1 Reflections on Ludwig von Bertalanfy s General System Theory: Foundations, Development, Applications Shelia Guberman PiXlogic, Los Altos, CA, USA Something is rotten in the state of Denmark. W. Shakespeare: Hamlet, Prince of Denmark. Abstract: The purpose of this paper is to analyze the basic concepts of General System Theory (GST) as they were described by Ludwig von Bertalanfy almost 50 years ago in the book General System Theory: foundations, development, applications (1968). All excerpts below are from the above mentioned book. This study follows our previous paper dedicated to the analysis of M. Wertheimer s book Productive thinking published also a number of decades ago. In that paper we show how deep and wise the author was in his ideas, how much mistakes we could avoid if we pay more attention to them, how much we can still learn from that book for high-tech development in computers and internet. With the same attitude we started the analysis of L. von Bertalanfy s book on one of prominent lines of research in XX century - General System Theory. I. Source. B. did a great job attracting attention of the scientific community to important and complicated problems in understanding the Nature. It was the aim of classical physics eventually to resolve natural phenomena into a play of elementary units. This however is opposed by another remarkable aspect. It is necessary to study not only parts and processes in isolation, but also to solve the decisive problems found in the organization and order unifying them, and making the behavior of parts different when studied in isolation or within the whole.(p.31). We are forced to deal with complexities, with wholes or systems. (p.5). Here are the main notions, which attracted B s attention in the 20s and 30s and which promised (in his view) the resolution of modern scientific and technical problems: the whole, the parts, the units, the complexity, the organization. That was similar to the problems raised earlier by the Gestalt psychology. The difference was that the gestaltists treated the whole as a psychological phenomenon, the product of our mind, but Bertalanfy thought the whole exists independently and tried to create a regular scientific discipline - GST. At the same time B understood that Gestalt psychology showed the existence and primacy of psychological wholes. (p.31) It is true that the time of Gestalt-psychology flourishing was very different from the time of rapid growth of General Systems Theory. The beginning of the century was the Silver age of arts (particularly in painting, poetry and theater). The time of GST s raise the middle of the century was the hour of triumph of science (particularly in nuclear physics, electronics and molecular biology). That partially explains Bertalanfy s choice to look for the solutions on complexity problems, wholeness and so on in the exact sciences: GST is a logico-mathematical science of wholeness (p.256) II. Definitions.
2 We postulate a new discipline called General Systems Theory. Its subject matter is the formulation and derivation of those principles, which are valid for systems in general (p.32). GST faces a lot of difficulties trying to define the basic notions. Here are some examples of definitions from his book with our remarks (in cursive). 1. GST is a logico-mathematical science of wholeness (p.256) 2. GST is a general science of organization and wholeness(p.288). 3. System theory is a broad view (p.vii). 4. New science (p.xvii). 5. System thinking (p.xix). 6. The GST is scientific exploration of wholes and wholeness, which, not so long ago, were considered to be metaphysical notions. Novel conceptions have developed to deal with them ( wholes and wholeness S.G), such as dynamical system theory, cybernetics, automata theory, system analysis by set, net, graph theory and others (p.xx). But that claim does not fit the reality: none of these disciplines had defined such notions as whole and wholeness. 7. GST is a general science of wholeness (p.37). 8. Systems philosophy organismic outlook of the world as a great organization (p.xxi). That kind of definition leads nowhere. 9. What is to be defined and described (?) as system is not a question with an obvious or trivial answer. It would be readily agreed that a galaxy, a dog, a cell and an atom are real systems, and logic, mathematics and music are conceptual.systems (p. XXI). One would agree that the first are real, and the latter are conceptual objects, but one can t agree that they are real systems and conceptual systems because the notion of systems was still not defined. 10. Systems approach (p.4) 11. We are forced to deal with complexities, with wholes or systems. (p.5) 12. The general theory of hierarchical order obviously will be a mainstay of GST. 13. Gestalt psychology showed the existence and primacy of psychological wholes. (p.31) 14. Systems, i.e. complexes of elements standing in interaction (p.33). That definition is not acceptable because there is no definition for complexes of elements. 15. Systems are sets of elements standing in interaction (p.38). 16. System of elements in mutual interaction (p.38). (defining systems by systems ). 17. Organisms are, by definition, organized things. 18. If we are speaking on systems, we mean wholes or unities. Some of his definitions could not serve as definitions as they use other undefined terms (for example, complexes of elements standing in interaction (14) or system of elements in mutual interaction (p.45). When the definition is non-contradictory ( sets of elements standing in interaction, (15), it is so broad that any arbitrary set of objects in the universe becomes a system. Such definitions single out no objects for investigation. B made an effort to substitute the definition of system by means of another basic notion: If we are speaking on systems, we mean wholes or unities (11). But wholes, or wholeness, or unities were also never defined. The attempt to define other basic terms was also inadequate: Organisms are, by definition, organized things. Bertalanfy completely understood the situation: What is to be defined as system is not a question with an obvious or trivial answer (p.xx).. He admitted the problem and sometimes softened the rank of the GST using such terms as system thinking (p.xix), systems philosophy (p.xix), systems approach (p.4) Because of the difficulties in straight definition of basic notions B tried to define the matter of the theory and the basic notions by examples. On the first page of his book he claimed: An introduction into the field is possible in two ways. One can either accept one of the available models and definitions of systems
