Euclid and the Instructive Challenges of Anomalies

 George Anastaplo

Every empirical law has the disquieting quality that one does not know its limitations. We have seen that there are regularities in the events of the world around us which can be formulated in terms of mathematical concepts with an uncanny accuracy. There are, on the other hand, aspects of the world concerning which we do not believe in the existence of any accurate regularities. We call these initial conditions. The question which presents itself is whether the different regularities, that is, the various laws of nature which will be discovered, will fuse into a single consistent unit, or at least asymptotically approach such a fusion. Alternatively, it is possible that there always will be some laws of nature which have nothing in common with each other. At present, this is true, for instance, of the laws of heredity and of physics. It is even possible that some of the laws of nature will be in conflict with each other in their implications, but each convincing enough in its own domain so that we may not be willing to abandon any of them. We may resign ourselves to such a state of affairs or our interest in clearing up the conflict between the various theories may fade out. We may lose interest in the “ultimate truth,” that is, in a picture which is a consistent fusion into a single unit of the little pictures, formed on the various aspects of nature.

 – Eugene P. Wigner (1960)


 We very much depend upon– or, at least, we are certainly accustomed to– the regularity of things, beginning perhaps with that overarching “regularity” of the daily “rising” and “setting” of the sun. A solar eclipse, if not properly understood, can shake us up. By “properly understood” is meant that accounting for the apparent irregularity of an eclipse which sees it as the occasional conjunction of various well-established regularities.

Persistent anomalies pose a challenge, if not even a threat, for us. An anomaly can be defined as a “deviation from the common rule.” The “anomalous,” we are told, is derived from Greek words meaning literally “a departure from the same”– or “inconsistent with or deviation from what is usual, normal, or expected.” (Webster’s Ninth New Collegiate Dictionary)

There are no doubt anomalies all around us that we simply do not notice, but when noticed can be fairly easily explained. For example, I was struck, at a concert last Sunday, by something I had been many times exposed to but had never explicitly noticed: The National Anthem was played, for which, of course, everyone present stood up. Well, not everyone: those among us most intensely involved with the music, the ensemble rendering it, remained seated.

That anomaly happens to be one of those that may be “fairly easily explained,” once one has stopped to think about it. But it can suggest to us that there are likely to be other anomalies which we simply do not notice or which (and this can be far more troubling), once noticed, may be difficult, if not impossible, to explain.

The geometry to which we are accustomed, most of which was formulated more than two thousand years ago, can be said to conceal anomalies that have profound implications for our understanding of things. But these anomalies tend to be so well concealed that we can be quite comfortable in our responses to the usual geometric drill.

Geometry, we are told, is a word derived from the Greek terms geo and metron, terms having to do with earth and measure. And so we are given the definition, “a branch of mathematics that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids.” Further on we are told that geometry is “the study of properties of given elements that remain invariant under specified transformations.” (Webster’s Ninth New Collegiate Dictionary)

It is often said, and may well be true, that geometry was first developed for practical use, in support of such activities as allocating and maintaining parcels of land. For most people, such uses of geometry are sufficient. One can conduct oneself effectively and lead a decent, happy life without the refinements that the accomplished geometrician offers. Similarly, one can make an effective use of, say, language without being a linguist.

But there are other uses for geometry besides the most common and the most practical. A study of it can help us see what it means to know– and to see how the truth about matters can be established, with the identification of vital premises and the development of consequences from various combinations and operations in arguments. One can come thereby to see, perhaps even to “feel,” what understanding means.

Also, we are told, mathematics can be like music, something to be enjoyed for its own sake, for its beauty. This enjoyment can be deepened by an appreciation of the techniques used to produce what is being admired, whether it is playing an instrument or developing a proof. And this can do much for the soul, especially when one is subjected to the discipline of the craft upon which the activity depends.


Among the ancient practitioners of geometry, there stands out the name of Euclid (who flourished about three hundred years before the Christian era began). There had evidently been among the Greeks, for two to three centuries before Euclid, considerable experimentation with geometrical problems. In the distant past there was the somewhat mythical figure of Pythagoras; closer at hand had been such accomplished mathematicians as Plato and Aristotle. Elsewhere in the ancient world– in Egypt and in India, and perhaps as well in China– there had also been serious students of geometry.

