“Let it have been drawn”: the role of diagrams in geometry

The use of diagrams in geometry raise questions about the place of the physical, the sensory, the human in mathematical reasoning. Multiple sources of evidence speak to how these dilemmas were tackled in antiquity: the linguistics of diagram construction, the state of drawings in the oldest extant manuscripts, commentaries of philosophers, and implicit assumptions in mathematical proofs.

Transcript

Diagrams. What are their role in geometry? Some people like to think that the logic of a geometrical proof doesn’t need the diagram. Mathematics is supposed to be pure and absolute. Diagrams seem connected to the visual, the intuitive, that makes it kind of psychological, and perhaps therefore even subjective.

Certain people don’t like that association one bit, so they try to minimise the role of diagrams. Maybe diagrams are just crutches to help those with weaker minds, whereas a perfect logical reader could follow the proof from the text alone. Some people like to think so. It’s a dogma that fits modern tastes.

But, historically, that interpretation is a pretty poor fit. In some ways, classical geometry appears to have embraced visuality rather than tried to replace it with abstract logic.

There are signs of this attitude in the very language of Greek geometry. The word for proving is the same as the word for constructing: grafein, to draw. To prove something is literally to make it graphic. And a theorem, in ancient Greek, is a diagramma, a diagram. Instead of the Pythagorean Theorem the Greeks would say the Pythagorean Diagram.

Indeed there is always one diagram for each theorem in Greek mathematics. That’s a very rigid rule. In modern mathematics we often find it natural to have several pictures for some proofs, and no pictures at all for many other proofs. Just do what comes natural to explain the particular content. But not the Greeks. One theorem, one picture: this rule was extremely firmly ingrained in their conception of geometry.

And not only in geometry, in fact. Euclid follows this rule slavishly even when he writes about number theory. For example, he proves (Elements VII.30) that if a prime number divides a times b, then it divides either a or b. A very important theorem that is still proved in every modern book on number theory. But no modern book would include a picture for this. It’s just not a visual thing at all, so it makes little sense to draw a picture to go with it.

But Euclid does. The numbers that he is talking about he draws as line segments. The bigger the number the longer the line. But this has little to do with his proof. The proof is not visual. It’s just as abstract as the ones in the modern books. So the diagram doesn’t really do anything. And it’s like that theorem after theorem after theorem: Euclid has these useless diagrams that are basically irrelevant to the content. But he insists on the rule “one theorem, one diagram” even where it doesn’t really seem to serve any purpose.

At least it doesn’t serve any purpose in terms of capturing or visualizing the steps of the proof. Maybe it has other purposes. One purpose could be to signal that number theory is subsumed by geometry. The number 5 really just means a line segment of length 5 units, Euclid seems to be saying with these diagrams. So since Euclid has established the foundations of geometry, and number theory so to speak lives within geometry, then it follows that Euclid has established the foundations for number theory as well. Number theory doesn’t need separate foundations since it is subsumed by geometry. Maybe this is what Euclid is trying to emphasize with his pictures of numbers.

Or maybe Euclid needs pictures because he doesn’t have algebra. A modern proof of theorems like these are very dependent on algebraic notation. If p divides ab, then p divides a or b. In the course of the proof you keep referring to relationships between these number all the time. Suppose p divides ab but not a. Etc., etc. It would be hard to get all that across without algebraic symbols.

If you have a picture you don’t need algebra, because you can point. Instead of the letters a, b, p you have line segments of different lengths that you can point to and say: suppose that one dives that one. You don’t need algebraic symbols or letters, because you are pointing to a picture. The mode of presentation is oral; you have your audience in front of you, and you have drawn the diagram in the sand with a stick, and you point to it as you reason your way through the proof.

You might say: But Euclid does have labels, like A, B, C, etc. So he is referring to entities by letter or label designation, not merely by pointing visually. Well, maybe. But one could argue that that’s not really what Euclid’s A, B, Cs mean.