3 (?) and rigorously derive the consequent theory. The other approach which is followed in present book is to start from problems as they have arisen in the various sciences, and to develop it in a selection of illustrative examples (p. XVII). Such an approach was developed in the second half of the XX century into a technology known as Pattern Recognition. The main idea of Pattern Recognition is as follows. In many fields of science and business the decision making is complicated. It means that there is a lot of information but we don t know the laws that determine the outcome, so, we can t make the right decision. That is typical for medicine, geology, economy and so on. The Pattern Recognition technology gives a chance and a tool to improve the decision-making by learning based on experience, on cases with known outcome. The mandatory demand of that technology is the representation of all examples has to be in the same terms, so, they can be compared and a general decision rule can be created. That was not the case in the reviewed book. The example of system concept using geometrical objects (pp ) has no chance to be expressed as a system of differential equations from classical theory of dynamic systems. If so, how can anybody generalize them? Bertalanfy showed a lot of common sense in judging GST. Bertalanfy used a quotation from Ashby to describe the state of art in creating the new science. Ashby has admirably outlined two possible ways or general methods in systems study: Two main lines are readily distinguished. One, already well developed in the hands of von and his co-workers, takes the world as we find it, examines the various systems that occur in it zoological, physiological, and so on and then draws up statements about the regularities that have been observed to hold. This method is essentially empirical. The second method is to start at the other end. It considers the set of all conceivable systems and then reduces the set to a more reasonable size (p. 94). It is remarkable that Bertalanfy didn t deny Ashby s view on his work as non-theoretical, but essentially empirical. Moreover, he admits that his approach to the mathematically minded will appear naïve and unsystematic. Ashby itself choose the second way. He started with the definition of system but didn t succeed. As showed his definitions were not general enough. At first, he recruited areas of science that use the word system but use them as a neutral word (non scientific term): 1) systems of differential equations, 2) open systems (well defined physical meaning), 3) nervous system, control system and so on. The point is that none of them deal with wholeness. Secondly, Bertalanfy pointed out the difficulties in a particular area of science and declared a priori that system approach will resolve the problem. These areas mainly represent the sciences which are poorly formalized (biology, sociology, economics, psychiatry, etc). III. Detailed analysis (mathematics, physics, computers, cognition). Bertalanfy used a huge number of examples, which illustrated how the system approach works in science and technology. We will discuss in details some examples, which are more familiar to the author. III. 1 - Elementary Mathematics. The basics of system concept are explained in the very beginning of Chapter 3. We will quote the full text of the example (in bold) and follow it with our remarks (in cursive). In dealing with complexes of elements (i.e. systems according the definition) three different kinds of distinction may be made The quotation marks on elements and use of may be made expression instead of definite There are reflect author s understanding that no precise statements can be declared and that the example is weak.
4 .. i.e. 1 according to their number; 2. according to their species; 3. according to the relations of elements. The following simple graphical illustration may clarify this point (Fig.1) with a and b symbolizing different complexes. Fig. 1 If we forget for a short while that all that has to be about GST, we will find a well-known psychological problem. Three similar problems are represented. In each problem two sets of elements are given. The question is: what is the difference between the two sets in each case? The answers are easy and obvious. In the first case two sets differ in the number of elements. In the second case two sets differ in the nature of the elements. In the third one they differ in geometry. It is obvious that the set a is the same in all cases and all the differences we can see exist in our perception only. In cases 1 and 2 the complex may be understood as the sum of elements considered in isolation The use of words understood and considered shows that Bertalanfy was going to analyze a perception problem. It means that he was investigating the system not in the outside world but in our mind. Also it is impossible to understand the meaning of sum of elements as the notion of sum was never defined in the context of GST. In case 3, not only the elements should be known, but also the relations between them. What is the reason to know something about the system? The answer is simple: for describing the difference between the given two systems (complexes of elements). Characteristics of the first kind may be called summative, of the second kind constitutive. A question arises: characteristics of which objects are we talking about: elements or complexes of elements (systems)? The nature of the element is by definition a characteristic of an object. The number is a characteristic of a set of elements (it could not be applied to a single element). The relation between the objects is a characteristic defined for pair of objects. As a result one can t use the proposed classes of characteristics (summative and constitutive) to classify any kind of objects. We can also say that summative characteristics of an element are those which are the same within and outside the complex. As we note above, between the two summative characteristics there is only one characteristic of an element the nature of the element. According to the definition this characteristic is independent of the place or the position of the element because it is a characteristic of the element in principle. So, the last definition becomes a tautology.