Much had been done well before Euclid among the Greeks. He, it seems, undertook to arrange in a proper, even “classic,” order much of what had been discovered by his predecessors. He was somewhat like those Twentieth Century musicians who used the simple folk songs and traditional music of their peoples to compose their quite sophisticated symphonies.

Thus, Euclid seems to have been a gifted compiler who, in his refinements and organization of the work of his predecessors, elevated the subject by displaying a sense of the whole in a way that it may never have been presented before. One can wonder whether Euclid recognized that he did for geometry what Homer evidently did for the stories he had inherited about the Trojan War, the gods, and the great men and women of old.

Euclid has had his champions across millennia. One of the wittiest was Charles Dodgson (whom we know as Lewis Carroll). Dodgson, in his booklength “play,” Euclid and His Modern Rivals (1879), is particularly concerned to preserve the Euclidian mode of organizing the subject, particularly as it is to be taught in the schools of his day. He thereby challenges those who undertake to rearrange the propositions to be considered.

Fundamental alternatives to Euclidian geometry, alternatives associated with the names of Lobachevsky and Riemann, were already being developed. But they do not seem to be critical to Dodgson’s primary concern in his treatise. Nor does he seem to be concerned about those challenges that were to be developed thereafter about the very nature of mathematics, with a greater emphasis eventually placed in modernity upon self-consciousness (and hence upon construction?) than upon discovery.


Euclid had a gift for minimizing the number of definitions, axioms (or common notions), and postulates upon which he depended, even as he was aware of what did have to be postulated rather than proved (and particularly the parallel lines postulate).

He was poetlike in his economy and in his arrangements of elements. The order of the proposition reflects his talent. There are said to be two peaks in his scheme of things: the theorem about the properties of right triangles associated with the name of Pythagoras and the theorems about the five regular solids associated with the name of Plato.

Thus, Euclid tells a coherent story, leaving out superfluous details. Not everything has to be made explicit. Variations upon his themes can be left to others to develop. One can discern in Euclid the intuition that he used in deciding what to use, how, and when so as to present a sufficient and compelling whole.

Dodgson, in his championing of Euclid, has him say in his own defense (Dodgson, [Second Edition, 1885], p. 23):

Now the Propositions relating to Pairs of Lines may be divided into two classes, the first covering the ground occupied by my Axiom 10 (“two straight Lines cannot enclose a space”) and my Propositions I. 16, 17, 27, 28, 31; the second that occupied by my Axiom 12 and Propositions I. 29, 30, 32.

There are problems with this array, to which I will return, but first let us consider what is then said by Dodgson’s “Euclid”:

Those in the first class are logical deductions from Axioms which have never been disputed; the second class has furnished, through all ages, a battlefield for rival mathematicians. That some one of the Propositions in this class must be assumed as an Axiom is agreed on all hands, and each combatant in turn proclaims his own special favourite to be the one axiomatic truth of the series, insisting that all the rest ought to be proved as Theorems.

It is critical to notice here the recognition that if one undertakes to prove this or that axiom, then one has to accept as axiomatic this or that proposition which had formerly been proven by the use of any axiom that is now being converted into a proposition. We are thereby reminded that thinking does depend upon premises which cannot themselves be demonstrated, with the more gifted thinkers being those whose intuition is such as to permit identification of the most plausible and productive (and the fewest?) premises to be accepted and built upon.

We can, upon returning to the Dodgson passage about Axiom 10 (“two straight lines cannot enclose a space”), notice that there are significant variations in the Euclidian manuscripts that we have inherited.  Here, for example, is how Sir Thomas L. Heath deals (in his monumental edition of Euclid) with Dodgson’s Axiom 10 (Heath, The Thirteen Books of Euclid’s Elements [Second Edition, 1925], vol. 1, p. 232):

Other Axioms Introduced After Euclid’s Time

 (9) Two straight lines do not enclose (or contain) a space.