When Euclid calls things alpha, beta, gamma, it is perhaps inaccurate to translate this as A, B, C. Because it would also mean 1, 2, 3, or first, second, third. The Greeks wrote numbers this way, using the letters of the alphabet. Alpha meant 1, beta meant 2, and so on. So perhaps we shouldn’t think of Euclid’s ABC as algebraic designations. Perhaps it simply means “the first point,” “the second point,” and so on.

This makes it seem a lot closer to the pointing hypothesis. Perhaps the standard way for mathematicians to explain their reasoning was to point to a picture and say “this one,” “that one,” and so on. Then to encode this in writing they used alpha, beta, gamma, to mean “the first one I mentioned,” “the second one I mentioned,” and so on.

If this is right, then the letters in the English version of the Elements are a bit deceptive. They seem more algebraic, more modern, than they really are. From that point of view, diagrams in number theory make some sense.

In fact, in early modern geometry, in the 17th century, you sometimes see people labeling points in diagrams 1, 2, 3 instead of A, B, C. Because they thought this was the right way to translate Greek into Latin. Euclid’s alpha is really a 1, and so on. They were more sensitive to Greek culture back then. Nowadays people have forgotten about that stuff.

Here’s another fun linguistic-cultural perspective on diagrams in Greek geometry. The language in which Euclid describes constructions is quite odd. “Let the circle ABC have been described.” The language of Greek mathematics “makes the author and temporality disappear from a proof,” as one historian has put it. Euclid is not saying that he’s drawing the diagram, and he’s not telling the reader to draw the diagram. He’s just sort of commanding the diagram into existence.

You know the book of Genesis in the Bible: “Let there be light,” God said, and there was light. Euclid uses literally the same kind of construction. It’s exactly the same verb form as in the Ancient Greek version of the Bible. Just as God makes heaven and earth by merely pronouncing that they exist, so Euclid makes geometrical objects appear just ordering them to be. It’s not “I draw” or “you draw” but “let it have been done.”

You could read this as supporting a Platonic conception of mathematics. Euclid is distancing himself from actual drawing. The objects of mathematics just are. They are not something you or I have to make.

But here’s a counter argument to this interpretation. Netz argues that actually Euclid’s grammatical construction merely reflects a purely practical circumstance of the Greek tradition. Namely, that Greek mathematicians had to prepare their diagrams in advance due to technical limitations of the visual media available. Here’s what Nets writes:

“Of the media available to the Greeks none had ease of writing and rewriting. [Standard media were papyri and wax tablets, and, for larger audiences, such as Aristotle’s lectures,] the only practical option was wood painted white. None of these [ways of representing figures] is essentially different from a diagram as it appears in a book. The limitations of the media available suggest the preparation of the diagram prior to the communicative act---a consequence of the inability to erase. This, in fact, is the simple explanation for the use of perfect imperatives [such as] ‘let the point A have been taken’. It reflects nothing more than the fact that, by the time one comes to discuss the diagram, it has already been drawn.”

That’s Netz’s interpretation, and if he’s right then Euclid’s grammatical choice reflects only incidental cultural circumstances and says nothing about philosophical commitments.

So “let it have been done” just means “I did it yesterday”. It doesn’t mean that geometry is set apart from concrete action and that doing has no place in mathematics.

It’s fascinating how the same aspect of the text takes on such a different meaning when cultural context is taken into account, compared to a purely philosophical reading. In fact, let me tell you about another striking aspect of Greek manuscripts which is also like that. Namely, the way diagrams are drawn in manuscripts of Greek geometry.

Diagrams in manuscripts of Greek mathematical treatises are very often very poorly drawn. They are oversimplified and crudely schematic. Ellipses, parabolas, and hyperbolas are represented as pieces of circles and so on. Very poor pictorial accuracy.

Also the simplicity and specificity of the diagrams often obscure important mathematical points. For example, the figure for the Pythagorean Theorem is often drawn in manuscripts with the two legs of the triangle being equal, even though the theorem holds for any right-angle triangle. The diagram thereby gives the misleading impression that the theorem is less general than it really is.