5 Constitutive characteristics are those which are dependent on specific relations within the complex More precisely, the constitutive characteristics are not dependent on specific relations but they are the relations themselves (according to Bertalanfy description above). for understanding such characteristics we therefore must know not only the parts, but also the relations. It seems that there is a tautology once more, because the constitutive characteristics itself are the relations. Returning to the examples represented in Fig.1, it has to be noted that these examples lead to deeper understanding of the problem. The essence of the problem can be found by analyzing the set a only. Let us describe the set a so that one can reconstruct the image using the following description: Four small white circles located on a horizontal line at equal distances situated in the center of the page. That description potentially contains all possible oppositions between the set a and any other set. To get an opposition one has to change one of the descriptors. For example, changing four to five gives the difference in number; converting white to black creates the difference in the nature of elements; changing horizontal line to vertexes of rectangle creates the difference in relations between elements. One can see that there are not three but four kinds of possible distinctions between sets. The fourth distinction is the difference in the relation of the set a as a whole to the environment. In our case that distinction is represented by the direction of the line (horizontal or vertical) and the position on the page (in the center or at the margin). So, the emergent features (in this case it is the geometrical relations between the elements) don t arise from nothing in a mystical process of creating the whole, but are chosen from the description generated in our mind by the given image. Finally, it has to be mentioned that in the 60s, M. Bongard has published a book on pattern recognition , which contained a file of 100 cases similar to the three cases shown in Fig.1. He questioned if it is possible to develop a program that could find the distinction rule in each puzzle (some of them are not trivial to a human eye). Shortly he and V. Maximov succeeded in developing such a program. III. 2 - Communication and computers. The book contains a number of references to the field of communication and computer science. Some of them sound naive and don t support the author s idea. III The general notion in communication theory is that of information (p.41). The fact is that the definition of quantity of information exists, (it was introduced by C. Shannon) yet there is no scientific definition of information. III Examples can easily be given where the flow of information is opposite to the flow of energy, or where information is transmitted without a flow of energy or matter. The first is the case in a telegraph cable, where a direct current is flowing in one direction, but information, a massage, can be send in either direction by interrupting the current in at one point and recording the interruption in another (p.42). It is not correct to think that in the cable the current move is only in one direction - from the battery to the far end. It is a fundamental law of electricity that the same amount of current that goes from the buttery to one wire comes back through the returning wire. That is why the cable line has two wires. III For the second case (when information is transmitted without a flow of energy or matter), think of the photoelectric door openers: the shadow, the cutting off of light energy, informs the photocell that somebody is entering (p.42). Of course, the mystical communication without energy and
6 matter don t exist (at least in science). The point is that the absence of light in the photocell is a signal only when that event is followed by a period of presence of light, which, unfortunately, demands energy. The next reference to the communication field concerns the measurement of information. To represent the idea of the bit as a unit for measuring information, Bertalanfy wrote: Take the game of Twenty Questions, where we are supposed to find out an object by receiving simple yes or no answers to our questions. The amount of information conveyed in one answer is a decision (!) between two alternatives, such as animal or non-animal. With two questions (?) it is possible to decide for one out of four possibilities. Thus, the logarithm (base 2) of possible decisions ( the number of possible decisions will be correct because it is impossible to calculate the logarithm of decisions. SG) can be used as a measure of information (p.42). This story sounds very similar to the classical one but small differences change dramatically the understanding of the issue. Here is the classical story. Let us suppose we have eight golden coins looking identical. Seven of them are really identical but one coin is lighter then others. The question is: if one has a simple scale, what will be the minimum number of weighs to find the false coin. The right answer is three: first, compare the weight of any four coins and the rest four coins. Secondly, use the lightest set of four, divide it in two even parts and put them on the scale. Finally, take the lightest pair of coins and compare them. So, the false coin was found in three measurements. Here is the difference between these two stories. In the classical case it is defined that one can ask only one question: Which set of coins is heavier?, and a tool is given that answers that particular question only. That means that the answer to the problem ( three weighs ) doesn t depend on the skill of the questioner and therefore is a good measure of information. The story represented by Bertalanfy is different in the point that the questions are not predefined. So, the result of the game (i.e. the number of questions used to get the answer) depend on the skill (or the luck) of the questioner and one can t be sure that nobody will find a shorter sequence of questions to reach the answer. Therefore in that case the number of questions can t serve as a measure of information. The classical story tells us that getting to the goal in the shortest way is necessary to divide the set we questioned in two even parts. That is why 2 was chosen as the base of the logarithm in the definition of the bit. As a matter of fact in the Twenty questions game the questions never divide the set given at the particular step of the game in two even parts. That means that the answer to such a question doesn t contain one bit of information. It is true that in the Twenty questions game a good question is the one that divides the set of potential objects into two approximately even parts. IV. The big mystery In the middle of the book, in the ocean of examples dedicated to creation of a new physico-mathematical science, i.e. GST, appears an island of completely different approach. Bertalanfy wrote 10 lines that deeply and dramatically explain the essence of the system s problem, in a way completely different from the rest of the book. IV. 1 - Bertalanfy dismissed the mysticism of emergent characteristics one of the basic statements in system analysis - in the following couple of sentences: The constitutive characteristics are not explainable from the characteristics of isolated parts (because they are characteristics of different kind of objects: the whole and the parts - SG). The characteristics of the complex, therefore, compared to those of the elements, appear as new, or emergent. So, according to Bertalanfy the emergent features are simply characteristics of the relationships between the parts. IV. 2 - Bertalanfy admitted that systems exist in our mind only: A system as total of parts with its interrelations has to be conceived of as being composed instantly. Yes! It is true: an instant transformation is impossible in the physical world - it is possible in our minds only.