 Proclus (p. 196, 21) mentions this in illustration of the undue multiplication of axioms, and he points out, as an objection to it, that it belongs to the subject matter of geometry, whereas axioms are of a general character, and not peculiar to any one science.  The real objection to the axiom is that it is unnecessary, since the fact which it states is included in the meaning of Postulate 1 [To draw a straight line from any point to any point]. It was no doubt taken from the passage in I, 4, “if… the base BC does not coincide with the base EF, two straight lines will enclose a space: which is impossible”; and we must certainly regard it as an interpolation, notwithstanding that two of the best MSS have it after Postulate 5, and one gives it as Common Notion 9.

(See, also, Heath, vol. 1, pp. 249-50.)  We notice, if only in passing, that Heath himself uses an edition of Euclid which has only five common notions (or axioms):

1.  Things which are equal to the same thing are also equal to one another.

2.  If equals be added to equals, the wholes are equal.

3.  If equals be subtracted from equals, the remainders are equals.

4.  Things which coincide with one another are equal to one another.

5.  The whole is greater than the parts.

We can be reminded by all this not only of Euclid’s sovereign intuition but also of the productive, although sometimes tedious, conversation which has taken place among geometricians across millennia, a conversation in which we are privileged to join here and there.


One can wonder about the uses of intuition generally, something that all of us have had to be quite adept in to get as far as we have in our lives, including in our ability to get to this place today.

Intuition is presented thus in a standard dictionary (Webster’s Ninth New Collegiate Dictionary):

intuition. n. [LL intuition-, intuitio act of contemplation, fr. L intuitus, pp. of intueri to look at, contemplate, fr. In-+tueri to look at] (15c)   1   a: immediate apprehension or cognition   b: knowledge or conviction gained by intuition     c: the power or faculty of attaining to direct knowledge or cognition without evident rational thought and inference    2: quick and ready insight

In geometry, as in mathematics generally, intuition can help one to select definitions, axioms, and postulates.  Intuition can also guide one in the selection of the propositions to be developed, as well as in the order that they should be presented.  (There could always be many variations here, as may be seen in both the Heath and the Dodgson volumes.)

Many of the propositions in Euclid seem, on preliminary inspection, to be plausible enough, consistent as they are with common experience.  What seems plausible at the outset is likely to be susceptible to demonstration.  After all, we do manage our affairs fairly reliably from day to day, relying upon our intuition as to what fits where and how.

But some of the propositions are not readily suggested by intuition.  The most spectacular among these is, of course, the Pythagorean theorem (I, 47), considered by some mathematicians to be “a miracle.” (For example, by Simcha Brudno.) Also remarkable are some of the elaborations of ratios in Book V and thereafter.  Then there are such intriguing propositions as that which establishes that the areas are equal of all triangles constructed on equal bases, no matter where the tip of the third angle is placed on a line parallel to that on which the base is placed (I, 38).

Of course, if something is true, there may well be many ways of proving it.  It is said, for example, that there have been hundreds of proofs developed for the Pythagorean theorem, with its spectacular assertion that the square on the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides.


It has been noticed that Euclid inherited, and then rearranged in a distinctively useful manner, much of what may be seen in his great treatise, Elements.  Among the things inherited by him was the discovery of that remarkable anomaly, the incommensurability epitomized by the elusive relation between the side and the diagonal of a square. This was evidently first discovered, at least among the Greeks, by the Pythagoreans.  (This discovery is used by Socrates in Plato’s Meno.)

One broad definition of “incommensurable” is this: “lacking a basis of comparison in respect to a quality normally subject to comparison.”(1590)  If we could understand why the discovery of incommensurability shocked the Pythagoreans (and others of like mind) as much as it is said to have done– it could once be called a “scandal”– if we could understand this, we would be better able than we might otherwise be to understand ancient thinkers. (I mention, in passing, that the theologically-minded might have recourse here to doctrines about the Mystery pervading the universe.)

The Pythagoreans had evidently regarded the universe as so organized numerically (whatever that may mean) that everything could fit together in a comprehensible whole, with all its parts somehow meshing mathematically.  This they thought to be so for us, however limited our knowledge of the whole might be at any particular time.

