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      From the earliest moments of human history, thoughtful men and women have struggled with the tension between the continuous, ever changing stream of sensation and the discrete and unchanging concepts with which language and memory records it. Throughout history, there have been those who concerned themselves more with the sensations, and those who cared more about the concepts. Plato (ca 428 -347), for example, is the archetype of those who favored concepts over sensations, arguing that all sensations -- all the information we receive from the world around us through our senses -- are errors, misinformation designed to lead us away from the truth.

    The truth, according to Plato, is stable, unchanging, perfect, and cannot be learned from studying our experiences. Truth resides not in this world, but in an ideal world, a world of ideas, where only abstract, perfect and unchanging concepts --concepts like “truth”, “beauty”, “justice”, along with abstract, perfect, immaterial circles, triangles, and other perfect geometric figures. Plato suggests that we once resided in that perfect world, but were cast out due to a fundamental sin we committed in the darkest regions of the past. We are now imprisoned in this world of illusions, and we can learn nothing from observing it; rather our sensations will lead us away from the truth. The best we can hope is that we can remember something from our previous sojourn in the world of ideas, but to do so we must close ourselves off from the distractions of this world of illusion and change.
Not all Greeks abided by Plato's injunction to avoid careful study of the world of experience. Aristarchus of Samos (ca 310-430 BC), for example, who is often considered the first person to suggest that the sun, and not the earth, was at the center of the solar system, developed methods to measure the size of the earth, moon and sun, and the distances among them, although his primitive instrumentation led him to underestimate these distances greatly. Aristarchus was himself a representative of a community of scientists and mathematicians including Egyptian and Babylonian astronomers, architects, and engineers who understood that measurement is always and only a process of comparison to a standard. As Edna Kramer (1902-1984) remarks, “All measurement, whether it is concerned with distance or time or weight or electric current or intelligence or beauty, is merely comparison with a standard.”1

    This distinction between the platonists -- those who believe that knowledge is absolute, categorical and certain, and the scientists -- those who believe knowledge is comparative, relative and uncertain -- is the most fundamental distinction in epistemology.
  Plato's World  

    In Plato's world, knowledge, represented by the Greek word episteme, is perfect, unchanging, categorical and hierarchical. It is not derived from empirical study of the world we live in, but rather recollected from our experiences in the world of ideas before our fall from grade. The latinization of Plato’s concept of education comes from the Latin word ducere (to lead) and e (from, out of) expresses Plato’s belief that knowledge is found within the mind, not in the world. Plato’s student, Aristotle (384-322 BC), agreed with Plato that knowledge was perfect, unchanging, categorical and hierarchical, but did not agree that the source of knowledge was some idealized world of ideas, but rather the world itself. Instead, Aristotle asserted that all existing things were made of two principles; primary matter and substantial form. Matter is restless, constantly changing, indefinite, and has no characteristics apart from its restlessness. Form is absolute, unchanging and perfect. It is matter which makes a thing exist, but it is form which makes it what it is (to ti isti, later translated to the Latin essence.)

    When we observe nature, it is these forms which we abstract from our experience, and these abstract, unchanging and perfect forms are the object of our knowledge. The object of knowledge is the same for Aristotle as for Plato, but its origin is this world rather than the world of ideas. The continuous change we observe is simply the everlasting process of restless matter throwing off and taking on forms, which are themselves unchanging and absolute.

    In this absolute world of perfect forms, a man is a man because he has the form “man.” (Aristotle always said “man,” because he did not believe women were human, but rather a domesticated animal like a dog or cat.) Aristotle also invented a method of reasoning within this categorical system called the syllogism, a system based on patterns of inclusion in sets of nested categories. In the illustration, Arnold Schwarzenegger, Michael Jackson, Frederick Chopin and Wolfgang Mozart are all members of the category “All Men,”, and all men are members of the category “mortal”, therefore Arnold, Michael, Frederick and Wolfgang are also members of the category “mortal”.
    In the Aristotelian model, it doesn’t make sense to consider the extent to which each of these men exhibits the characteristics of manhood, since the system is absolute and categorical. The form “man” in Aristotle’s system has the attributes “rational” and “animal.” Insofar as he is a man, every man is identically rational and animal. The form “man” is unchanging and perfect. We abstract the meaning of “man” from our observation of many individual men, and, although the restlessness of the primary matter which establishes their existence obscures the perfection and changeless nature of the form “man,” we are able to form a concept of the pure form independent of its manifestation in any individual.