7 IV. 3 - Bertalanfy uncrowned the holy banner of system approach: The whole is more then the sum of its parts. Bertalanfy wrote: If, however, we know the total of parts contained in a system and the relations between them, the behavior of the system may be derived from the behavior of the parts. In the last statement Bertalanfy explained that when the partition is wrong (i.e. when the whole is divided into inappropriate pieces), the whole can t be understood and reconstructed from pieces. But if the whole is divided into right parts, the whole can be appropriately reconstructed from these parts. This transforms the cardinal problem of relationships between whole and parts into the problem of the appropriate partition of a given object. In other words Bertalanfy came to the problem of finding an adequate language of description. IV. 4 - Bertalanfy explicitly admitted that only Gestalt psychology showed the existence and primacy of psychological wholes (p.31). Then the biggest surprise follows. In his book Bertalanfy repeated dozens of times how great GST is and how it will completely reform the modern science - without defining the matter of research and any basic notions. But here, after the paragraph with cardinally important statements about whole and parts, about systems and emergent characteristics, he wrote: These statements are trivial! V. The cause of the GST' success Despite all the difficulties in defining the subject of GST and its basic notions, the system movement has spread around the world and became an important part of the XX century scientific life. What is the cause of that phenomenon? V. 1 - There are an infinite number of phenomena that are extremely important in the mankind s life but are very complicated and for now can t be solved by scientists: in biology, sociology, geology, economy, culture, politics, war, cosmology and in science itself. It is a real challenge and a powerful stimulus for a creative mind to find a solution for either one of such problems, or a general solution for a broad class of problems. L. von Bertalanfy creates an environment that promotes the will to work on complicated problems. It is no wonder that many talented scientists respond to that call. In time a lot of new constructions of GST appeared based on different basic notions but none of them succeeded, and the reason was the same: all authors attempted to create GST as a solid physical-like or mathematical-like science. It supposes that any dependence on our conciseness can t be accepted. Therefore none of the theories catch the notion of whole, which is a product of our mind. V. 2 - From the very beginning Bertalanfy tried to penetrate a lot of scientific areas, particularly poorly formalized areas (such as biology, medicine, psychology, sociology) and actively developing branches of science and technology (cybernetics, computers) that didn t yet have a solid scientific background. Scientists always want to represent their work in a frame of a solid scientific theory, particularly on the mathematical base. In the last two hundred years it was a common view that the particular science contains as much science as much it contains mathematics. The GST gave them some theoretical frame that emphasized the complexity of the problem, offers a set of pseudo-scientific terms, mathematical formulas and notions, and linked them to fashionable cybernetics, computers, game theory, etc.
8 V. 3 - Because GST community proclaims its revolutionary essence, it was favorable to all nontraditional hypothesis, theories and approaches. Therefore, many innovative scientific papers, which were rejected by solid professional and of course conservative magazines and conferences, were welcomed on the GST forums. At these forums free discussions flourished and real progress in science took place. It is a very symptomatic fact that 100% of the examples of systems in the book are natural objects objects that are not constructed by humans. The reason is that in human creations the structure of the object, the parts constituting the object (the whole ) are known. So, there is no question on how to describe the object, nor on which parts the object consists of. It means that in these cases there is not any system problem and Bertalanfy understood it very well. From that point of view, it also follows that the complexity is not a necessary attribute of a system. The system can be simple (for example a simple drawing) but still raises the problem of partition. As we show above, the proper definition of a system in GST is so broad that it covers any set of things in the universe and therefore is not acceptable in a physical theory. But as soon as one accepts the point of view on systems as a psychological problem, a Gestalt problem particularly, the unlimited definition of that notion becomes legitimate because then a system is a product of our mind and we are free to investigate any set as a system. VI. 50 years later How does GST look today? VI. 1 - The number of definitions multiplies every second author proposes a new definition. It demonstrates that everybody understands the necessity of definitions and nobody is satisfied with any previously proposed ones. The following examples are taken out of a single book (Proceedings of Third European Congress on Systems Science, 1996): - Set of communicating components according one and only one type of communication (p. 91). - A set of actions from a source on a sink represented by a messenger (p. 427). - This system is defined from a set S of the internal states, a set I of the inputs an from application F of the Cartesian product of S x I in S (p. 775). - Usually, we use the word Holistic in an abstract meaning, which says little or often nothing (p. 59). The number of examples may be increased. As a matter of fact mostly each definition is formulated a way which connects the content of the particular paper with the word system. Here is a prominent example of devaluation of the system theory notions: From the systemic way of knowing it s possible recognizing concepts as singularity, individuality, variety, dynamism, evolution, hierarchy, patterns, order-disorder, entropy and as many terms of our systemic approach we like to use (p. 58). O, sancta simplicita! A lot of papers use the word system in the title, in the introduction and/or in the conclusion of the paper. A very interesting paper on training of restaurant s waiters use the word system only once in the last sentence: Can the existing system evolve toward authoring shell? (p. 767). At the same time the average quality of the papers is good a lot of interesting and fresh ideas from all around the science. VI. 2 - Many authors express their dissatisfaction on the state of art and admit that GST is still in the beginning of the way. The debate on GST that evolved in the last few years shows the development of different points of view and opinions about the definition of system (p. 348).
