Visual and Spatial Analysis. Eds. B. Kovalerchuk, J. Schwing, Springer 2004. Chapter 7 TOWARD VISUAL REASONING AND DISCOVERY Lessons from the early history of mathematics Boris Kovalerchuk Central Washington University, USA Abstract: Currently computer visualization is moving from a pure illustration domain to visual reasoning, discovery, and decisions making. This trend is associated with new terms such as visual data mining, visual decision making, heteroge- neous, iconic and diagrammatic reasoning. Beyond a new terminology, the trend itself is not new as the early history of mathematics clearly shows. In this chapter, we demonstrate that we can learn valuable lessons from the history of mathematics for visual reasoning and discovery. Key words: Visualization, visual reasoning, visual discovery, history of mathematics. 1. INTRODUCTION In Chapter 1, visuals were classified in three ways: (1) illustration, (2) reasoning, and (3) discovery. Illustration demonstrates the essence of enti- ties involved and presents a solution statement without showing the underly- ing problem solving reasoning process. Reasoning sets up explanatory rele- vance of entities to each other and discovery finds relevant entities. These categories form the creativity scale shown in Figure 3 in Chapter 1 where illustration and discovery are the two extremes in this scale with many in- termediate mixed cases. Reasoning occupies the middle of this scale. In Chapter 1, all three concepts have been illustrated with the Pythagoras Theo- rem: (1) visualization of the theorem statement, (2) visualization of the proof process for the theorem’s statement, and (3) visualization of the discovery process that identifies the theorem’s statement as a hypothesis. In visual decision making the listed categories have their counterparts: 154 Chapter 7 • visualization of a decision • explanation of a decision • visualization of the process of discovery of a decision. Computer visualization is moving from being pure illustration to reason- ing, discovery, and decision making. New terms such as visual data mining, visual decision making, visual, heterogeneous, iconic and diagrammatic rea- soning clearly indicate this trend. Beyond a new terminology, the trend itself is not new as the early history of mathematics clearly shows. In this chapter, we demonstrate that we can learn valuable lessons from the history of math- ematics. The first one is that all three aspects had been implemented in the ancient times without the modern power of computer graphics: 1) Egyptians and Babylonian had a well developed illustration system for visualizing numbers; 2) Egyptians and Babylonian had a well developed reasoning system for solving arithmetic, geometric and algebraic tasks using visual- ized numbers called numerals; 3) Ancient Egyptians were able to discover and test visually a non- trivial math relation, known now as the number π. How can we learn lesson from this history? How can we accelerate the transition from illustration to decision-making and problem solving in new challenging tasks we face now using history lessons? At first, that history should be described in terms of visual illustration, reasoning and discovery. This will give an empirical base for answering posed questions. Traditionally texts on the history of mathematics have different focus. This chapter could be viewed as an attempt to create such an empirical base for a few specific subjects. The first lesson from this analysis is: inappropriate results of illustration stage hinder and harm the next stages of visual reasoning and decision mak- ing. Moreover, this can completely prevent visual reasoning and decision making, because these stages are based on visualization of entities provided in the illustration stage. The most obvious example of such a case is exhibited by Roman numer- als. These numerals perfectly fulfill the illustration and demonstration role, but have very limited abilities to support visual reasoning for arithmetic (summation, subtraction, multiplication and division). Hindu-Arabic numer- als fit reasoning tasks much better. The second lesson is that the most natural visualization that seems iso- morphic to real world entities is not necessarily the best for reasoning and decision making. The Ptolemy Geocentric system was isomorphic to the ob- served rotation of the Sun around the Earth, but eventually it became clear that it does not provide advanced reasoning tools to compute the orbits of other planets. 