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Seeing and saying

Gordon Fisher

1. Prologue

Once upon a time, probably some tens of thousands of years ago, people began to speak. Before that, people no doubt looked and saw, listened and heard, touched and felt, sent and received gestures, and communicated with themselves and others. But they didn’t talk about it. Then gradually they began to try to express with languages what they saw and heard and touched, and so on. People are still trying to do this, with limited success.

“No language can be anything but elliptical, requiring a leap of the imagination to understand its meaning in its relevance to immediate experience.” (A. N. Whitehead)

I am going to speculate about ways Euclid and other geometers have used languages to give directions for drawing and imagining certain kinds of diagrams which show us something about ways we see things. One need not turn to formal geometry to give directions of this sort. Sketches can be used to show ways to get from one place to another, how to build things, where to look for objects in the sky like stars and planets, and so on. But there’s something special about geometric diagrams. For one thing, it is the custom of geometers to give directions not only for drawing or imagining certain kinds of directions. They also give directions for establishing relations between collections of diagrams, called proofs.

2. On the way to Euclid

Long ago, some people took to making pictures of certain things they saw. After some thousands of years, people took to trying to use pictures to draw what they spoke of. That is, they started trying to picture sounds, linking sight with hearing and saying by writing. Besides drawing pictures of what they saw, they began to draw pictographs of sounds of speech.

http://ancientscripts.com/chinese.html

“A pictograph is a depiction of a material object. Chinese characters mostly originated from picture writing. In other words, most Chinese characters were originally pictures of objects. However, there is a fundamental difference between pictographs and pictures: the former, usually rough sketches of objects(e. g. ¤é "sun" , ¤ë "moon" , ¤s "mountain" , ªe "river" , ¤H "man" , ¤j "big" ) or consisting of a characteristic part only (e.g. ¤û "ox", ¦Ï "sheep" ), are much simpler than the latter. More important is that pictographs are associated with definite meanings and pronunciations, and have become symbolic, and as a result of increasing simplification and abstraction, pictographs of the later ages are quite different from their originals. Compared with those in the Oracle-Bone Inscriptions, pictographs in the Regular Script are no longer picture like. In a sense they are not really pictographic, but simply symbolic.”
http://faculty.virginia.edu/cll/chinese/introduction.html

“One of the oldest and most complete diagrams from Euclid's Elements of Geometry is a fragment of papyrus found among the remarkable rubbish piles of Oxyrhychus in 1896-97 by the renowned expedition of B. P. Grenfell and A. S. Hunt. It is now located at the University of Pennsylvania. The diagram accompanies Proposition 5 of Book II of the Elements, and along with other results in Book II it can be interpreted in modern terms as a geometric formulation of an algebraic identity - in this case, that ab + (a-b)²/4 = (a+b)²/4 (although the relationship between Euclid's propositions and algebra, which he did not possess, is controversial).”
http://www.math.ubc.ca/~cass/euclid/papyrus/papyrus.html

Euclid had many predecessors, and many commentators on his work, and it’s hard to say what he saw and wrote about on his own, what he wrote about that he derived from others, and what others have added to what he said and which he is given credit for. For simplicity, I will talk as if some one person named Euclid wrote what Thomas Heath translated into English under the title The Thirteen Books of Euclid’s Elements (1926).

Euclid proposes in his preliminary assumptions (postulate, axioms) to write about points which have no parts, lines which have length but no breadth, lines which are straight because they lie evenly on themselves, surfaces which only have length and breadth, and solids which have length, breadth and depth. Since we never see the elements with which Euclid built his system with our outward looking eyes, there has been much commotion about how we do see these elements, if we can properly say that we do see them with our eyes. Plato proposed, in the Republic and elsewhere, that these elements somehow exist, but that beings in their everyday worlds can’t ever perceive them. They are said to be Ideals, or Forms, etc. Aristotle criticized such views, for example in the Metaphysics. A. N. Whitehead is notorious for having said that “the safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato.” It seems rather strong to classify all of Aristotle’s works as consisting of footnotes, not to mention the innumerable people who have philosophized since Plato’s time. However, Plato’s works have cast a long shadow (see his Republic).

