Why construct?

Euclid spends a lot of time in the Elements constructing figures with his ubiquitous ruler and compass. Why did he think this was important? Why did he think this was better than a geometry that has only theorems and no constructions? In fact, constructions protect geometry from foundational problems to which it would otherwise be susceptible, such as inconsistencies, hidden assumptions, verbal logic fallacies, and diagrammatic fallacies.

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Ancient Greek geometers were obsessed with constructions. Why?

Euclid’s Elements spends almost as much time showing how to draw geometrical figures as it does proving theorems about them. In fact, it seems Euclid thought drawing was a prerequisite for proving. For instance, the first theorem involving squares is the Pythagorean Theorem. In the proposition right before it, Euclid explains in detail how to construct a square by ruler and compass. The same goes for every other geometrical entity ever used in the Elements: first you construct it, and only then can you say anything about it. Without constructions there can be no geometry, Euclid seems to be saying.

And not only Euclid. All the best Greek geometers had their own signature constructions. Three famous construction problems dominated higher geometry for centuries: doubling the cube, trisecting the angle, squaring the circle. The long list of mathematicians who contributed their own distinctive solutions to these problems is a who’s who of everybody who was anybody in ancient geometry.

What fundamental motivations—what philosophy—drove ancient Greek geometers to this fixation with constructions? Why did Greek mathematicians think it was a good idea to spend hundreds of years trying to make an angle the third of another, or a cube twice the volume of another, in dozens of different ways? Why did they so stubbornly bang their heads against the same wall for century upon century? What sin could be so grave that they imposed on themselves such a Sisyphean task?

Why indeed make things at all? And why do so only sometimes? Why meticulously articulate recipes for transferring line segments by ruler and compass, only to then suddenly move entire triangles like it’s nobody’s business in the very next proposition, as Euclid seemingly does? (When he uses superposition to prove triangle congruence in Proposition 4.)

Euclid knew what he was doing, in my opinion. Constructions were a deliberate strategy to guard against fundamental threats to the reliability and rigour of geometry. If our house is built on rotten pillars it’s only a matter of time before it comes crashing down.

Some ancients critics of geometry indeed identified some ominous cracks in its foundations. Remember, the quarrelsome Greeks, they questioned everything with zeal. Some people tried to take down geometry. They were determined to show that it was pseudo-knowledge that was by no means as certain and exact as the mathematicians claimed.

Geometers had to deal with such external attacks. More so than in other centuries and cultures, mathematicians in Ancient Greece were under constant critical-philosophical attack. They had to formulate a defense. And they did, in my opinion. This is where their obsession with constructions comes from.

So what were these philosophical attacks, to which constructions were the answer? There are a number of them. Here’s one:

False diagram fallacies. If you draw diagrams that are slightly off, you can easily fool yourself when doing geometry. There’s a famous example for instance where one proves that any triangle is isosceles. The conclusion is obviously absurd. But it is “proved” in a way that looks just like any other proof in Euclid.

The false proof is made possible by a subtle error in the diagram. The proof involves bisecting one of the angles of the triangle, and then raising the perpendicular bisector of the opposite side. These two lines meet somewhere. That’s drawn in a plausible-looking way in the diagram. The proof then proceeds based on the diagram, just as Euclid does in his proofs.

But the way the diagram was drawn was erroneous. The two lines were drawn as meeting inside the triangle when in fact their true intersection would be outside the triangle. This is a subtle issue that is easy to miss. So when we reasoned based on the diagram we made some hidden assumptions that we were hardly even aware of.

The rest of the proof was typical Euclid-style stuff. So this example shows that a small and subtle mistake in the way we drew a diagram can destroy the certainty of geometrical reasoning. All the other steps of the proof were very carefully justified, just as Euclid always justifies each of his steps. But that was all for nothing since the subtle error in the diagram poisoned the well and destroyed the whole thing.

The Greeks were evidently well aware of this type of problem. Plato mentions it explicitly: “geometrical diagrams have often a slight and invisible flaw in the first part of the process, and are consistently mistaken in the long deductions which follow.” (Cratylus, 436d) Plato is exactly right. “A slight and invisible flaw” at the outset is enough to ruin “the long deductions which follow.”

We even know for a fact that Euclid himself wrote a (now lost) treatise on fallacies in geometry which is likely to have dealt with these kinds of issues. So the Greeks were clearly well aware of this threat to geometrical certainty. What did they conclude from this?

