Seeing and saying
“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
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.
“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
, ªe "river"
, ¤H "man"
, ¤j "big"
) or consisting of a
characteristic part only (e.g. ¤û "ox",
), 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
“One of the oldest and most
complete diagrams from
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
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
lines of the sort
Rather than worrying about the ontological status of
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
Here is a copy of a figure given by Nathaniel Miller in his
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
(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. (1st 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. (2nd 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. (3rd 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. (4th diagram)
(5) He does the same using the center of the second circle. (5th 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 5th 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
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
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
Historians are fond of repeating that
Let us see how
Let us look at Proposition 1 and what
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
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
For axiomatic purposes with modern intentions there would have
been no use for Postulate 3 : wherever
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
(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
(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.
My main point in quoting Mueller’s conclusions has been to
indicate how already in antiquity it was recognized that some of
But how was it that a theorem of
Here is a discussion by David E. Joyce of logical gaps in
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.
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
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.
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):
geometry in general, and
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
However . . . most informal geometric proofs still use
diagrams and more or less follow
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.
Fowler, David. The Mathematics of Plato’s Academy. Clarendon Press. 2nd edn, 1999.
of Greek Mathematics.
Thirteen Books of Euclid’s Elements.
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).
Commentary on the First Book of Euclid’s Elements.
Trans. Glenn R. Morrow.