  The World of Science  
      In addition to the word “episteme”, which meant a knowledge at once abstract, perfect and unchanging, the Greeks had another word, “scia”, which represented another kind of knowledge, the comparative, practical, changing and relativistic knowledge one could gain about the world of everyday experience. Both Plato and Aristotle considered this an inferior kind of knowledge, one that was appropriate to lesser human beings, such as tradesmen, carpenters, brickmasons, and the like, and not to the highest men, the philosophers. It is this kind of knowledge that is embraced by the world of science. Plato and Aristotle’s predecessors, particularly the Sophists, believed that no certain knowledge could come from the world of experience, and that all knowledge about the world of experience was relative and uncertain.

    In sharp contrast to the absolute, categorical and hierarchical world of Plato and Aristotle, the world of science is relative and comparative. An object is not large, but large compared to some other thing. A work of art is not beautiful in an absolute sense, but it is more or less beautiful than some standard. And men are more or less rational or more or less animal compared to other men, rather than in an absolute sense. Furthermore, the world of science, being based on observations made by fallible humans on changing and evanescent phenomena is always subject to uncertainties of measurement and subject to revision by later, more precise observations. As Einstein says, “as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”

    It goes without saying that the Platonic model, which recommends isolation from the world of experience, is not suitable to the development of an empirical science. Aristotle’s rejection of Plato’s antipathy for empirical observation was not in itself enough to overcome the problem, however, because it retains the idea that the goal of science is absolute, categorical, unchanging truth. The development of physical science as we know it required the rejection of Aristotle’s categorical, syllogistic model and the development of the comparative model. Although early Egyptian, Babylonian, Assyrian, Hindu and other architects and astronomers used comparative measurements, most writers mark the beginning of serious, self-conscious science in the modern sense with Galileo Galilei (1554-1642).

    Galileo’s experimental work is filled with outstanding examples of the comparative method of measurement. His experiments with buoyancy, for example, show him adding enough salt to a glass container of water so that a pea will float midway between the top and bottom of the water. Adding 1 pinch of salt will cause the pea to rise “3 finger breadths.” Here Galileo is comparing the distance the pea rises to the width of his fingers. He is alleged to have observed the swinging of chandeliers in church and timed them against his pulse beat.

    In his even more famous experiment with falling bodies, Galileo rolled a ball down a groove in an inclined plane. As he did so, he hummed a rhythmic song, and marked the position of the rolling ball on the beats with chalk. He then placed movable frets across the chalk marks, then rolled the ball repeatedly, adjusting the positions of the frets until the ball bumped over them exactly on the beats. He then had only to measure the distances among the frets to determine how far the ball fell in equal intervals, and from this he was able to deduce the exponential law of falling bodies. In this case he is comparing distances fallen to the rhythmic pulse of music.
  Galilean Transformations and Standards of Measure  

    The sophists believed not only that observations and their resulting knowledge were relative, but also that they were personal and individualistic, so that the observations of one observer need not correspond to the observations of any other. This radical individuality of knowledge was at the base of the sophists' belief that no knowledge was possible, but only opinion. The goal of science, however, is not a personal, idiosyncratic knowledge unique to each individual, but rather a public, shared, objective knowledge about which all observers agree. If Galileo and another observer compare motions of any object to their respective pulse beats, they will disagree to the extent that their pulse rates are not the same. If the observations made while comparing some motion to the pulse beat of the observer reveals something exciting, the excited observer’s pulse rate will increase and change the outcome unpredictably.