9 The systemic view of the 1950s-1970s may not help us much in the 1990s, for the former may be more introductory than substantively revolutionary (p. 884). None of Bertalanfy s principles are mentioned. VI. 3 - The idea that systems are products of our mind, that the system problem is a problem of adequate description raised in B s writings in the 60s. Now it is still in discussion. Matter and mind are getting closer and closer, language and objects may find their previous unity. I think that is a systemic goal we are going to achieve (p. 59). VI. 4 - GST forums still attract not only interesting, talented and serious works, but also some superficial papers that would be rejected by the professional magazines for fair reasons. Since the author of the present paper has been involved during 40 years in Pattern Recognition business I will mention only one paper from the same volume. It contains a full page of images (p. 25) that have to illustrate the power of a new kind of neuronal network (synergetic computer). On the first figure 10 portraits in gray are shown. Five more images represent different parts of one of the portraits (marked G). The claim was: our computer could recognize faces from fragments. This result is trivial because to perform the search the computer uses fragments of the image stored. Any substantial part of an image is unique, i.e. has a unique distribution of black and white pixels. Therefore it is only a matter of time to find the appropriate image in the database for any computer. This way any image can be identified using its part. The real problem is to identify a portrait in the database using a different picture of the same person. The next statement in that paper was the following: our computer recognizes deformed faces. The corresponding figure contains three portraits (from ten mentioned above) and three rectangular grids slightly deformed: The grids show how the images were transformed. Two of them were deformed substantially, and one slightly. It turned out (by chance, of course) that the target image was the least destroyed one. It is appropriate to mention that the idea that the problems of recognition can be solved by developing new sophisticated algorithms died long ago. The practice shows that most recognizers produce the same results if applied to the same data. The only way to improve the results substantially is to change the description of the objects. This point binds pattern recognition to the systems problems. VI. 5 - The issue of complexity was one of the main concerns of GST and still is. Let us describe the problem in plain words. The words complex system mean that one can t predict the behavior of the system well enough. One cause of that could be the big number of parts in the system (for example, following Bertalanfy, the three bodies problem ). We know the laws of interaction between the parts, but we didn t find an analytical solution, and the computations on the computer are consuming expensive. This case is not interesting for GST. GST concentrates on systems with descriptions that have failed to consider the relations between parts. The idea is that taking in consideration these relations will resolve the problem, i.e. one will be able to predict the behavior of the system. As a matter of fact that means that the description of the system has to be changed: a set of new parameters has to be involved. That way of resolving the problem contains a possibility of failure: the partition of the system could be wrong. In that case we will no find adequate rules of interactions between parts. That is the most common situation in the investigation of complex systems: the existing partition of the given system into parts is not adequate. From author s personal experience the resolution of a number of complex problems takes place only after new descriptions of the systems at hand (oil exploration, earthquake location, speech compression, and handwriting recognition). It seems that the huge enterprise the Institute of complexity in Santa Fe (Arizona, USA) did no progress in investigating culture, economy, ecology, etc, because the leading idea was to use non-linear equations and supercomputers to deal with the existing descriptions of these systems. To my knowledge the problem of new descriptions was never raised in that place.