Today, the issue of diagram fallacies is taken to show how dangerous it is to rely on visual and intuitive assumptions. The solution is to purge geometry of any kind of reasoning based on diagrams. In the late 19th century this view was expressed forcefully by leading geometers, and it has remained the mainstream view ever since. “A theorem is only proved when the proof is completely independent of the diagram,” as Hilbert said for example. Instead of relying on pictures, geometry must be made to proceed through purely logical deduction.

But this is not the only possible diagnosis and treatment of the problem with the erroneous proof. Another point of view is to say: the problem is not that the proof relied too much on diagrammatic reasoning, but that it did so too little. The problem is not that the proof is insufficiently divorced from visual considerations, but that it is too divorced from them. The example doesn’t show that diagrams are dangerous even if they are just schematic accompaniments to otherwise logically solid proofs, but rather that diagrams are dangerous when they are merely treated as such.

The solution is not to place less emphasis on diagrams, but more. That is, to demand diagrams to be not merely schematically sketched but in fact precisely constructed according to the most exacting standards and rigorous proofs that these constructions accomplish the configurations in question. This would indeed prevent errors of this type from occurring. No one adhering to this mode of doing geometry would ever find themselves reasoning about false diagrams like the one in the above example.

This diagnosis of the source of error in the false proof above leads immediately to the conclusion that precise constructions of angle bisectors, bisectors of segments, and perpendicular lines are foundationally very important, and that no proof must ever be formulated without constructive recipes for all entities occurring in it having been established beforehand.

And this is exactly what we find in Euclid’s Elements. Without fail, Euclid always meticulously shows how to construct all entities involved in all of his propositions. And all the constructions needed to ensure that we end up with the correct figure rather than the deceptive one in the above example are carefully spelled out as core propositions right at the heart of the Elements: how to bisect an angle (Proposition 9), how to bisect a line segment (Proposition 10), how to raise a perpendicular from a point on a line (Proposition 11), how to drop a perpendicular from a point to a line (Proposition 12).

In other words, right off the bat of the Elements, Euclid carefully explicates precisely the tools needed to solve the false diagram problem mentioned by Plato. Coincidence? I don’t think so. Euclid knows the problem. Euclid knows how to solve it. That’s why he’s obsessed with constructions.

Or rather, it’s one of the reasons. There are other, equally compelling grounds to base one’s geometry on constructions.

Constructions are related to existence issues. It is impossible to conduct a serious axiomatic study of geometry without paying attention to existence issues. For example, do squares exist? Existence is separate from definition. Euclid defines what a square means in his definitions. But that doesn’t mean there are any. You could also define what “unicorn” means. That doesn’t mean unicorns exist.

You might think it’s obvious: of course there are squares, any child realizes that, you can see it with your own eyes. But it’s more subtle than you might think. In fact the existence of squares implies the parallel postulate. Wallis showed this in the 17th century. You could replace the Euclid’s parallel postulate with the assumption that you could make a square on a given line segment. Then you could prove Euclid’s parallel postulate and all his other theorems based on that assumption. So it’s no small matter to assume that there are squares.

Therefore, any investigation that aims to elucidate the fundamental assumptions of geometry cannot treat any object whose existence has not first been either proved or explicitly postulated. To do otherwise would be to render the entire enterprise of axiomatic geometry useless and moot, since it would open a back door through which any number of hidden assumptions can creep in. The point of an existence proof for squares, then, would not so much be to establish that there is such a thing as squares, but to ensure that any foundational assumptions involved in supposing the existence of squares have been systematically accounted for.

Another example of this type occurs in Legendre’s attempt to prove by contradiction, using only the first four postulates of Euclid, that the angle sum of a triangle cannot be less than 180 degrees. His proof implicitly assumes that given two intersecting lines, and a point not on those lines, it is possible to draw a line through that point that intersects the two given lines. This assumption does not hold in hyperbolic geometry. Therefore Legendre’s attempted proof is worthless, since the contradiction did not come from the assumption he intended to refute, but from an innocent-seeming existence assumption introduced along the way in his argument.

This shows once again the danger of letting even the most harmless-looking existence or construction assumptions proliferate without explicit control. Inconsistencies can arise from even the most inconspicuous of assumptions. The moral of the story is that the mathematician must stick to a minimalistic set of stringently controlled construction principles, whose consistency should be as unquestionable as possible.

Issues of this nature were recognised in antiquity. Quite possibly, even the specific issue of Legendre’s assumption may have been investigated in works that are no longer extant, such as the lost treatise On Parallel Lines by Archimedes. At any rate, closely related issues emerge explicitly in the treatments of parallels by Simplicius and Al Jawhari.

