical world. Of course, conversation has more than verbal effect such as seeing each other's face but we cannot ignore the fact that there are many people who enjoy pure text format communication, and some of them even fall in love only in the network. Though much of the conversation among young ladies in Japan do not seem to have much content for me, it has much information for them. Information can be expressed in naive language. The content does not always demand consistency, logical clarity nor organization. It is the task of each person in the conversation to respectively judge what is worth knowing for him and to organize what he heard into an applicable form.

Conversation has another aspect; that is exchange of ideas. What we have to bear in mind when we discuss the ideas we have is to distinguish the properties that involve values or cultural dependency, and propositions shared by both of us. However, here again it is not always necessary to seek logical properties.

The keyword may be satisfaction. In classical economics, human beings were treated as universally rational being (economic man). However, it is shown by computation theory that in certain circumstances, Nash equilibrium is not realized in polynomial time in a 2-player game. There are many other examples that show the 'bounded rationality' of human beings is essential when we consider human thoughts. Soft systems approach, poly-agent system are one of the examples that I will study with respect to communication and decision making under bounded rationality. I hope I will be able to contribute to some extent in the future.