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 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.

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