So you might think: aha, clearly the Greeks didn’t care about the diagrams. They are poorly executed, poorly thought through. So diagrams couldn’t have been an important part of geometry then.

Well, not so fast. The diagrams are drawn this way in the manuscripts that exist today. But who wrote these manuscripts, and when? In fact, the oldest manuscript of Euclid’s Elements that exists today is closer to us in time that it is to Euclid. It’s from the Middle Ages. A thousand years ago. That might seem ancient enough, but Euclid lived thirteen hundred years before that.

There was no printing press until the 15th century, so for well over a thousand years the book had to be copied by hand. You had to hire a scribe to write the whole thing out.

Manuscripts are fragile. The Greeks wrote on papyrus. It takes a miracle for a roll of papyrus to survive more than two thousand years. Just think of books from the 19th century, maybe some old book from your grandparents. They are already falling apart, and that was only a hundred years ago. Imagine storing that for twenty times as long. It will fall apart on its own, and that’s not even counting the risk of fires, or floods, or insects, or wars, and so on.

So few documents from Greek times survive to this day, and hardly any of those are mathematical. Only the tiniest little scraps of mathematics from antiquity itself are still around. And they are not enough to say anything about how the Greeks dealt with diagrams.

We only have these later copies. Or better put: a copy of a copy of a copy of a copy and so on. Our oldest manuscript may very well be, who knows, maybe twenty or thirty copying steps away from Euclid’s original.

The state of the diagrams in these manuscripts perhaps says more about the copying and the copyists than it does about Greek geometry. The scribes who copied these manuscripts probably often knew little or no mathematics. They probably had some training as scribes; training in Greek, in writing. Perhaps they mostly copied literary texts or whatever.

So they were probably pretty good at copying text, but not at copying diagrams. It’s pretty straightforward to copy text if you know the language. An A is an A. You can’t really misinterpret it.

Diagrams are a lot more subtle. Often you can only understand what aspects of a diagram are essential by studying the text, the logic of the proof that goes with it. But the scribes would not have done this. They were hired copyists, not research students. They didn’t study the content, they just blindly copied it for a paycheck, like a photocopier.

This is enough to explain why the diagrams are so simplistic. It is natural in such a context of copying that the diagrams gradually degenerate and converge to more simplistic versions. This is the predictable outcome of repeated copying by generations of scribes largely ignorant of mathematical content. For a very simple reason: an ignorant copyist can easily misinterpret a subtle diagram in a simplistic way while going the other way around, toward a more subtle and exact diagram, could only be done by someone with a solid understanding of the mathematical content, who would restore the diagram based on what the text suggests.

For example, in the case of the Pythagorean Theorem, a scribe might get a version of the figure where the two legs look approximately similar and then mistakenly assume that exact equality was intended. He then copies it this way, and specificity is introduced. Now others will keep copying this simplified diagram. No one will restore more generality in the diagram, because that would require revising the figures based on mathematical understanding, which was not the task of copying scribes.

There’s a fun paper on this by Christian Carman in a recent volume of Historia Mathematica. Carman tested this hypothesis with his students. He had them go in a circle and copy a mathematical diagram from one another, like the children’s game Chinese whispers or telephone where you whisper something, then they try to pass it on, and so on. By the time the message has made it full circle it has become something else. It’s the same with diagrams.

You can see also how the specificity aspect emerges from this. The original diagram might show two lines meeting at an angle of, say, 75 degrees. Copying is a bit imperfect, so maybe someone copies it more like 82 degrees. Then the next guy thinks: well, this is probably supposed to be 90 degrees, they just drew it a little bit wrong. So they make it 90 exactly. Then from that point on everybody copies it as 90 degrees. Because exactly 90 degrees looks a lot more intentional than 82. This is why the process almost always goes toward more specificity.

So we cannot conclude anything about ancient philosophy of mathematics from the way diagrams are drawn in the manuscripts. This aspect of the manuscript sources is very likely an artefact of transmission that says nothing about ancient geometry.