http://www.users.globalnet.co.uk/~loxias/plato/caveframes.htm

Teachers and users of geometry commonly say that perceptible dots drawn on some surface are points. One might infer from this way of speaking that Euclidean points exist in some way analogous to the way perceptible dots do. Plato may have done something like this. Similarly, teachers say that certain kinds of perceptible streaks they draw are Euclidean lines. Some say that some people can visualize lines without breadth or depth inside of us with some inner apparatus. Some people call such an inner apparatus a mind’s eye, and speak of seeing lines without using our outward looking eyes. However, as far as I know, no one seriously claims that with a mind’s eye, a person can see lines of the sort Euclid defines the way we can see streaks with our outward-looking eyes, unless perhaps they happen to be using see to mean understand in a certain sense.

Rather than worrying about the ontological status of Euclid’s points, lines, etc., I propose instead to regard Euclid to have been writing about measurements, and designs, and sights of celestial bodies and ships from shores, and tools of trade and war. As for points, I like to interpret Euclid’s assumption that a point is that which has no parts as saying that we can choose to look at certain things and ignore the length, breadth and width they are seen to have when we look at them. For example, if one is interested in finding out how far it is from our earth to its sun, one can draw perceptible circular disks, think of them as picturing the earth and sun, and then make streaks between these pictures and think of them as, say, pictures of rays of light (or lines of sight) between the earth and the sun.

http://members.aol.com/satrnpres1/astronomy/unit1a.html

One can measure one face of an Egyptian pyramid by stretching a piece of rope so it lies evenly above and parallel to the edge. That much of the piece of rope which is between the two points may then be taken to be a measure of that edge. By ignoring the breadth and depth of the rope, we can view the part of the rope just above the edge as one of Euclid’s straight lines. We may wonder how it is that we can ignore breadth and depth of what we look at, but we don’t have to puzzle ourselves about how it is that there can be entities with length but without breadth and depth, like there can be pyramids like there are in Egypt. If we wanted to build another pyramid with a bottom edge like that of the one we measured, we could tie knots in a piece of stretched rope lying just above the two ends of the edge, and use that much of the rope between the two knots as a guide. The knots may be understood as one of Euclid’s points by ignoring its length, breadth and depth.

Egyptian Rope-stretchers

http://www.civilization.ca/civil/egypt/images/fback3b.jpg

Euclid proceeds to use the points, lines, etc., that he has introduced to show how to construct various plane figures. His first proposition uses two circles to construct an equilateral triangle. Here is what he wrote in Greek, followed by Heath’s translation.

Here is a copy of a figure given by Nathaniel Miller in his Euclid and His Twentieth Century Rivals (2007). The successive diagrams in this figure correspond strikingly, step by step, to the proof of Proposition 1 of Book I given by Euclid in the Elements, as quoted above from the translation by Heath.

Imagine an equilateral triangle being traced out in sand by an Egyptian rope-stretcher, as a first step in building a pyramid, as follows:

(1) He drives a stake into the ground, and having determined a favored direction, holds the rope taut, uses it as a guide for tracing out a straight path in this direction, and drives a second stake where the free end of the rope is. (1^st diagram of the figure)

(2) Holding the rope taut at the free end, he walks around the first stake and traces out a circle with a stick whose center is at the first stake. (2^nd diagram)

(3) He frees the rope from the first stake, ties one end of it to the second stake, and with the rope held taut, traces out a second circle, which he sees crosses the first circle at two locations, at which he drives two more stakes. (3^rd diagram)

(4) He fastens the rope taut between the center of the first circle and one of the locations where the circles cross, and traces a straight path between these two locations. (4^th diagram)

(5) He does the same using the center of the second circle. (5^th diagram)

At this point, the rope-stretcher is finished, and one can see that he has indeed constructed an equilateral triangle. I expect that most people who have read what I’ve written above will now (if they couldn’t before) be able to see by just looking at the 5^th of Miller’s diagrams that the triangle has almost equal sides, since all three sides are radii of circles which have the same radius. That is, the triangle is equilateral enough for practical purposes.