On a more conceptual level, Aristotle pinpoints the same type of fallacy in the work of some “who suppose that they are constructing parallel straight lines: for they fail to see that they are assuming facts which it is impossible to demonstrate unless parallels exist. So it turns out that those who reason thus merely say that a particular thing is, if it is.” (Prior Analytics, 65a) Indeed. Aristotle is right. Circular assumptions are easy to make, especially with respect to existence issues and subtle foundational questions in the theory of parallels.

Aristotle draws the obvious conclusion that existence issues must be controlled by either explicit postulates or existence proofs. “What is denoted by the first [terms] is assumed; but, as regards their existence, this must be assumed for the principles but proved for the rest,” says Aristotle (Posterior Analytics, 76a). For example, “what a triangle is, the geometer assumes, but that it exists he proves.” That’s Aristotle again (Posterior Analytics, 92b). He’s quite right.

Constructions are a way to ensure existence. Euclid’s first proposition proves that equilateral triangles exist. His 46th proposition proves that squares exist. And so on.

But there’s much more to constructions than merely establishing existence. Construction also establish consistency. That is, it shows that objects are not self-contradictory.

For example, suppose I add to Euclid’s Elements the definition: “A superright triangle is a triangle each of whose angles is a right angle.” Then its angle sum is three right angles by definition, but also two right angles according to a theorem of Euclid’s. So two right angles equal three right angles—an obvious contradiction.

The definition of a superright triangle is disturbingly similar to that of an equilateral or isosceles triangle, and applying Euclid’s theorem to it sounds just like the kind of thing we do in geometry all the time. So it casts doubt on the entire enterprise of geometry. How do we know that the propositions of the Elements are not one or two steps away from leading to contradictions? The geometers must reply with some definitive criterion that explains why none of their theorems are susceptible to this kind of error.

In a way it is clear what the problem is. There are no superright triangles. Hence one can consider the problem solved by ensuring the existence of the objects one speaks of. One way of accomplishing this would be to say: Only constructive definitions, that imply a recipe for making the object defined, are permitted in mathematics. This is clearly not the path taken by Euclid, however. For instance, Euclid defines a square at the outset but only shows how to produce one much later, in Proposition 46, based on substantial previous results.

Another strategy would be to demand that we cannot make a propositional statement about a particular class of objects unless we have first shown beforehand that the class in question is nonempty. Thus the types of inferences made in the false argument are only warranted if supported by suitable existence proofs, and that is why theorems about triangles cannot be applied to superright triangles, but can be applied to equilateral and isosceles triangles, which Euclid indeed proves exist by means of constructions.

But existence is not the only aspect that should be emphasised here. Another important lesson from the superright triangle example is the danger of defining objects through multiple conditions. A superright triangle is defined as: having three sides; having one right angle; another right angle; ant yet another right angle. The first two conditions were fine. It was taking all of them together that was impossible. The more conditions you add, the greater the risk of ending up with an inconsistency.

Another example of this is to say: Let ABC be a triangle such that: one angle is a right angle; the sides next to the right angle have lengths 4 and 7; the third side has length 9. Actually I have taken this example from a 16th-century geometry textbook. But the book messed up. Some of these conditions would have been fine on their own, but all of them taken together are inconsistent. You cannot make a right triangle with those side lengths. Those numbers contradict the Pythagorean Theorem.

Hence defining or introducing an object through a list of specifications of its properties is unacceptable. Doing so would leave the door wide open for possible inconsistency to enter mathematics, and hence ruin the claim to certainty of mathematical reasoning.

A rigorous mathematical theory needs a systematic guarantee that such errors cannot be committed. Constructions are a way to provide such a guarantee. Instead of introducing objects by a list of properties, construction builds it up step by step. Thus properties can no longer be ascribed to an object merely by decree. Rather they must be introduced by a rigorously controlled stepwise process. Each step in this process involves the application of a construction postulate or a demonstrated construction proposition or theorem, which means that assumptions and conditions of validity are carefully monitored and reduced to a few axiomatic principles.

One could argue that the challenge posed by the superright triangle fallacy is not convincingly solved by the insistence on existence proofs. This solution diagnoses the problem as effectively just another variation on the existence issue discussed before, which Aristotle mentioned and so on. But one can readily see it as pointing to a deeper problem. It arguably casts doubt on the credibility of verbal logic altogether. While it is clear that being more careful about existence issues would eliminate the particular problem of the superright triangle, it is not clear whether this is the only problem with relying on verbal logic.