The incommensurability which has already been noticed– with respect to the diagonal and side of a square– meant that there could be intimately related lines which could not be measured evenly by the same unit, no matter what unit was used.  Thus, Euclid could say, the side and the diagonal of a square are commensurate in areas, but not in lengths.

This discovery can be particularly disquieting if it should suggest that it is difficult, if not impossible, to connect the changing to the permanent, to connect becoming to being.  This can be especially troubling if we sense that most of what we observe is indeed changing (or becoming) – and if we further sense that we can truly know only the permanent (or being).

In short, incommensurability suggests that there is a fundamental “disconnect” in our apprehensions of things. (The irrationality associated with the incommensurability may be seen in the square root of 2 and in π, or in the relations among aspects of a circle.) How disconcerting such a “disconnect” can be may be suggested by our recollection of how much we depend, even in our everyday affairs, upon a reductio ad absurdum approach in weeding out plausible solutions to our problems. Does the recognition of incommensurability undermine our repudiation of the absurd? (The Heisenberg “uncertainty principle” does not seem, at least to me, an instance primarily of incommensurability, but rather an instance of recognition of the effect that observation itself, if not also the observer, may have on things observed.)


It is difficult, perhaps impossible, for us to grasp how the Pythagoreans and others saw things, and hence precisely what the understanding of the universe was that may have been shaken by the discovery of incommensurability.

The Zeno of the paradoxes, it seems, exploited the difficulties people can have combining, in one “picture,” both being and becoming. But however intriguing as well as exasperating Zeno can be, we do not really doubt that the swift-footed Achilles can overcome the plodding tortoise.

Still, permit me to suggest the destabilizing effect that something like “incommensurability” may have. We all know the story of the blind men who tried, by the use of touch alone, to describe the elephant they encountered. But, it can be said, the diversity of their conjectures is not an instance of incommensurability, but rather simply of unwarranted extrapolations from limited experiences.

Imagine, then, the effort to tunnel a road through a mountain, with crews boring from opposite sides. Consider the distress experienced when the two borings pass each other, rather then meeting. Although this may have some of the “feel” of incommensurability, it still is not it. Rather, it can be understood as merely a matter of faulty surveying. (This may be more like the Heisenberg Uncertainty Principle.)

Something of the “mystery” of incommensurability is suggested by the following exercise. You receive your bank statement, whereupon you add the debits charged against you. But, suppose, when you add the column of debits up, you get one total, when you add it down, you get another. I suspect many of us have had this experience. But, suppose further, this discrepancy persists no matter how often you do this (and even if you use a calculator).

With this fanciful exercise we may be closer to what the ancient geometers felt when they first encountered “incommensurability.” We can get even closer perhaps with the following story: Suppose that in this building, the third floor could be reached only by those who enter the building through the North entrance, while the fourth floor could be reached only by those who enter the building through the South entrance– even though (and here is the challenging part) the second floor, where we are, can be reached from either the North entrance of the South. That is, the paths of the North-door people and the South-door people somehow meet, even though there cannot be a crossover from one stream to the other. (All this could be complicated further by a condition which permits individual exits from the building only alternatively, one by one, North and South, without any evident means of communication between the two exits. We have, with this further complication, borrowed a puzzle from modern Quantum Theory, to which we will return.)

The anomalies that I have been imagining, in an effort to suggest the effect that the discovery of “incommensurability” may once have had, are familiar to us from an old joke. A stranger asks a local whether he had ever been to a certain town. Yes, he had. Well, then, would he give the stranger directions for getting there. The joke is, of course, that the local man, after beginning one way, then another, to give the requested directions, gives up by saying, “You can’t get there from here!” (Another version of this, testifying to how tantalizingly absurd this kind of situation is, may be seen in a column in the New York Times earlier this week [Thomas L. Friedman, “Chicken á la Iraq,” New York Times, March 5, 2003, p. A27]. The columnist wrote, “This reminds me of the joke about the man who gets lost and asks a cop for directions, and the first thing the cop says is, ‘Well, you wouldn’t start from here.’”)

Incommensurability, it seems, may have had something of the absurd about it for the more learned of the Greeks. One could not get from here to there– one could not describe one thing in terms of the other– even within the confines of a right triangle. And this seems to have been, for some of the Greeks, most disquieting, even though Plato could make something of a joke about it in his Meno.