So we still don’t know what Euclid thought about diagrams. We know what Plato thought. His opinion was reportedly that mathematicians who “descended to the things of sense” were “corrupters and destroyers of the pure excellence of geometry.” That’s how Plutarch describes Plato’s opinion. So basically an anti-diagram agenda.

But there is no evidence that mathematicians shared these sentiments. On the contrary, the combative way in which this view is presented in the sources clearly show that they were far from a consensus opinion. Plato himself openly puts his view in diametrical contrast with that of the geometers. Here’s what he says in the Republic (VII 527):

“No one with even a little experience of geometry will dispute that this science is entirely the opposite of what is said about it in the accounts of its practitioners. They give ridiculous accounts of it, for they speak like practical men, and all their accounts refer to doing things. They talk of squaring, applying, adding, and the like, whereas the entire subject is pursued for the sake of knowledge [and] for the sake of knowing what always is, not what comes into being and passes away.”

Again, Plutarch reports on the same conflict and makes it crystal clear that Plato’s views on geometrical method was diametrically opposed to that of the leading mathematicians of his day. Here’s what Plutarch says: “Plato himself censured Eudoxus and Archytas and Menaechmus for endeavouring to solve the doubling of the cube by instruments and mechanical constructions.”

So not only is there no evidence that any notable Greek mathematician was a Platonist, but the Platonic sources themselves clearly and openly admit that their view is an ideological extreme that was not widely shared, especially not among mathematicians.

So what’s the alternative? If the mathematicians were not Platonists, what were they? Maybe the didn’t care about philosophy at all. Here’s how Netz puts it:

“Undoubtedly, many mathematicians would simply assume that geometry is about spatial, physical objects, the sort of thing a diagram is. The centrality of the diagram meant that the Greek mathematician would not have to speak up for his ontology. The diagram acted, effectively, as a substitute for ontology. One went directly to diagrams, did the dirty work, and, when asked what the ontology behind it was, one mumbled something about the weather and went back to work.”

That’s what Netz thinks, and it seems consistent with Plato’s rants against the geometers that they would have been disinterested in these questions indeed.

But I think Netz is selling the mathematicians short. I do not believe that Greek mathematicians “simply assumed” these things, and could only “mumble something about the weather” if pressed on the issue. I suspect that, on the contrary, Greek mathematicians had a philosophically sophisticated defence of their ontological stance, based on the operationalist ideas that we discussed before.

Let’s see how this plays out in a concrete mathematical example. From a modern point of view, the right way to do geometry is as a formal axiomatic-deductive system. The Greek tradition has often be interpreted as aspiring toward, but falling short of, this ideal. According to this view, Euclid’s Elements was a brave and admirable attempt at a formal treatment of geometry, especially for its time, but that it contains some fundamental flaws stemming from Euclid’s inability to fully avoid implicit reliance on intuitive and visual assumptions.

Operationalism, by contrast, embraces visual reasoning and keeps abstract logic at arm’s length. This arguably fits the Greek geometrical tradition better than modern formalistic conceptions of geometry. Indeed it is well known that Greek geometry sometimes bases inferences on diagrammatic considerations that are not explicitly formalised.

The most famous example is Proposition 1 of the Elements. In this proposition, the existence of a point of intersection of two circles is tacitly assumed but can arguably not be formally justified from Euclid’s definitions and postulates.

The modern mathematician rejects anything not obtained through logical deduction from formal axioms. The operationalist classical geometer rejects anything not obtained through concretely defined operational procedures. We can formulate the difference between the two points of view in terms of what kind of audience the geometer is trying to convince. If we adopt the modernistic point of view, we can picture the audience of a mathematical proof as a veritable logic-parsing machine. The mathematician feeds in statements, in the form of symbolic strings in a suitable formal language, one by one, and the machine tests whether each statement follow from the one before it based on basic logical inference rules or previously established theorems. This point of view fits very uneasily with classical geometry for a range of reasons, including the use of diagram-based reasoning.