The remaining four of Miller’s diagrams correspond to applications of a definition and a common notions in Euclid’s proof, as given by Heath.

Note that nowhere have I assumed that the desired length of the sides of the equilateral triangle has been specified using numbers. You can just start with a piece of rope which looks to you to be as long as you want it to be.

3. Only logical

Evidently put off by Plato’s relegation of what we can see with our eyes to a never-never land of Forms beyond our senses, Aristotle talked about forms as structures or patterns of individual sightings. There is matter, which has no form(s) to speak of, and then there are combinations of forms and matter which are things that can be sensed (sensible things), like the particular chair I’m sitting in (on?) right now, and my daughter’s two dogs, Pal and Buddy. This suggests thinking about Euclid’s Elements as describing a way of using diagrams to exhibit forms of perceptible material objects, and of talking about relations between such forms.

Aristotle composed a system of useful rules for making true statements out of other statements known or believed or assumed to be true. This is known as Aristotelian or classical logic or syllogistics, a primary ancestor of later axiomatic methods A commonplace tradition has it that Euclid made conscious use of Aristotle’s logic to establish relations between diagrams which exhibit different forms. This has been challenged. For example, Abraham Seidenberg writes:

Historians are fond of repeating that Euclid developed geometry on an axiomatic basis, but the wonder is that any mathematician who has looked at The Elements would agree with this. . . . . . Could it be that, by insisting on the axiomatic bases, we are viewing The Elements from a false perspective and see its accomplishments in a bad light? This is precisely what I intend to prove.

The Greeks of Euclid’s time had the axiomatic method; Aristotle’s description of it can be considered a close approximation to our own. Or better yet, one may consider Eudoxus’ theory of magnitude as presented in Book V of The Elements: the procedure there disclosed is pretty much in accordance with our view of what an axiomatic development should be. It is known, however, that The Elements is a compilation of uneven quality, so that even with the definitions, postulates, and common notions of Book I, it is unwarranted to assume that Book I is written from the same point of view as Book V. . . . . .

Let us see how Euclid uses Postulate 3 [a circle can be drawn with any center and radius]. The very first proposition of Book I is To construct an equilateral triangle on a given straight line, say AB. With centers A and B, circles are drawn with radius AB. The circles intersect in a point C and ABC is a required triangle. But the question as to the existence of C is not raised. And here the polemic begins. We are invited to believe that Euclid had some subtle insight into the nature of geometry (or of reasoning) when he postulated that circles can be drawn, yet overlooked the obvious in Book I, Proposition I.

Let us look at Proposition 1 and what Euclid says in a straightforward way. Postulate 3 says nothing about existence: it says one can draw a circle. And Proposition 1 does not ask us to prove the existence of an equilateral triangle, but to draw one. Anyone who has thought about a construction problem knows very well that there is a difference between existence and construction. Even granted that an equilateral triangle on AB exists, there is still a trick to finding one: and Euclid shows us this trick.

A proposition of the form: There exists an equilateral triangle on any given straight line is a theorem, whereas a proposition of the form: To construct an equilateral triangle on a given straight line is a problem. We recognize the difference between a theorem and a problem, and so did Euclid. He does not label his propositions as theorems or problems, but each proposition ends either with the words "which was to be proved" or the words "which was to be done", so we have but to look at the last words of Proposition 1 to have incontestable evidence that Euclid considered it a problem.

Proposition 1 does indeed (if we allow the intersection) give the existence, but it also gives something more. As to the existence itself, there already were in ancient times opposing views. According to the school of Menaechmus, geometric objects, say equilateral triangles, exist because we produce them: just as a chair exists because an artisan has constructed it, so an equilateral triangle exists because the geometer constructs it. The followers of Speusippus, the successor of Plato at the Academy, on the other hand, held that geometric objects are eternal things, and hence not brought into being: it is better to say these objects exist. If we follow Speusippus, then the circles will meet, and Euclid is vindicated of every fault, at least through Proposition 1. The followers of Speusippus, however, in accordance with the view just expressed, insisted on calling all propositions "theorems" (rejecting the designation "problem" for the constructions). Hence it looks as though Euclid was inclining toward Menaechmus. In any event the issue as to whether the circles meet was not contemplated in ancient times.