10 VI. Conclusion In his work Ludwig von Bertalanfy pursued the line of investigation started in the beginning of the century by Gestalt Psychology. The issue was the relations between whole and parts. That issue has a long history going back to Aristotle. The Gestalt psychology considers the problem as a psychological one, assuming that both whole and parts are products of our mind, our way to perceive the world. Bertalanfy s intention was to create a mathematical science of wholeness, which does not depend on our mind, since a "solid science" should not. Unfortunately, he didn t succeed in resolving these problems. His fate is similar to Einstein s: he formulated the Unified Field Theory but didn t succeed in resolving it. The fact is that Bertalanfy saw the approach, the approach that continue the Gestalt s psychology line, which treated whole and parts as products of our mind. He formulated shortly but consistently a number of statements that filled the new wine-skin with new wine. It is a mystery why Bertalanfy rejected that way. In author s personal experience it was the second mystery of science which has happened during the second part of the XX century. The first one took place in German atomic project during the World War II. German physicists and chemists famous for their scrupulosity and precision have failed to measure the correct diffusion length of neutrons in pure graphite. They found the length equal to 35 cm though the correct answer was 70 cm. It meant that absorption of neutrons in graphite is too big. As a result the graphite pile, as a shortest way to get the chain was rejected. Will these mysteries be ever resolved in the future?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
J. T. M. Miller, Department of Philosophy, University of Durham 1 Methodological Issues for Interdisciplinary Research Much of the apparent difficulty of interdisciplinary research stems from the nature
Principles of Programming Languages Prof: S. Arun Kumar Department of Computer Science and Engineering Indian Institute of Technology Delhi Lecture no 7 Lecture Title: Syntactic Classes Welcome to lecture
Unpacking Division to Build Teachers Mathematical Knowledge Melissa Hedges, DeAnn Huinker, and Meghan Steinmeyer University of Wisconsin-Milwaukee November 2004 Note: This article is based upon work supported
P ROCEEDINGS VOLUME I SPACE SYNTAX TODAY THE REASONING ART: or, The Need for an Analytical Theory of Architecture Professor Bill Hillier and Dr Julienne Hanson University College London, London, England
Why Computer Science? Robert H. Sloan University of Illinois at Chicago I have two teenage daughters. The older one is in college, and is studying computer science (CS). The younger one is in high school,
COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5
Atomic structure This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,
2.2. Instantaneous Velocity toc Assuming that your are not familiar with the technical aspects of this section, when you think about it, your knowledge of velocity is limited. In terms of your own mathematical
! Revised August 26, 2013 7:44 AM! 1 Chapter 1: Introduction Copyright 2013, David A. Randall 1.1! What is a model? The atmospheric science community includes a large and energetic group of researchers
Chapter 1 Chemistry: The Study of Change This introductory chapter tells the student why he/she should have interest in studying chemistry. Upon completion of this chapter, the student should be able to:
Cryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Lecture No. # 11 Block Cipher Standards (DES) (Refer Slide
ON EXTERNAL OBJECTS By Immanuel Kant From Critique of Pure Reason (1781) General Observations on The Transcendental Aesthetic To avoid all misapprehension, it is necessary to explain, as clearly as possible,
Investigation in Learning and Teaching of Graphs of Linear Equations in Two Unknowns Arthur Man Sang Lee The University of Hong Kong Anthony Chi Ming Or Education Infrastructure Division, Education Bureau
Extracted from Strategic Planning for Political Parties: A Practical Tool International Institute for Democracy and Electoral Assistance 2013. International IDEA, Strömsborg, 103 34 Stockholm, Sweden Phone
Lecture 25 Maher on Physical Probability Patrick Maher Philosophy 270 Spring 2010 Form of pp statements Experiments and outcomes Let an experiment be an action or event such as tossing a coin, weighing
Elementary School Mathematics Priorities By W. Stephen Wilson Professor of Mathematics Johns Hopkins University and Former Senior Advisor for Mathematics Office of Elementary and Secondary Education U.S.
The Philosophical Importance of Mathematical Logic Bertrand Russell IN SPEAKING OF "Mathematical logic", I use this word in a very broad sense. By it I understand the works of Cantor on transfinite numbers
Philosophy and Requirements Educational Goals The purpose of the UT Martin undergraduate educational experience is to prepare all students for the opportunities and challenges of a dynamic world. The combination
MATHS B-DAY 2006 Friday 24 November IN THE HANDS OF TIME The Maths B-Day is sponsored by and Maths B-day 2006-1- Wiskunde B-dag 2006 0 Introduction The maths B-day assignment this year is totally focused
Transcription of Questions and Answers from Virtual Open Event of Wednesday 23 rd February 2011 Topic Question Answer Administration Administration Administration Administration/Entry Requirements Is it
Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing
IFNA Mission The International Federation of Nonlinear Analysts (IFNA) is a not-for-profit educational and research oriented organization (society), that was founded more than 25 years ago, with the ambitious
Numbers and Operations in Base 10 and Numbers As the chart below shows, the Numbers & Operations in Base 10 (NBT) domain of the Common Core State Standards for Mathematics (CCSSM) appears in every grade
Joshua Baker Philosophy 123 Professor Slinker Laplace's Demon By finishing the work began by Sir Isaac Newton in mathematics, and further applying it to physics and astronomy, French physicist and mathematician
Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question
Performance Assessment Task Which Shape? Grade 3 This task challenges a student to use knowledge of geometrical attributes (such as angle size, number of angles, number of sides, and parallel sides) to
VISUAL ALGEBRA FOR COLLEGE STUDENTS Laurie J. Burton Western Oregon University VISUAL ALGEBRA FOR COLLEGE STUDENTS TABLE OF CONTENTS Welcome and Introduction 1 Chapter 1: INTEGERS AND INTEGER OPERATIONS
Pascal s wager So far we have discussed a number of arguments for or against the existence of God. In the reading for today, Pascal asks not Does God exist? but Should we believe in God? What is distinctive
EXPLAIN THE EFFECTIVENESS OF ADVERTISING USING THE AIDA MODEL SAHAR GHARIBI a DR.SEYED YAHYAH SEYED DANESH b DR.KAMBIZ SHAHRODI c Abstract The main objective of this research, Explain the effectiveness
Courses for Grade 11 Students All students are required to select eight (8) courses: ADVANCED ENGLISH 11 (compulsory or other level) Advanced English 11 is an intensive program of study that offers a challenging
The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards: Make sense of problems
Variables and Hypotheses When asked what part of marketing research presents them with the most difficulty, marketing students often reply that the statistics do. Ask this same question of professional
Proposal Writing: The Business of Science By Wendy Sanders The NIH System of Review The NIH has a dual system of review. The first (and most important) level of review is carried out by a designated study
Demonstrating Understanding Rubrics and Scoring Guides Project-based learning demands a more progressive means of assessment where students can view learning as a process and use problem-solving strategies
Chapter 8 Information, Entropy, and Coding 8. The Need for Data Compression To motivate the material in this chapter, we first consider various data sources and some estimates for the amount of data associated
Grade 5 Math Content 1 Number and Operations: Whole Numbers Multiplication and Division In Grade 5, students consolidate their understanding of the computational strategies they use for multiplication.
Verbatim Memory Tool for Memorizing the Apostle Paul s Epistle to the Ephesians KJV Created by Donald L. Potter Using Mike Shead s Verbatim Converter April 23, 2012 www.donpotter.net Ephesians KJV: Verbatim
What mathematical optimization can, and cannot, do for biologists Steven Kelk Department of Knowledge Engineering (DKE) Maastricht University, NL Introduction There is no shortage of literature about the
Department of Geography GEO 271 Everything is related to everything else, but near things are more related than distant things. - Waldo Tobler s First Law of Geography 1.1 Research in Geography [Meaning
IS YOUR DATA WAREHOUSE SUCCESSFUL? Developing a Data Warehouse Process that responds to the needs of the Enterprise. Peter R. Welbrock Smith-Hanley Consulting Group Philadelphia, PA ABSTRACT Developing
Research into competency models in arts education Paper presented at the BMBF Workshop International Perspectives of Research in Arts Education, Nov. 4 th and 5 th, 2013. Folkert Haanstra, Amsterdam School
CALCULUS COURSES AT THE COMPUTER SCIENCE FACULTY, THE UNIVERSITY OF INDONESIA Kasiyah MACHMUDIN Department of Mathematics, University of Indonesia Computer Science Faculty, University of Indonesia firstname.lastname@example.org
Glencoe correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 STANDARDS 6-8 Number and Operations (NO) Standard I. Understand numbers, ways of representing numbers, relationships among numbers,
9 th Grade Physical Science Springfield Local Schools Common Course Syllabi Course Description The purpose of the Physical Science course is to satisfy the Ohio Core science graduation requirement. The
CHAOS LIMITATION OR EVEN END OF SUPPLY CHAIN MANAGEMENT Michael Grabinski 1 Abstract Proven in the early 196s, weather forecast is not possible for an arbitrarily long period of time for principle reasons.
Reinhardt University Name: Francesco Strazzullo Group: Faculty Major Selection Summary Saved Majors Careers That Match Mathematics Saved Occupation Name Mean Salary Bank and Branch Managers $113,730.00
Parallelogram This problem gives you the chance to: use measurement to find the area and perimeter of shapes 1. This parallelogram is drawn accurately. Make any measurements you need, in centimeters, and
The power of money management One trader lost ($3000) during the course of a year trading one contract of system A. Another trader makes $25,000 trading the same system that year. One trader makes $24,000
Physics - programme subject in programmes for specialization in Dette er en oversettelse av den fastsatte læreplanteksten. Læreplanen er fastsatt på Bokmål Laid down as a regulation by the Norwegian Directorate
Mathematics on the Soccer Field Katie Purdy Abstract: This paper takes the everyday activity of soccer and uncovers the mathematics that can be used to help optimize goal scoring. The four situations that
Compass Interdisciplinary Virtual Conference 19-30 Oct 2009 10 Things New Scholars should do to get published Duane Wegener Professor of Social Psychology, Purdue University Hello, I hope you re having
In this chapter, we present the theory of consumer preferences on risky outcomes. The theory is then applied to study the demand for insurance. Consider the following story. John wants to mail a package
Quantity/potential-related elementary concepts in primary school teacher education Federico Corni, Enrico Giliberti, Cristina Mariani Faculty of Education, University of Modena and Reggio Emilia, Italy
Aquinas Third Way Last time we had arrived at the following provisional interpretation of Aquinas second way: 1. 2. 3. 4. At least one thing has an efficient cause. Every causal chain must either be circular,
The Basics of Graphical Models David M. Blei Columbia University October 3, 2015 Introduction These notes follow Chapter 2 of An Introduction to Probabilistic Graphical Models by Michael Jordan. Many figures
Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.