The operationalist point of view, on the other hand, envisions the audience of a mathematical proof differently. A Euclidean proof is addressed at a person with a ruler and compass. This person is every bit as critical as the logic machine of the modernists. He is hell-bent on trying to argue against us at every stage. But our strategy for convincing him to nevertheless concede the truth of our theorems is not by appeals to formal logical inferences. Instead we make him draw things. We build our results up from simple operations with ruler and compasses. In this way we put our critic in a difficult position. He is forced to either agree with us, or to deny a very specific, concrete claim about a very specific, concrete figure that he himself has drawn.

For instance, what is the person with the ruler and compass supposed to say regarding the intersection of the circles in Proposition 1 of the Elements? He just drew the two circles himself on a piece of paper. It would be ridiculous for him to claim that there is no justification for the assumption that they intersect. They clearly intersect right there in front of his eyes, and it was he himself who drew it using tools whose validity he had admitted.

Since operationalism gives absolute primacy to the concretely constructed diagram, the sceptic has no other foothold from which to reject the proof. The logic machine of the modernist paradigm would catch the gap in Proposition 1 at once, and shoot down our proof. But operationalist mathematics is not susceptible to that kind of critique. Geometrical proofs are claims about what happens when you carry out concrete constructions. Constructed diagrams is all there is, so the only way to question a geometrical proof is to question what it says about a concretely constructed diagram. The sceptic cannot hide behind sophistical logic and vague generalities, but is forced to either concede the validity of the proof or deny something so obvious that he will look ridiculous.

The conception of a proof as addressing a sceptic fits the Greek context well. It’s just like a Socratic dialogue. You extracting concessions from a determined opponent in incremental steps. Just as Socrates does in the dialogues of Plato. And just as disputants would aim to do in a stage debate of the kind the Greeks loved.

I think one could argue that the diagrammatic inferences Euclid permits are precisely those that such a sceptic, who has drawn the diagram himself, could not reasonably doubt. This fits well with Kenneth Manders’ observation that Euclid permits diagrammatic inferences only of properties of the diagram that are invariant under minor variations or imperfections in the drawing process.

For example, in Proposition 1 of the Elements, the equality of the legs of the triangle can of course not be established merely by visual inspection of the diagram; rather, these equalities have to be derived from postulates and definitions, as do all exact properties of diagrams in Euclid’s geometry. Indeed, a sceptic could very well question whether such properties hold, despite having just constructed the diagram himself. The equality of the legs is not immediate from the diagram in and of itself, but only follows when we remind ourselves that we used the same radius for both circles and so on. You could draw the diagram without keeping such things in mind. You could not, however, draw the diagram without directly experiencing one circle cutting unequivocally right through the other one.

Operationalism relies on diagrammatic reasoning only in this restricted sense. It attributes foundational status to diagrams in certain respects, but of course it does not go so far as to say that the truth of propositions or veracity of solutions to problems can be verified merely by measurements in a diagram. Of course such things have to be established by rigorous demonstration, which is obviously the main preoccupation of Greek mathematical sources.

What Plato says about inferring geometric truths from diagrams remains true also for operationalists. This is a quote from the Republic (VII 529): “If someone experienced in geometry were to come upon [diagrams] very carefully drawn and worked out, he’d consider them to be very finely executed, but he’d think it ridiculous to examine them seriously in order to find the truth in them about the equal, the double, or any other ratio.”

Indeed, exact properties such as ratios cannot be inferred from diagrams, no matter how carefully drawn, just as Plato says. But the operationalist enterprise does not rely on such epistemic overreach. Instead, its use of diagrammatic reasoning is much more restrictive and limited to essentially qualitative or topological or inexact inferences from diagrams.

So operationalism makes sense of Euclidean practice with regard to diagrammatic reasoning. It eliminates the need to attribute to Euclid a big logical blunder in his very first proof, or the need to denigrate the more visual aspects of Euclid’s reasoning as lowly intuition and an imperfect form of mathematics. Instead it articulates a philosophy of mathematics that incorporates this aspect of Euclidean mathematical practice into a coherent and purposeful whole.