    In order for observations to be compared across observers, the comparative method requires that those observers use the same comparative standard, or else understand the relationships among various different standards. If observers measure the distance rolled down an inclined plane while singing different songs, their observations will not agree. By June 12, 1215, the understanding that unstandardized measures lay at the root of disagreement and dispute had gained such currency that the 35th item of the Magna Charta demands a single, standard set of weights and measures: “There shall be one measure of wine throughout all our kingdom, and one measure of ale, and one measure of corn, namely the quarter of London; and one breadth of dyed cloth, and of russets, and of halberjects, namely, two ells within the lists. Also it shall be the same with weights as with measures.” As the understanding of the role of comparative standards in developing consensus about observations grew, so did the collective resources devoted to developing measurement standards. A great milestone in international understanding was the International Treaty of the Meter, in 1875, which established the basis for international agreement as to comparative measurement standards which continues to this day.

    Even though multiple observers make observations using standardized measurement scales, it’s still possible for their observations to

disagree solely as a result of the different orientations of their individual frames of reference. As can be seen in the figure, for example, an observer at P1 will see, from left to right, A, B, C in that order, while and observe e at P2 will see the array C,A,B from left to right. These differences are solely the result of difference of reference frame, even though both observers are viewing the “same” three objects.

    The situation is even more complicated if the observers are in motion relative to each other. If an observer in a train drops a ball, s/he will see the ball drop directly down to the floor in a straight line. An observer on an embankment next to the moving train will observe the ball fall in a parabola. Neither of these is absolutely correct, but the observations of either observer can be transformed to match the observations of the other by a simple translation. Even more complicated is the case of observers rotating relative to one another. If an observer on a merry-go-round drops a ball, the ball will appear to move off in a straight line from the center, but, to an observer on the ground, the motion will again appear more complicated. Here, a rotation will suffice to bring the two observations into correspondence. It is a great discovery of classical mechanics that all relative motions due to differences in reference frames can be transformed away by a combination of translations, rotations and reflections (galilean transformations.)

    It is these sets of standards, along with developments in the mathematics of transformations across reference frames, such as the Galilean Transformation for ordinary space and time, and the Lorenz Transformation, which solve the problem of idiosyncratic, individual knowledge that can’t be compared from observer to observer that so troubled the sophists. By means of standardized comparative measurements and transformations across individual reference frames, the experiences of different observers can be compared. As Einstein says, those about which we agree, we call real.

  Aristotle’s Social Science  
      While physical scientists following the methods of Galileo made explosive progress in describing experience, social scientists continued to work with the antiquated Aristotelian concepts. Since Aristotle’s model was categorical, it could not describe processes precisely, which led to some enormous observational errors. Aristotle theorized, for example, that objects fell at a rate proportional to their weight, so that a heavy object would fall faster than a light object. This, of course, is not true, as Galileo’s law of falling bodies shows, but Aristotle’s law stood almost 2000 years.

    Many writers make light of the fact that the most learned minds of Europe and Asia were unable to demonstrate that Aristotle’s Law of Falling Bodies was false for 2000 years by suggesting that air resistance can account for the discrepancy; after all, in air, a feather does fall more slowly than a hammer. But, in fact, the discrepancy between Aristotle’s law and everyday observation is enormous. Consider a paper clip that weighs half a gram, and an encyclopedia that weighs 5 kilograms. The book is 10,000 times heavier than the paper clip, and Aristotle’s law demands that it fall 10,000 times faster than the paper clip. If one lays the paper clip on top of the book, and drops the book from the height of a person standing on a desk, it will be obvious to every observer in the room that the paperclip remains in contact with the book at all times, falling at an identical velocity. See these NASA vids showing objects dropped in space too.

    How could this extraordinary disagreement between theory and observation survive 2000 years? Could it just be that no one made any observations of falling bodies? Absolutely not. In 1066, at the Battle of Hastings, gunpowder was introduced into Western warfare. For the following 500 years, military research enlisted the best minds of Europe in the analysis of the trajectory of mortar rounds and artillery shells. Yet it was Galileo who determined that the trajectory of a projectile was parabolic, almost six hundred years later. It is unlikely that the military researchers were unmotivated or unintelligent over that 500 plus year period. Much more likely is that the new comparative methods of Galileo made it possible to see phenomena that were not accessible to those using an Aristotelian categorical model.
  The Comparative Model  
      To illustrate the difference between the Aristotelian categorical model and the comparative scientific model, consider the following definition of several men. In the following questionnaire, respondents are asked to report how dissimilar or “far apart” some men and attributes are from each other. Since all measurement is the comparison to some standard, respondents are told to use the distance between sensitive and strong as a standard dissimilarity, and to estimate all other dissimilarities as ratios to that standard. The result is a square matrix of dissimilarities which can be converted to a spatial representation by well known mathematical procedures.