I should immediately add that various physicists and mathematicians I have consulted recently about the phenomenon of incommensurability have been far less troubled by it than the Pythagoreans and others may once have been. Indeed, it is hard to interest them in it even as a curiosity.

For one thing, they have long been accustomed to such anomalies. Besides, they believe they have the means available, in modern mathematics, for figuring around such gaps. Their responses, especially those of the physicists, are related to the way they understand what they are doing. Even though refined measurements are made by them of multitudes of tiny movements, they do not expect to come up with exact figures, but rather with approximations. Indeed, because of the elaborate devices needed for their experiments, these scientists can be as much practical-minded engineers as they are theoretically-minded physicists. Indeed, also, they seem to make much more of process and less of a sovereign comprehensive and comprehensible rationality.

Most of what the physical scientists do today is as much constructing as it is discovering. They rely upon and illustrate thereby the modern tendency to believe that we can only know what we make, not what we find. All this is reflected in the observation by someone quite adept in these matters: “I don’t know what an electron is if I don’t describe it mathematically.” (See Alin Connes, ed., Triangles of Thought [2001], p. 35.) And, it seems, the mathematics now available is such that algebra can do what geometry could not, providing, in effect, an end-around maneuver with respect to such challenges as the incommensurability problem. But consider Bertrand Russell’s caution, “Physics is mathematical not because we know so much about the physical world, but because we know so little: it is only its mathematical properties that we can discover.” (George Anastaplo, The Artist as Thinker [1983], p. 252)

It should be noticed that the much-vaunted precision of modern science has its lighter moments, that precision which substitutes for a grasp of comprehensible principled relations among the things studied. Thus, I heard an astrophysicist report, a couple of days ago, that a critical condition in a collapsing star he was describing had occurred 71 million years BC. I suppose that if I, in turn, wanted to be really precise, I would have to say that that critical condition occurred 71 million, two thousand and three years ago (assuming, of course, that “BC” begins with the start of a millennium).  (Talk, Ryerson Institute, The University of Chicago, March 5, 2003)


However reconciled we may be to our experiences with incommensurables, we do want things to hang together, more or less, yearning as we do for what I have roughly called “comprehensible principled relations.” It may even be a yearning for identification with the Oneness of all things.

Thus, we expected the Pioneer space ship, which has traveled billions of miles since it was launched decades ago, to encounter the same “laws of physics” outside our solar system that it did when being launched. It was, by the way, only recently that signals from that spaceship became too faint to be received here on earth. Evidently, “the rules of the game” that Pioneer encountered have remained the same throughout its generation-long mission.

Incommensurability no longer seems to trouble us, at least in its classic forms. But there are other anomalies which can trouble some of those who are versed in what is going on in modern physics. Thus, there have been findings in Quantum Physics which Albert Einstein himself resisted, protesting that God did not play dice with the universe. The finding which particularly troubled him is suggested by what I have posited about the rigorous sequence of departures from the North and South doors of this building. Even Niels Bohr, who was evidently on the other side from Einstein in this controversy, observed, “Anyone not shocked by quantum theory doesn’t understand it.” (I mention, in passing, something even more shocking, the notion that all the matter in the universe can be considered [if only “mathematically”] to have once (only once?) been collected in a point, to a “singularity,” whatever that may mean.)

Perhaps it is partly because of limitations connected with our mortality that our desire for a solid grasp of things is bound to be frustrated. The “incommensurability” gap in the sought-for complete picture may be somewhat like the perhaps permanent gaps in contemporary science. A century ago, such a gap could be seen in the question whether light was a wave or a particle. (Today, physicists will say that the particle-characteristics of light show up when one kind of probing is done, while the wave-characteristics of light show up when another kind of probing is done.)

In all of these matters, some guidance may be provided by another ancient, Lucretius. For it was he (perhaps like Epicurus before him) who believed that sense could not be made of the foundations of things unless an occasional unpredictable “swerve” could be posited in the ceaseless movements of the eternal atoms.


The tension between being and becoming is something that we, as human beings, do somehow manage to live with, if not even (at times) to benefit from.