We know for a fact that logical paradoxes and fallacies figured prominently in Greek thought in the classical era. Some of these are clearly relevant to mathematics, such as Zeno’s paradoxes of motion. But there are others.

The liar paradox arguably shows that natural-language propositional logic is incoherent. It shows that verbal logic allows propositional statements to be formulated that are inherently contradictory. “This statement is false” or “I am lying” are examples of such statements. If the statement is true, it follows that it is false. And if the statement is false, then it follow that it is true. So there is no way of assigning a truth value to such a statement without ending up with a contradiction. This kind of thing clearly poses an issue for a logic-based conception of mathematics, not least in connection with proofs by contradiction.

Another example of a paradox discussed in ancient times was that of the horn: What you have not lost, you have; but you have not lost horns; therefore, you have horns. Here again the blind, mechanical application of logical inferences in a quasi-algebraic manner leads to an absurd conclusion. As with the superright triangle fallacy, it is possible to attribute the problem to some specific cause: in this case not so much an existence issue as a certain misleading ambiguity in the first premiss. Furthermore, the fallacy may be regarded as “obvious.” But trying to defuse the paradox in these ways does not solve the core issue exposed by the paradox, namely that “blind” logic, in and of itself, seems to lead to erroneous conclusions.

This multitude of logical paradoxes arguably validates the suspicion that when we supplemented verbal logic with existence proofs we had perhaps not gotten to the bottom of all its problems yet.

It would not have been out of character for the Greeks to have taken radical steps to protect themselves from logical fallacies and paradoxes. The situation may be somewhat comparable to the discovery that the square root of 2 is irrational. This would have been in the very early days of Greek geometry and we don’t know much about it for certain. But a development more or less along the following lines has often been imagined.

In the beginning, the Greeks seem to have blissfully assumed that arithmetic and geometry would always be in natural harmony. The square root of 2 discovery ruined this by showing that natural geometric entities such as the diagonal of a unit square could not be represented by “numbers” (that is to say, rational numbers). The Greeks had burned their fingers and would not make that mistake twice: their response was an extreme foundational purge that eradicated any foundational status of arithmetic and then some. From that moment on, everything is at bottom geometry. Even where a modern mind wishes to see algebra, Euclid and the other Greek mathematicians insist on geometrical formulations with a pedantry bordering on paranoia.

The historical evidence, or absence thereof, of such a “square root of 2 crisis” is a much-debated issue among historians. But the basic point—that Greek mathematicians may very well have gone to great lengths to protect themselves from foundational objections—is plausible. It was a time when the foundations of any subject was constantly under attack from rival philosophers, and people were ready to go to the ends of the earth to rebut such charges.

Extreme action in response to paradoxes that call the bedrock of mathematics into question is a quite plausible scenario. And arguably one with a strong historical precedence in the form of the square root of 2 case. It is squarely within the realm of historical possibility that such a context may have led to the radical proposal of denying any reliance on abstract logic in mathematics and instead founding all of geometry on concretely constructed figures.

So verbal logic is dangerous. It invites paradoxes, like the liar paradox and the paradox of the horns. It has no guard against reasoning about inconsistent objects such as superright triangles. Making constructions, rather than logic, the foundation of geometry solves these problems.

It is suggestive in this connection that Euclid’s proofs are all “purely quantifier free.” That is to say, they never make assertions of the form “there exists” or “for all.” From the point of view of modern mathematics, which is a logic-oriented mathematics, those phrases are fundamental and are used all the time. These phrases are so commonly used that mathematicians do not even have the patience to spell out these two-syllable expressions every time they use them. So they have made up special symbols to abbreviate them. A backwards E and an upside-down A.

I mentioned Hilbert, a leading pioneer of the modernist movement, around 1900. I mentioned that Hilbert wanted to translate all visual information or any inferences based on diagrams into purely logical form. That leads precisely to formulations with those favorite phrases of the mathematician: “these exists” so-and-so; “for all” objects of such-and-such a class, this and that property holds.

That’s the language of modern mathematics. But not of Euclid. He never uses that manner of speaking which is so natural to logic-oriented mathematics. This fits very well with the hypothesis that Greek mathematicians vehemently rejected the notion that their reasoning was based on syllogistic or propositional logic. Instead they relied on constructions. In fact, they did so in part precisely because logic is so problematic.

Or so I have claimed. As usual we cannot know for certain. Euclid didn’t say why he’s so obsessed with constructions. But I think this is a good way of making sense of it.