Please estimate how different or "far apart" each of the following

words or phrases is from each of the others. The more different,

or further apart they seem to be, the larger the number you should

write. To help you know what size number to write, remember


If two words or phrases are not different at all, please write

zero (0). If you have no idea, just leave the space blank.

Thank you very much for your help.



COL. ----------------------------------------------------------

0102 9-17 RATIONAL and ANIMAL _____

0103 18-26 RATIONAL and STRONG _____

0104 27-35 RATIONAL and SENSITIVE _____

0105 36-44 RATIONAL and ARNOLD S. _____

0106 45-53 RATIONAL and CHOPIN _____

0107 54-62 RATIONAL and MOZART _____

0108 63-71 RATIONAL and MICHAEL J. _____

0109 72-80 RATIONAL and YOURSELF _____



COL. ----------------------------------------------------------

0203 9-17 ANIMAL and STRONG _____

0204 18-26 ANIMAL and SENSITIVE _____

0205 27-35 ANIMAL and ARNOLD S. _____

0206 36-44 ANIMAL and CHOPIN _____

0207 45-53 ANIMAL and MOZART _____

0208 54-62 ANIMAL and MICHAEL J. _____

0209 63-71 ANIMAL and YOURSELF _____

0304 72-80 STRONG and SENSITIVE _____



COL. ----------------------------------------------------------

0305 9-17 STRONG and ARNOLD S. _____

0306 18-26 STRONG and CHOPIN _____

0307 27-35 STRONG and MOZART _____

0308 36-44 STRONG and MICHAEL J. _____

0309 45-53 STRONG and YOURSELF _____

0405 54-62 SENSITIVE and ARNOLD S. _____

0406 63-71 SENSITIVE and CHOPIN _____

0407 72-80 SENSITIVE and MOZART _____



COL. ----------------------------------------------------------

0408 9-17 SENSITIVE and MICHAEL J. _____

0409 18-26 SENSITIVE and YOURSELF _____

0506 27-35 ARNOLD S. and CHOPIN _____

0507 36-44 ARNOLD S. and MOZART _____

0508 45-53 ARNOLD S. and MICHAEL J. _____

0509 54-62 ARNOLD S. and YOURSELF _____

0607 63-71 CHOPIN and MOZART _____

0608 72-80 CHOPIN and MICHAEL J. _____



COL. ----------------------------------------------------------

0609 9-17 CHOPIN and YOURSELF _____

0708 18-26 MOZART and MICHAEL J. _____

0709 27-35 MOZART and YOURSELF _____

0809 36-44 MICHAEL J. and YOURSELF _____

      Figure 2 is the spatial representation of responses to the questionnaire constructed by the Galileo computer program and displayed with ThoughtView™ The dimensions (x,y and z) are the principle axes of the configuration of points. In the map shown in Figure 2, objects are close to other objects that are like them, and far from objects from which they differ. Direction (up, down, left right, in, out) is of no significance. The grid is supplied only to help visualize the distances among the objects, and has no other significance. The orientation of the space is also of no significance, so it doesn’t matter which objects are at the top, or how far from the grid they are. The inter point distances are invariant under rotation, translation and reflection, so turning the space upside down or sideways does not affect it’s meaning.

Figure 2: Galileo Map of Men and Attributes

    As can be seen in the map in Figure 2, this representation of men is fundamentally different from the Platonic/Aristotelian view. First, it is comparative rather than categorical. The various men are more or less “rational” and more or less “animal” depending on how close they are to those terms in the map. Mozart and Chopin are closer to “sensitive” and “rational” than the other men, while Arnold is closer to “strong” and “animal” than the other men. It is fundamentally relative, since we can only say two men or attributes are “far apart” compared to some other dissimilarity. Secondly, it is based entirely on the observations of the respondents, and represents their experience rather than objective truth. Third, it is subject to uncertainty and changes, since all measurements contain some degree of error, however, small, and represent a snapshot of a world in flux.