For axiomatic purposes with modern intentions there would have been no use for Postulate 3 : wherever Euclid says to draw a circle with center A and through B one has but to say instead: "Consider the circle with center A that passes through B". Hubert, who in Chapter I of his Foundations of Geometry follows Euclid pretty closely, has no axiom corresponding to Postulate 3, or anything like it. (Seidenberg, 1974/1975)

In a similar vein, Ian Mueller writes:

(1) Aristotle's formulation of syllogistic in the fourth century is basically independent of Greek mathematics. There is no evidence that he or his Peripatetic successors did careful study of mathematical proof.

(2) Similarly, the codification of elementary mathematics by Euclid and the rich development of Greek mathematics in the third century are independent of logical theory.

(3) Likewise, Stoic propositional logic, investigated most thoroughly by Chrysippus in the third century, shows no real connection with mathematical proof.

(4) Subsequent to Chrysippus, hoi neōteroi [the young ones] considered various new forms of argument, including the unsystematically conclusive. Some of these new forms of argument may have come from mathematics. However, as the name 'unsystematically conclusive' suggests, no attempt was made to provide a logic for these arguments.

(5) Around the end of the second century B.C. Zeno of Sidon (and perhaps other skeptics and Epicureans) tried to undermine mathematics by pointing out gaps in proofs. Posidonius replied to Zeno, in many cases denying the existence of the gaps. But Posidonius also recognized that some geometric arguments, which resemble unsystematically conclusive arguments, depended on unstated principles. He considered the unstated principles self-evident and therefore called the arguments valid on the strength of an axiom. However, he made no progress in developing a logic to apply to these arguments. The debate over the need for further axioms in geometry continued for centuries and affected the text of the Elements itself.

(6) The reawakening of interest in Aristotle's works in the first century B.C.⁸¹ produced a Peripatetic reaction to Posidonius's analysis of ordinary mathematical argument. Aristotle's general remarks about the universality of the categorical syllogism became a dogma to be defended at all costs. Unsystematically conclusive arguments were made systematic by adding a universal premiss and attempting to transform the result into a categorical syllogism. The attempt was uniformly a failure.

(7) In Galen's Institutio Logica there is a more balanced view of unsystematically conclusive arguments, which Galen calls relational. Relational arguments depend for their validity on an additional axiom which is usually universal and usually categorical, but relational syllogisms are distinct from both categorical and hypothetical syllogisms. However, there is no evidence that Galen made any attempt to formulate a logic of relational syllogisms. [Mueller,1974, pp. 35-70]

My main point in quoting Mueller’s conclusions has been to indicate how already in antiquity it was recognized that some of Euclid’s proofs are “unsystematically conclusive”. I take it that the “unsystematically” refers to the fact that there are gaps in the proofs with respect to the techniques of written formulations of logic of the time of the sort based on explicitly stated axioms and explicitly stated syllogistic derivations of theorems.

But how was it that a theorem of Euclid could have been considered “conclusive” if its proof was not securely based on logic of the time? As a person who taught university mathematics for 40 some years, it seems to me to be obvious that people of the time who found Euclid’s proofs conclusive, albeit not fully supported by Aristotelian logic, were convinced by what they saw with their own eyes and by diagrams modeled after what they saw in the manner of Euclid and other geometers – what they saw either by outward looking or by imagination (inward imaging).

Here is a discussion by David E. Joyce of logical gaps in Euclid’s so-called proof of Theorem I.1:.

Critiques of the proof

It is surprising that such a short, clear, and understandable proof can be so full of holes. These are logical gaps where statements are made with insufficient justification. Having the first proof in the Elements this proposition has probably received more criticism over the centuries than any other.