People have various ways of coping with what might otherwise disturb them unduly because of such tension. One way is by recourse to dreams, in which irreconcilables are brought together, if only temporarily, but still often with salutary purgative effects thereafter. We can be taught by our dream experiences how difficult it is to say, with certainty, what the now is and what the here is. (I continue to find puzzling the inability of dreamers to recognize explicitly at the moment the troubling absurdities they can encounter in dreams.)

At the root of this difficulty of saying what the here and now are may be the age-old question, “Did the world have a beginning, or is it always?”

If the world had a beginning, it can be argued, it must be simply inexplicable, except to the extent that its Creator guides human beings in their understanding of things. Vital to this kind of guidance is the mystery sometimes associated with the anomalies we encounter. (The “phenomenon” of incommensurability may be one way of noticing the mystery that human beings have invoked n various ways.)

If, on the other hand, the world has always been (and hence will always be?), it can be argued, this suggests that the ways (including the causes) of the world may always be discernible. Even so, there may always still be, as well, the difficulty of grasping in an unchanging way the things that are constantly changing. This difficulty may be evident in our awareness of incommensurability. It may be evident as well in the Socratic insistence upon the importance of being aware of one’s ignorance.

However all this may be, Euclid himself, in his serene competence, steadily worked, by a sustained illustration of what it means to know– he worked to limit the scope of the natural limitations that human beings must confront and yet can learn to live with, at least to some extent and for a while.


I return, in closing, to our concert of last Sunday. The soloist on that occasion– the principal trumpeter with the Chicago Symphony Orchestra– showed us (as Euclid does) what it means to be truly competent in one’s discipline. He was, in his solos, daring and yet surefooted.

The amateur ensemble seemed after his performance, to play markedly better, as if inspired and rededicated. It seemed, that is, that their souls had been touched. Or was it that the audience had been moved, by the gifted soloist, to hear more of what was intended by the ensemble, if not also intended even by the composer himself?

Euclid, too, may thus move us amateurs when we chance to walk with him awhile, listening to his reassurances about what we can know, about the way we can know it, and (perhaps most vital) what we can know about what we can and cannot know– and perhaps even about why all this should be as it seems to be.


This talk was given in the Basic Program First Friday Lecture Series, The University of Chicago, at the Cultural Center, Chicago, Illinois, March 7, 2003.

George Anastaplo is Lecturer in the Liberal Arts, The University of Chicago, Professor of Law, Loyola University of Chicago; and Professor Emeritus of Political Science and of Philosophy, Dominican University. His most recently published book in 2003 was But Not Philosophy: Seven Introductions to Non-Western Thought (Lanham, Maryland: Lexington Books, 2002), which includes a discussion of Stephen Hawking’s A Brief History of Time (1998).

The epigraph is taken from Eugene P. Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Communications on Pure and Applied Physics (New York: Interscience Publishers, Inc., 1960), vol. 1, pp. 1, 11. See, also, Jagdish Mehra, The Collected Works of Eugene Paul Wigner (Berlin: Springer-Verlag, 1992), Part B, Volume VII, pp. 41, 246, 301, 305, 314, 321, 245, 503, 515. “The relation of teachers to students hasn’t changed very much [in my time]. The attitude in physics is different. People become much more specialized. Last week I read Physical Review Abstracts, and every other abstract I could not understand. Perhaps it is partly the jargon– the technical expressions. It hurts me, and I’m afraid it will hurt physics.” Ibid., p. 77 (1973 conversation). See, as well, Hellmut Fritzsche, “Of Things That Are Not,” in John A. Murley, Robert L. Stone, and William T. Braithwaite, eds., Law and Philosophy: The Practice of Theory (Athens, Ohio: Ohio University Press, 1992), vol. I, p. 3.

It should be instructive to recall here an observation by Alexis de Tocqueville in his mid-Nineteenth Century text, The Old Regime and the Revolution (University of Chicago Press, 1998, p. 161):  “It seems as if the French people were like those supposedly elementary particles inside which, the more closely it looks, modern physics keeps finding new particles.  “(Recall also the Bertrand Russell caution recalled in Section VII of the essay.)

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