Figure 4: Regions of Uncertainty around Concept Locations

    Figure 4 shows the same space as Figure Three, but each concept is surrounded by a cube that represents the standard error of the estimate of its coordinates. Statistically speaking, there is about a 68% likelihood that the concept is located within its corresponding cube. As more and more measurements are added, these cubes can be made as small as desired, but they can never be made zero; in this sense, there is always some uncertainty about exactly where each concept is located.

Figure 5: Distances between 5 men and four attributes

    Although the comparative method gives up the hope of being perfect, absolute and unchanging, it is, in fact, much more informative that the categorical model. First, each man is defined relative to the several attributes selected here, and defined as having more or less of them, rather than having them or not, as the categorical model demands. (See Figure 5.) This model is much more informative than the categorical model, which can only say whether a man has an attribute or doesn’t. While it can never be perfectly accurate, it can be made more and more precise without limit.

    Secondly, the comparative model allows for the comparison of observations of multiple observers. In a categorical model, people chop up the continuous flow of experience into categories, which they name. But it is well known that different cultures, languages and even individuals break up their experiences differently, and that some will call “orange” what others call “red,” just as some will say “potato” and others “potahto.” These differential reference frames influence both the perception of the world and any communication about it, so it is impossible to compare the experiences of any two observers.

    Consider the case of emotions. One native English speaker defined the principle human emotions as the following: FEAR, LOVE, HAPPINESS, JOY, ANGER, HATE, SADNESS, SORROW, ANXIETY , JEALOUSY , and LUST. A native Hindi speaker defined the basic human emotions as: DAR, PREM, KHUSHI, AANAND, KRODH, GHRINAA, UDASEE, DUKH, CHINTA, JALAN and VAASNA. Are these the same? Native English speakers disagree about the meaning of the English words, as native Hindi speakers do about the Hindi words. Anyone not bilingual in English and Hindi has no hope of comparing the two lists, or the emotions of the two people who made the lists.

    Using a Galileo complete pair comparison questionnaire like the one used for men, above, it’s possible to ask each respondent to compare each emotion to all the others, and him/herself to each of the emotions (i.e., reporting “how close” s/he is to each of the emotions). The result will be an emotional space for the English speaker, and another for the Hindi speaker:

Figure 6: Emotions of an English Speaker Figure 7: Emotions of a Hindi Speaker

    In these maps, as with the map of men and their attributes, the respondents are closer to the emotions to the extent that they are experiencing them, and the emotions are close to other emotions that are similar to them and far from those that are seen to be different. Within each map it is therefore possible to compare each emotion to every other emotion, and to see the relationship of the individual (MYSELF or MA) to each of the emotions.

    It is not possible, however, to compare the maps to each other, because the maps do not share a common orientation. We are accustomed to the idea that the top of most geographic maps is always oriented to the North, but it was not always so. This convention developed over centuries of cartography, and represents a conventional agreement among mapmakers. No such convention exists for Galileo maps, so the orientation of each map is arbitrary.

    This problem is easily solved, however. Since Galileo coordinates are made by exactly the same procedures as ordinary scientific practice, the lessons learned by Galileo and his successors apply to Galileo maps as well. Any set of Galileo coordinates can be transformed onto any other set of Galileo coordinates by a combination of rotations, translations and reflections. Using the Galileo program Microrot, the coordinates of the English speaker and the Hindi speaker can be projected onto the same coordinate reference frame, as figure 8 shows:

Figure 8: English and Hindi words for 11 emotions.

    As in all Galileo spaces, the closer two words are, the more nearly they mean the same thing (two words at exactly the same point would be identical.) Within the Galileo frame of reference, differences in apparent meaning due to different reference frames can be transformed away and true2 differences can be expressed on common reference frames.


1 Kramer, Edna E., The Main Stream of Modern Mathematics, The Scholar’s Bookshelf, Princeton, NJ, 1988, p.42

2 “True” differences are differences not due to effects of differential reference frames, and not “true” in a platonic sense.

page updated May 30, 2012

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