Why does the point C exist? Near the beginning of the proof, the point C is mentioned where the circles are supposed to intersect, but there is no justification for its existence. The only one of Euclid 's postulate that says a point exists the parallel postulate, and that postulate is not relevant here. Thus, there is no assurance that the point C actually exists. Indeed, there are models of geometry in which the circles do not intersect. Thus, other postulates not mentioned by Euclid are required. In Book III, Euclid takes some care in analyzing the possible ways that circles can meet, but even with more care, there are missing postulates.

Why is ABC a plane figure? After concluding the three straight lines AC, AB, and BC are equal, what is the justification that they contain a plane figure ABC? Recall that a triangle is a plane figure bounded by contained by three lines. These lines have not been shown to lie in a plane and that the entire figure lies in a plane. It is proposition XI.1 that claims that all parts of a line lie in a plane, and XI.2 that claims that the entire triangle lies in a plane. Logically, they should precede I.1. The reason they don't, of course, is that those propositions belong to solid geometry, and plane geometry is developed first in the Elements, also, no doubt, plane geometry developed first historically.

Why does ABC contain an equilateral triangle? Proclus relates that early on there were critiques of the proof and describes that of Zeno of Sidon, an Epicurean philosopher of the early first century B.C.E. (not to be confused with Zeno of Elea famous of the paradoxes who lived long before Euclid), and whose criticisms, Proclus says, were refuted in a book by Posidonius. The critique is sound, however, and the refutation faulty.

Zeno of Sidon criticized the proof because it was not shown that the sides do not meet before they reach the vertices. Suppose AC and BC meet at E before they reach C, that is, the straight lines AEC and BEC have a common segment EC. Then they would contain a triangle ABE which is not equilateral, but isosceles. Zeno recognized that in order to destroy his counterexample it was necessary to assume that straight lines cannot have a common segment. Proclus relates a supposed proof of that statement, the same one found in proposition XI.1, but it is faulty. Proclus and Posidonius quoted properties of lines and circles that were never proven and never explicitly assumed as postulates.

The possibilities that haven't been excluded are much more numerous than Zeno's example. The sides could meet numerous times and the region they contain could look like a necklace of bubbles. What needs to be shown (or assumed as a postulate) is that two infinitely extended straight lines can meet in at most one point. (Joyce, see bibliography)

Of course, if a builder were to follow the directions given in Euclid’s proof by marking out the circles using a piece of a rope in something like the way I proposed in Section 2 of this article, he would see that the circles meet as claimed, and would not expect to have to justify what is in front of his eyes. Oddly enough, looking at the diagram which accompanies Euclid’s Proposition 1 can mislead you into concluding that all of the steps of Euclid’s written proof have been linked to what he has presented so as to be fully justified by written premises which he asks you to accept, perhaps because you can interpret them as corresponding to actions you can perform using such instruments as pieces of rope and people who use the rope to make something you can see. That you can be misled in this way has often been taken as evidence that looking at pictures can deceive you, and that it would be nice not to have to rely on pictures at all when composing a written proof.

In this connection, Nathaniel Miller says in his Euclid and His Twentieth Century Rivals (2007):

Euclidean geometry in general, and Euclid’s proofs in particular, have mostly fallen out of the standard mathematics curriculum. This is at least in part because Euclid’s Elements, which was viewed for most of its existence as being the gold standard of careful reasoning and mathematical rigor, has come to be viewed as being inherently and unsalvageably informal and unrigorous.

One key reason for this view is the fact that Euclid’s proofs make strong use of geometric diagrams. For example, consider Euclid’s first proposition, which says that an equilateral triangle can be constructed on any given base. While Euclid wrote his proof in Greek with a single diagram, the proof that he gave is essentially diagrammatic, and is shown in Figure 1. [See Figure 1 above. As discussed in Section 2 of this essay, the successive diagrams in this figure correspond strikingly, step by step, to the proof given by Euclid in the Elements, as quoted from the translation by Heath.]