Ok, so we have seen a number of specific considerations that point toward the foundational importance of constructions. Let’s bring these ideas together into a single philosophy. In fact, there is such a philosophy. I call it operationalism.

Operationalism is a term most closely associated with a 20th-century movement in philosophy of science that grew out of relativity theory and quantum mechanics. But several of its key ideas are much older and more universal. I propose that this rich tradition in philosophy of science was largely foreshadowed in Greek philosophy of geometry. The key commitments and motivations of modern operationalism and related traditions could very plausibly have been precisely mirrored in Greek geometrical thought.

One of the leading modern defenders of operationalism is the Harvard physicist Percy Bridgman, a Nobel Prize winner. Here’s how he formulates the core principle of operationalism: “we mean by any concept nothing more than a set of operations; the concept is synonymous with the corresponding set of operations.”

Bridgman had physical concepts in mind, but it works equally well for geometry. For example, what does “triangle” mean? The operationalist answer is that “triangle” means: the figure obtained when drawing three intersecting lines with a ruler. This diagram is not a drawing of a triangle, or a physical instantiation of the formal concept of a triangle, or in some other way subordinated to or derived from some purer concept of triangle. No, a diagram resulting from these operations simply is what a triangle is. The act of drawing itself is the root meaning of the concept of “triangle.” The act of drawing is the foundational bedrock on which any claim about triangles ultimately rests. When Euclid says “let ABC be a triangle,” he strictly speaking simply means: draw one line, then another, then another (making three points of intersection).

In the same way, what is a line? Take a piece of string and pull the ends; that’s a straight line. What is a circle? Take a piece of string and hold one end fixed and move the other end while keeping the string taut; that’s a circle. What does it mean for two things to be equal? Put one on top of the other; if they align, and neither sticks out beyond the other, then they are equal. As Euclid says in Common Notion 4. What is a right angle? Cut the space on one side of a line into two equal pieces; that’s a right angle. As Euclid says in Definition 10. And so on.

Every statement Euclid makes in the Elements can be read as a statement about operations or the outcome of operations. Not every geometry book is like that. Far from it. Most geometry treatises of later eras do not allow themselves to be interpreted in operationalist terms.

Consider for example the parallel postulate, Postulate 5. This postulate is very convoluted and hard to read the way Euclid states it. It goes like this: “if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.”

Basically, this says that if two straight lines are heading toward each other then they meet. And the postulate includes a criterion for checking whether two lines are heading toward each other or not. Namely: cut across them with a third line, and check the angles it makes. Less than 180 degrees on one sides means the lines are inclines toward each other, so they will eventually meet on that side, says the postulate.

In fact it is clear already from the Elements itself that Euclid could have used a simpler, equivalent statement in place of this complicated thing. Such as: given any line and any point not on this line, there is no more than one parallel to the line through that point.

Why did Euclid opt for his much more convoluted formulation of the postulate? From the point of view of modern mathematics, his choice is strange, as witnessed by the majority of more modern treatments that much prefer the formulation in terms of existence of parallels. But from an operationalist point of view Euclid’s choice makes perfect sense. Euclid’s version of the postulate is purely about operations: if you draw two lines, and discover by an operational test that they stand in such-and-such a relation, then if you extend them such-and-such a thing will happen. Everything is formulated in terms of actions that the geometer performs. The existence formulation, on the other hand, is incompatible with operationalist principles. It only makes sense in some kind of preformationist framework that assumes that all the objects of geometry are already “out there,” independently of any geometer.

Similarly, Euclid doesn’t say “there are infinitely many prime numbers” but rather: if you have a list of prime numbers, you can make a larger list of prime numbers (Elements, Book IX, Proposition 20). This achieves the same thing but without needlessly entangling itself with the quasi-metaphysical assumption that “the set of all prime numbers” is a preexisting entity whose properties we are proving theorems about. There is no need for mathematics to make assumptions of that type. Doing so would only invite attacks from philosophical sceptics.

Operationalism avoids the dubious ontological assumption that the totality of all objects of geometry are somehow already at our disposal. The modern formulation of the parallel postulate assumes that mathematics can, so to speak, survey the totality of all lines through a particular point and make proclamations about this infinite set. Operationalism doesn’t make such an assumption.

We can also read for example the Pythagorean Theorem this way. Again, this theorem does not say that every element of the infinite set of all right-angle triangles has a particular property. Rather, operationally speaking, it says: if you have drawn a right-angled triangle, and if you then draw the square on each of the sides, then the areas of those particular squares are related in such-and-such a way. Until you have drawn a right-angled triangle, the theorem can be said to have no content.