So that’s one way to argue that the so-called logical gap in Euclid’s Proposition 1 is not a gap at all. It’s only a gap if you want geometry to be completely reduced to formal logic. From the point of view of operationalism it is not a gap.

There are other ways to try to save Euclid’s proof. More conservative ways. If you know some modern mathematics it’s a fun game to play to try to read all kinds of things into Euclid’s definitions.

For instance, Euclid’s definition of a circle specifies that it is contained by a single curve, and that it has an inside, and by implication also an outside. In Proposition 1, when you draw the second circle, it is evident that the second circle will have some points inside and some points outside the first circle. So you could argue that, topologically, there’s no way a continuous curve could go from the inside of a closed curve to the outside of it without crossing it. Therefore the existence of the intersection point can be regarded as implied by Euclid’s definition rather than a logical gap.

If you are a modern mathematician you might reply: well, that depends on the underlying field! The argument works for the plane of real numbers, but not if the underlying field is that of rational numbers only. Then indeed the intersection does not exist. So Euclid would have to specify the underlying field before the argument based on inside and outside could work.

Interestingly, one could argue that Euclid sort of does this actually. Because he says in Definition 3 that “the extremities of lines are points.” Now if you wear your modern glasses, you can read this as saying that lines contain their limit points. So the Euclidean plane is a complete metric space. So that rules out the argument based on the rational numbers.

Well, if you know modern mathematics it’s fun to think along these lines, but for my part I vote for the operationalisation reading of Euclid as the more historically plausible way of saving Euclid’s proof of Proposition 1.

Here’s an objection though to the operationalist interpretation. The so-called generality problem. Geometrical theorems are about entire classes of objects---infinite sets of them. For instance, the angle sum of all triangles. Yet all geometrical proofs in the classical tradition are always illustrated with, and reason based on, one particular diagram. The standard way to defend geometrical reasoning against this challenge is to say that geometrical proofs concern only properties that hold generally and do not rely on incidental properties that hold only for the particular diagram. This view was expressed already by Proclus.

Operationalism suggests a very different way of dealing with the generality problem: it denies the premiss that there is such a thing as “all triangles” in the first place. Before you have put your pen on the paper, there is no geometry. There are no lines, no circles, no triangles. We do not make the metaphysical assumption, as the modernists do, that there is some preexisting universe of these things “out there” about which geometry looks for universal truths.

From this point of view, the “problem” of generality ceases to exist. The theorem is not: there is an infinitude of triangles and all of those have angle sum 180 degrees. Instead it is: any triangle has angle sum 180 degrees. Which really means: if you put your ruler down and draw a line segment, then another one, then another one, then the angles of that one triangle has angle sum 180 degrees. The theorem has no other meaning than that. And the proof is not a logical schema talking about an infinite class of objects. Rather, it is a set of instructions for the sceptic to carry out that will convince him, regardless of which triangle he started with, that the theorem is true for that triangle. It is precisely the strength of the insistence on constructions to reduce everything from the abstract to the concrete in this way. We only talk about what we can see and draw and put on the table right in front of us. To do otherwise would be to engage in empty metaphysics, according to operationalism.

Greek geometry is remarkably consistent with such a reading. Indeed, as Netz has observed, Greek mathematical texts never explicitly claim generality beyond the concrete proof based on a particular diagram.

From a modern point of view, any reliance on diagrams in mathematics is inherently problematic, since mathematics is in essence independent of diagrams. On this view, diagrams are merely a secondary representation of mathematics, and furthermore one contaminated by intuition and other limitations. How, then, can diagrammatically based reasoning be a legitimate way of doing mathematics? That is, how could we ever be sure that what is true of diagrams is true of the “actual” content of mathematics? Operationalism does not answer the question but rejects it. There is nothing more “actual” than the diagram.

So the generality problem is dissolved since operationalism rejects the Platonist ontology of mathematics on which it is based. Nothing exists except what the geometer has constructed.