A formal proof is one in which all of the rules that can be used are set out in advance so carefully as to leave no room for interpretation or subjectivity, and in which each step of the proof uses one of these rules. . . . . . This idea of a proof is a descendant of the idea which found its first enduring expression in Euclid’s Elements; that proofs should proceed in logical sequence from axioms set out in advance. . . . . . Most formal proof systems have therefore been sentential – that is, they are made up of a sequence if sentences in some formal language. . . . . . Such a sentential axiomatization was given by David Hilbert in 1899, and since then, his axiomatization has replaced Euclid as the commonly accepted foundation of geometry.

However . . . most informal geometric proofs still use diagrams and more or less follow Euclid’s proof methods. . . . . . The formal sentential proofs given in a system like Hilbert’s are very different from these kinds of informal proofs. So, while Hilbert’s system provides a formalization of the theorems of geometry, it doesn’t provide a formalization of the use of diagrams or of many commonly used proof methods. . . . . .

The central aim of the present book is to show that they can. In fact, the derivation contained in Figure 1 is itself a formal derivation in a formal system called FG, which will be defined in the following sections of this book, and which has also been implemented in the computer system CDEG (Computerized Diagrammatic Euclidean Geometry). (Miller, 2006, pp. 1-5)

4. A Conclusion

The philosophical views of foundations of mathematics known as logicism proposed that mathematics, or at least so-called pure mathematics, can be based on sentential formal logic of the sort pioneered by Bertrand Russell and Alfred North Whitehead in their Principia Mathematica (1911-1913), and earlier by Gottlob Frege. These views may be characterized as attempts to found mathematics on languages, albeit formal or perfect languages in the tradition of the characteristica univeralis of Gottfried Leibniz.

It is sometimes said that such attempts were shown to lead nowhere because of the way Russell’s paradox overthrew Frege’s program for logicizing mathematics. However, as a practicing mathematician, I always took it that the paradox could be avoided by using Russell’s theory of types in some informal way, or John von Neumann’s method involving distinguishing between sets and classes. And as far as geometry is concerned, it is well known that Russell and Whitehead intended to continue their logical basis for mathematics so as to include it, but this was never done. On the other hand, after the publication of Principia Mathematica, Whitehead went off on his own, and among other things wrote works on projective and descriptive geometry.

The work of Miller points the way to a formal foundation of mathematics based on certain kinds of pictorial diagrams of a sort that can be drawn, and which may also be based on internal visualizations by people equipped to do so. My aim has been to lend support to the view that diagrams of this kind can be based on looking at and working with physical objects, as people did when building such structures as the pyramids of ancient Egypt, and that at least Theorem 1 of Book 1 of Euclid’s Elements can be viewed as a verbal description of diagrams which can guide such activities. This is not to say that logic plays no foundational role in mathematics, but only to say that pictures do play such a role.

Bibliography

Fowler, David. The Mathematics of Plato’s Academy. Clarendon Press. 2^nd edn, 1999.

Heath, Thomas. A History of Greek Mathematics. Clarendon Press. 1921. (Dover, 1981).

" " . The Thirteen Books of Euclid’s Elements. Cambridge U. P. 2^nd edition, 1926. 3 volumes. (Dover, 1956).

Joyce, David E. http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI1.html

Miller, Nathaniel. Euclid and His Twentieth Century Rivals. 2007. University of Chicago Press.

Mueller, Ian. "Greek Mathematics and Greek Logic," in Ancient Logic and Its Modern Interpretations, edited by John Corcoran, Reidel, 1974, pp. 35-70.

" " . Philosophy of Mathematics and Deductive Structure in Euclid’s Elements. 1981. MIT Press. (Dover 2006).

Proclus. A Commentary on the First Book of Euclid’s Elements. Trans. Glenn R. Morrow. Princeton U. P. 1999.

Seidenberg, Abraham. “Did Euclid’s Elements, Book I, Develop Geometry Axiomatically?.” Archive for History of Exact Sciences, vol. 14, 1974-1975, pp. 263-295.

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