Operationalism cuts away a huge amount of philosophical baggage, yet still allows us to retain virtually all mathematical content. Insofar as mathematical practice needs to be adapted when operationalism is adopted, this is in the form of explicating constructions for all objects dealt with. But, as we have seen, there are in any case strong internal reasons for mathematicians to adhere to this ideal.

Indeed, operationalist geometry is automatically protected from the fallacies discussed above in straightforward ways. The existence and false diagram issues are resolved because they could never arise in strict operationalist practice. And the verbal logic problems do not arise since verbal logic is not accorded any foundational role in operationalist mathematics. Thus operationalism very conveniently cuts off in one fell swoop numerous lines of attack of philosophical scepticism directed at mathematics, without the need for any sacrifices in mathematical content.

Operationalism is related to positivism. Science can only speak about observable facts, according to positivism. So positivism implies a strong adherence to a scientific worldview as the only source of knowledge, and a rejection of other humanistic or philosophical theories or belief systems. Science, positivists say, prudently restricts itself to what is actually knowable, while other forms of philosophy speculate futilely about the ultimate nature of things and all sorts of other concepts that transcend observable reality. On this view, much grandiose philosophising is wrongheaded and even strictly meaningless.

Here’s how Bridgman puts this point. “It is quite possible, even disquietingly easy, to invent expressions or to ask questions that are meaningless. It constitutes a great advance in our critical attitude toward nature to realize that a great many of the questions that we uncritically ask are without meaning. If a specific question has meaning, it must be possible to find operations by which an answer may be given to it.”

You know how all courses have to have “learning goals” these days? You can’t say: “this course is about quadratic equations.” Instead you have to say: “after completing this course, the student will be able to obtain the solutions to equations of the form blah blah blah.” So you must operationalise what it means to succeed in the course. You must state it in terms of what the student can do. Not in terms of just naming the topics.

Greek geometry was like that too. Everything is formulated in terms of doing. It’s not enough to just give names to things: you must make those names meaningful by explaining what you can actually do with them.

Here’s another quote from Bridgman: “Politics, philosophy and religion are full of purely verbal concepts. Such concepts are outside the field of the physicist. Only in this way can the physicist keep his feet on the ground or achieve a satisfactory degree of precision.”

So positivism and operationalism go hand in hand with an “us versus them”—scientists versus philosophers—type of attitude that is as much about rejecting other perspectives as it is about affirming its own principles.

It is possible that this dynamic was directly paralleled in antiquity. Ancient mathematicians would have felt that their geometry was a lot more grounded in reality than even quasi-science such as the four elements theory, not to mention more abstract philosophy such as, say, Aristotle’s doctrine of causes. Ancient mathematicians would have felt that their results were qualitatively different from philosophy in terms of reliability, objectivity, and many other dimensions. They may even have felt that much philosophy was empty gibberish. Perhaps this would have led them to articulate general methodological principles that would “explain” why their form of reasoning and knowledge was superior to that of the philosophers, as many scientists have been inclined to do ever since.

What methodological dicta might Greek mathematicians have seized upon to set their field apart from philosophy? Certainly not anything like the modern identification of mathematics with logic and axiomatic-deductive reasoning. Logic and deduction were already highly prized among Greek philosophers. If anything, they were too obsessed with deductive logic: Zeno’s argument that there can be no such thing as motion is one example among many of extreme faith in abstract deductive reasoning even when it is in blatant conflict with the most basic common sense. So ancient mathematicians could certainly not hope to stand out by their reliance on abstract deductive reasoning.

Axiomatics too was far from the exclusive purview of mathematicians; indeed it is obvious that basing one’s theories on a list of allegedly evident but ultimately unjustified axioms is very convenient for mathematicians and sophists alike. It may even be reasonable to say that the abundance of deductive philosophical systems that were clearly in conflict with one another would rather have been an incentive for the mathematics to insist that, unlike the philosophers, they did not rely on abstract logic.

Operationalism would have been an alternative readily at hand. Constructions had always been a central part of geometry, from the time of the Egyptian “rope-stretchers” whom the Greek identified as the originators of the field. Later theoretical developments, such as the irrationality of the square root of 2, had spoken in favour of taking geometry as the foundational bedrock of all mathematics. It would have been a short and natural step for the mathematicians to tie the foundations of their subject to their already ubiquitous ruler and compass.