This view re-emerged in modern mathematics for reasons independent of classical geometry. Here’s how famous Dutch intuitionist Brouwer puts it it his dissertation:

“Wheresoever in logic the word all or every is used, this word, in order to make sense, tacitly involves the restriction: insofar as belonging to a mathematical structure which is supposed to be constructed beforehand.”

There is no “all triangles.” There is only “all the triangles you have made.”

To be sure, many who are concerned about the generality problem will feel that operationalism “solves” the problem only by introducing further problems of equal or greater magnitude. For one thing, operationalism implies that “the very nature of meaning itself makes it impossible to get away from the human reference point,” as Bridgman puts it, since nothing exists or has meaning in geometry except through human agency. But operationalism denies that this is a problem, as Platonists would have it.

Regarding the generality problem more specifically, a modern mind may feel that the operationalist solution merely shifts the problem one step over. Even the operationalist is committed to a form of generality, in the sense that the proof of, say, the angle sum theorem must always work for any given triangle. Isn’t the operationalist mathematician still obligated to somehow justify that the proof has this form of generality, which is essentially the original generality problem in slightly different guise?

It is of course true that the proof is intended to be general in this sense, but officially the operationalist mathematician does not need to be committed to having proved that it is. The operationalist mathematician can simply say: I assert that such-and-such a construction will always have such-and-such an outcome; if you want to prove me wrong, feel free to try to come up with a counterexample.

Of course, psychologically the mathematician presenting a proof must be convinced that it will always work, for if a counterexample would be forthcoming he would be exposed as a fool. But this can be left to the discretion of the mathematician’s intuition. Internally, operationalist mathematicians are of course concerned with this kind of generality. But externally, as a reply to sceptical and philosophical challenges to the epistemological status of mathematics, there is no need for them to saddle themselves with the burden of claiming that their proofs themselves have inherent characteristics that strictly ensure such generality. Instead they can restrict themselves to presenting the proof as a challenge to any sceptic: apply these construction and inference steps to any one figure that fulfils the conditions stated, and you will find that you cannot credibly doubt the validity of any step, and hence you will become convinced that the proposition holds for that figure. It is possible, for the operationalist, to maintain that this is what a proof is.

One may well feel that this restrictive view of what a proof is sells mathematics short and fails to account for the nature and status of mathematical knowledge. However that may be, the fact remains that operationalism makes it possible to take such a stance. The restrictive view of the nature of proofs fits naturally with the operationalist conception of mathematical content and meaning, while it is incompatible with a Platonist conception of the nature of a mathematical theorem.

The restrictive view is a scorched-earth defensive position that can be useful when under philosophical attack. Saying that this is the only sense of mathematics one is willing to defend against sceptical attack does not preclude one from holding more expansive, Platonist beliefs in private. But it is a powerful way of cutting off lines of philosophical attack without changing the practice of mathematics substantially.

So, in conclusion, I have argued that Greek mathematicians were prepared to base geometry on actual diagrams. Despite their physicality, despite their links to human action and perception. Greek mathematics went against modern tastes in this respect.

One could argue against this by pointing to the crudeness of diagrams in surviving manuscripts, or the strangely passive language that Euclid and others used to describe constructions of diagrams. But we have seen that those things can better be explained as the result of cultural context rather than philosophy of mathematics.

The modern view that geometry should be studied through abstract reasoning not dependent on the visual and the physical also has ancient support in Plato’s philosophy. But Plato was not a mathematician. In the words of Francis Bacon, when “human learning suffered shipwreck [at the end of classical antiquity], the systems of Aristotle and Plato, like planks of lighter and less solid material, floated on the waves of time and were preserved,” while more mathematically advanced works were lost forever.

To understand ancient mathematics we must look beyond the surface. We must look beyond loudmouths like Plato. We must seek instead the assumptions conveyed implicitly in the way the mathematicians wrote their proofs. Based on this kind of evidence, a diagram-based mathematical practice can be plausibly reconstructed.