7.0 - Chapter Introduction In this chapter, you will learn improvement curve concepts and their application to cost and price analysis. Basic Improvement Curve Concept. You may have learned about improvement
Forex Trading Strategy Mistakes 3 X Common Trading Errors Which Cause Losses We all know that 95% of all Forex traders lose money and most of these traders have no hope of winning because they base their
Entry Level College Mathematics: Algebra or Modeling Dan Kalman Dan Kalman is Associate Professor in Mathematics and Statistics at American University. His interests include matrix theory, curriculum development,
Why STEM Topics are Interrelated: The Importance of Interdisciplinary Studies in K-12 Education David D. Thornburg, PhD Executive Director, Thornburg Center for Space Exploration email@example.com www.tcse-k12.org
Mechanics 1: Vectors roadly speaking, mechanical systems will be described by a combination of scalar and vector quantities. scalar is just a (real) number. For example, mass or weight is characterized
podcast Hello, and welcome to this JISC podcast interview. In this series we re speaking to people working on projects being funded by JISC s research data spring project to find out more about what they
The Miniature Guide to The Art of Asking Essential Questions by Dr. Linda Elder and Dr. Richard Paul Based on Critical Thinking Concepts and Socratic Principles The Foundation for Critical Thinking The
136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics
A Introduction to Matrix Algebra and Principal Components Analysis Multivariate Methods in Education ERSH 8350 Lecture #2 August 24, 2011 ERSH 8350: Lecture 2 Today s Class An introduction to matrix algebra
MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take
triplec 9(2): 460-465, 2011 ISSN 1726-670X http://www.triple-c.at Social Informatics Today and Tomorrow: Status, Problems and Prospects of Development of Complex Lines in the Field of Science and Education
Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture - 17 Shannon-Fano-Elias Coding and Introduction to Arithmetic Coding
The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards: Make sense of problems
Chapter 8 Systems of Equations 8. Systems of Linear Equations: Gaussian Elimination Up until now, when we concerned ourselves with solving different types of equations there was only one equation to solve
Such As Statements, Kindergarten Grade 8 This document contains the such as statements that were included in the review committees final recommendations for revisions to the mathematics Texas Essential
What is Holistic Financial Wellness? My 83 year old father doesn t like debt. Not one little bit. Growing up I was told many times Neither a borrower nor a lender be and that is the way he has lived his
1. CONTEXT Everywhere you turn these days, you will hear three buzzwords: SaaS (Software as a Service), cloud computing and Big Data. The first is a business model, the second a capacity model. The last
Sequences and Series Overview Number of instruction days: 4 6 (1 day = 53 minutes) Content to Be Learned Write arithmetic and geometric sequences both recursively and with an explicit formula, use them
Homeopathic times - Irish Society of Homeopaths - July 2002 Magnificent Massimo Manglialavori plunges into the heady world of mystery, structure and the importance of establishing an epistomological case
Composition as Explanation Gertrude Stein First delivered by the author as a lecture at Cambridge and Oxford, this essay was first published by the Hogarth Press in London in 1926 and revived in the volume
Gouvernement du Québec Ministère de l Éducation, 004 04-00815 ISBN -550-43543-5 Legal deposit Bibliothèque nationale du Québec, 004 1. INTRODUCTION This Definition of the Domain for Summative Evaluation
7th Grade Science Curriculum Overview Philosophy and Common Beliefs Science Curriculum Philosophy Statement Northbrook/Glenview District 30 utilizes a rigorous science curriculum built on essential questions,
Shape, Space, and Measurement- Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two- and three-dimensional shapes by demonstrating an understanding of:
raph Theory Problems and Solutions Tom Davis firstname.lastname@example.org http://www.geometer.org/mathcircles November, 005 Problems. Prove that the sum of the degrees of the vertices of any finite graph is
Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion
an introduction to VISUALIZING DATA by joel laumans an introduction to VISUALIZING DATA iii AN INTRODUCTION TO VISUALIZING DATA by Joel Laumans Table of Contents 1 Introduction 1 Definition Purpose 2 Data
Teacher s Guide For Core Biology: Animal Sciences For grade 7 - College Programs produced by Centre Communications, Inc. for Ambrose Video Publishing, Inc. Executive Producer William V. Ambrose Teacher's
GRADUATION AND DEGREE REQUIREMENTS GENERAL REQUIREMENTS FOR GRADUATION Upon successful completion of an approved plan of study and provided the following requirements have been fulfilled, the student will
Aesthetic Experience and the Importance of Visual Composition in Information Design By Tim Greenzweig When considering the design of information and information structures, the focus tends to gravitate