To the mathematicians it would have cost little to embrace all-out radical operationalism. Virtually all of mathematics was readily susceptible to being reframed in such a paradigm. It would have been a way of legitimating existing practice that would have necessitated little or no deviation from what they were already doing. Meanwhile, other branches of philosophy stood no chance of founding their teachings on an operationalist basis. So if the mathematicians were looking for a way to set themselves apart from the philosophers—to explain why their field had cumulative progress, universal agreement, and inviolable truths while philosophy had paradoxes and schools in constant disagreement with one another without any prospect of reconciliation—then operationalism would have been the obvious way to go.

Another virtue of positivism that could be held up as distinctive is that it restricts all knowledge claims to the domain of what is actually knowable in a straightforward empirical sense. Failing to adhere to positivism means making statements that are, by their very nature, empirically unverifiable and hence arguably unknowable almost by definition. Unlike most of philosophy, any statement of geometry is readily equated with a claim regarding certain empirical circumstances. Ancient mathematicians had a golden opportunity to highlight this natural attribute of their field as an epistemic virtue. They could pose to head-in-the-clouds philosophers the very difficult challenge of explaining what good a theory is if it has no “cash value” in the real world, in the form of empirically testable claims. And they could stress that geometry, by contrast, has no need to engage in that kind of theorising.

Related to this is the ideal of falsifiability. When the geometers claim that any triangle has an angle sum of two right angles, they are sticking their necks out. If their claim was false, it should be simple enough to find a counterexample. The operationalist formulation of geometry makes it possible to press this point very strongly. The theorem simply means: if you put a ruler down on a piece of paper and draw three intersecting lines, then cut out the three corners and put them point-to-point, then the three pieces fill precisely the angle on one side of a straight line. The very meaning of the theorem directly contains a concrete recipe for testing and potentially falsifying it.

Karl Popper, the philosopher of science, is the name most prominently associated with the philosophy of falsificationism. This was in the first half of the 20th century. To Popper, falsifiability is what set science apart from non-science.

As examples of non-scientific theories, Popper had in mind things like the theories of Marx and Freud, which were influential at that time. These theories had a sort of quasi-scientific appearance. They postulated fundamental laws and used these to explain many phenomena. But according to Popper it was pseudo-science. Because, no matter what phenomena were observed, they could always tell some story about how that fits with their laws.

So these theories pretended to have laws, but they were vague enough to allow many different possible applications, so that almost anything could be construed as consistent with these laws one way or the other. Just as astrological horoscopes in the newspaper make so-called predictions about the future, but in fact they are so vague that they can often be interpreted as having been correct no matter what happens.

This is why Popper emphasized the importance of falsifiability. For a prediction to be scientific, there must be clearly specified condition under which it would be regarded as having failed. The scientist must say: if such-and-such a thing happens, then I was wrong. Before making an experiment or observation, the scientist has already set down those criteria, that is to say, the conditions under which the theory must be regarded as having been falsified.

Non-scientific theories like those of Freud or Marx are not like that, according to Popper. Advocates of those theories use them to “explain” all kinds of things, but they never say: if such-and-such a thing were to happen, then that would prove me wrong and I would give up the theory.

Formulating geometry in terms of constructive operations is a great way of making it scientific in Popper’s sense. It makes the theorems of geometry directly testable. Euclid’s constructions are like lab instructions for carrying out such a hypothesis test. Do the construction and measure for yourself if it came out the way the theorem said.

Euclid’s parallel postulate is something that can be performed and tested in a very concrete way. It says: here’s what going to happen if you draw this kind of configuration.

Alternatives to the parallel postulate are not like that. Instead of the parallel postulate, you could say: Given a line and a point, there is precisely one parallel to the given line through the given point.

How would you test that? It’s in the form of a metaphysical statement, rather than in the form of a falsifiable scientific hypothesis. There is one and only one parallel. It’s like saying: There’s one and only one God. How can you verify that? How could you even prove it wrong? You can’t. Unlike scientific hypotheses, and unlike operationalist geometry, statements of that form do not come with a concrete set of operations one can perform to see if it works or not.

Operationalising geometry makes it falsifiable. It also makes geometry theory-independent. You do not need to accept the definitions and postulates of the mathematicians in order to perform this empirical test. Sceptics who try to criticise mathematics in general terms can thus be confronted with a concrete challenge: regardless of whether you accept any of our assumptions or modes of reasoning, we offer you hundreds upon hundreds of claims of the form: if you perform such-and-such concrete operations, then the outcome will always be one particular way rather than another. Feel free to prove us wrong, the mathematician can say. It would be impossible to meet the challenge and very difficult to try to dismiss it as illegitimate.