7. Toward visual reasoning and discovery 155 The third lesson is that if we want to design (invent) visualization that will survive reasoning and decision making tests later we should be able to formulate future reasoning and decision making tasks explicitly at the time when design of visualization as illustration is started. The forth lesson is that if we design (invent) visualization without a clear vision of future reasoning and decision making (problem solving) tasks the chance is no better than a flip of a coin that the visualization will survive reasoning and decision making tests later. It seems that the history of mathematics points towards the conclusion that most initial visualizations of numerals were invented for illustration, description and recording purposes. Their usefulness for reasoning and prob- lem solving was tested later. Those that survived, namely the Hindu-Arabic numerals, we use now. In essence, this history fits the idea of evolution with the survival of the fittest. In this chapter, we analyze the history of Egyptian and Babylonian nu- merals that support lessons learned presented above. 2. VISUALIZATION AS ILLUSTRATION: LESSONS FROM HIEROGLYPHIC NUMERALS To provide an illustration it is sufficient to visualize concepts involved in the solution. Let us consider a simple arithmetic example, 3535+1717=5252. There is a justified computational procedure for getting this solution 5252, but the expression 3535+1717=5252 does not show the reasoning steps that lead us to the solution. In this example, concepts visualized are input, output, the summation op- eration and equality relation. Such visualization tasks were successfully solved in ancient Egypt, Greece, India and Mesopotamia by developing symbols for numbers and to some extent symbols for operations and rela- tions too. 2.1. Egyptian numerals Hieroglyphic numerals. Table 1 shows Egyptian hieroglyphic numerals and some of their ideographic meanings [Allen, 2001a, Williams, 2002c, Aleff, 2003, Bertin, 2003]. 156 Chapter 7 Table 1. Symbolic Egyptian Hieroglyphic numerals 2 3 4 5 6 7 1 10 10 10 10 10 10 10 1 2 3 4 5 6 7 vertical heal snare lotus bent burbot kneeling figure Sun stroke bone, coil of flower finger fish, with raised vault rope Tadpole arms, Heh-god Compare this with Roman and Hindu-Arabic numerals presented in Ta- ble 2. Romans had also other numerals: V for 5, L for 50, and D for 500. Table 2. Roman and Hindu-Arabic numerals 2 3 4 5 6 7 1 10 10 10 10 10 10 10 1 10 100 1000 10000 100000 1000000 10000000 | X C M X C M Table 3 shows some alternative design of hieroglyphs from sources listed above. Left and right forms were used on the left and right sides of the text to provide visual symmetric view of the wall. Several fonts have been devel- oped for hieroglyphs. Table 3 uses Gardiner, Glyph basic, and Nahkt fonts [Bertin, 2003]. Table 3. Symmetric pairs and alternative glyphs Number Left form Right form 100 3 1,000 4 10,000 # 6 100,000 $ 7 7. Toward visual reasoning and discovery 157 Hieratic numerals. The Egyptian Hieratic numeral system is also 10 based. Hieratic symbols for 2 and 3 are repeated symbols for 1 ( | ), re- spectively (||, ||| ) and the symbol for 8 (=) is a repeated symbol for 4 (−). The same symbols || and ||| are used in Hieroglyphic and Roman sys- tems for 2 and 3. Other unique symbols in the Hieratic system are [Fried- man, 2003b]: 5 = , 6= , 7= , 9= , and 10= 2. This system also has unique symbols for 20,30,40,50,60,70,80,90 and 100. The symbol for 20 ( ′2 ) is directly based on symbols for 10 (2), and the symbol for 40 ( − ) is based on symbol for 4 ( − ). Symbols for 200, 300, 400 and 500 are based on the symbol for 100 ( ). They are drawn by adding one, two, three or four dots (.) above, 200 ( ), 300 ( ), 400 ( ), and 500 ( ) [Friedman, 2003b]. In Table 4, we show how numbers were composed using base glyphs presented in Table 1 using the idea of Hindu-Arabic decimal positional sys- tem that is a modern standard for number visualization. It is so common after several thousand years of use that we often miss the point that this is just one of the possible number visualization systems. It would be more transparent if we contrast it with a textual description of numbers. The textual description at most visualizes the sequence of sounds, e.g., the word “thousand” in phonetics based languages. Table 4. Egyptian hieroglyphs and operations [font from Williams, 2002c] Modern symbols = 1=100 10 102 103 Egyptian hieroglyphs 1 2 3 4 3 Modern 3105=5+100+3*10 3 (backward notation) 5 0 100 3*10 Egyptian 3105 111113444 11111 3 444 158 Chapter 7 In Egyptian numeral system, there is a flexible sequence and every deci- mal has its own symbol. Thus, some hieroglyphic numerals are shorter than in modern notation, for instance: 1000110 = 237 The hieroglyphic system is less positional than the Hindu-Arabic system that we are using now. The change of the sequence of the components does not destroy the value of the number: 237 732 1000110 = = but for the Hindu-Arabic numeral 1000110 the backward sequence 0110001 is not equal to 1000110. The Roman system is an intermediate system between the Egyptian and the Hindu-Arabic systems. This system is more positional than the Egyptian system and less positional than the Hindu-Arabic system. At first glance, Tables 1 and 2 show that both Roman and Egyptian systems provide simpler visualization for numbers than the Hindu-Arabic system.