The operationalist formulation of mathematical statements is reducible to straightforward recipes whose neutrality and objectivity is very difficult to deny. This is in stark contrast with many philosophical claims, which must often be bought into or rejected wholesale along with an entire theory because all the parts of the theory are interdependent. Even the very meaning of the concepts the theory uses is inherently bound up with the system as a whole.

Operationalism ensures that geometry is not like that. Operationalist geometry is not an entangled holism.

Here’s an analogy for this. Consider a casino. It has roulette and black jack and so on. You play with casino money. Plastic chips that only have meaning and value inside the casino. Once you leave the casino you can’t buy anything for those worthless poker chips.

Non-scientific thought-systems such as philosophy or religion are like the casino. Internally, they have all kinds of intricate laws and explanations for how everything fits together. And it’s easy to get caught up in the system once you buy into it. But to link it to the real world, you have to ask yourself: what’s the actual cash value of this stuff? That is to say, what could I actually do with any of this in the real world, concretely?

Operationalism is “cash value” geometry. It translates everything into real-world operations that anyone can perform. It’s cash money. You can use it directly and it works. It’s not casino money, which only makes sense if you accept the entire premise of the casino with all its internal rules.

Even someone who doesn’t believe in the postulates of Euclid, or doesn’t believe in geometrical proofs, etc. Even such a person can test these things. They can cut the corners off a triangle and see if they fit together the way Euclid says. Or they can draw squares on the sides of a right-angle triangle and see if the areas are equal the way Pythagoras says. Those are scientifically, concretely, real-world testable claims.

Let’s summarise. Operationalism safeguards mathematics against a multitude of plagues. It prevents us from reasoning about entities and concepts that are inconsistent, incoherent, non-existent, or imaginary.

Mathematicians would have had every reason to articulate such a philosophy. Greek antiquity was an age of sceptical philosophical attacks. Mathematics would have found itself under fire, and its enemies were no fools. The logic and rigour of mathematical proofs were by and large hugely impressive. Yet it had a conspicuous Achilles heel: a veritable self-destruct button that could bring the entire edifice crashing down at the slightest trigger. For if there was any way an inconsistency could slip into mathematical reasoning undetected, then everything that followed would immediately be rendered logically worthless. What guarantee do we have that this will never happen, or indeed that it has not already happened?

This vulnerability pertains especially to the way objects are introduced into mathematical discourse. It is safe to say “let ABC be a right-angled triangle,” but if you say “let ABC be a triangle with two right angles,” then you have introduced an inconsistency and all is lost. Then you can prove that 2 is equal to 1, and the entire credibility of mathematics collapses. So geometry needs to systematically guarantee that it could never commit an error of this type. In other words, it needs a meticulous gatekeeping policy that only allows the most carefully vetted entities to enter mathematical discourse.

Constructions are the answer to this problem. By insisting that geometry only speaks of entities that are constructed, the mathematician immediately knocks the legs out under boogeymen examples of inconsistent objects such as the superright triangle.

Constructions also ground mathematics in reality and gives a straightforward account of what geometry is and what geometrical statements mean. This can be used to set geometry apart from empty philosophy, from metaphysics, religion, astrology, all kinds of empty pseudo-science.

Philosophers of science in the 20th century spent a lot of effort trying to formulate the criteria that distinguished science from non-science. One of their answers was falsifiability: scientists bravely specify what would prove them wrong. They say: try this for yourself, and if it doesn’t come out the way I said I promise I will admit that I was wrong and that my theory should be rejected.

They also found that to follow through on this program it was important to translate abstract theoretical notions into observable real-world terms. Instead of merely speaking abstractly about for instance the concept of the force of gravity, it is necessary to translate the meaning of that theory into something doable, testable, such as: if you hang this led weight from this spring, then the spring will extend by so-and-so many centimeters. Things like that is what the concept of gravity comes down to in practical terms. This concreteness is essential to science, and essential to separate science from fancy games with words.

Euclid’s geometry is a perfect fit for all this stuff. It’s almost as if Euclid had read these 20th-century philosophers of science. Maybe Euclid and his friends had many of the same ideas. Maybe they too wanted to set their theory apart and explain why it was superior to other branches of philosophy. The way they based geometry on constructions is a perfect fit for making those kinds of arguments.

So there you go. These are many reasons to ground geometry in constructions. It is not for nothing that all depictions of Euclid shows him with ruler and compass in hand. These are no mere practitioner’s tools. They are in fact essential even to the theoretical foundations of geometry in numerous respects. That is what I have tried to argue.