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Opinionated History of Mathematics
44 minutes | 24 days ago
That which has no part: Euclid’s definitions
Euclid’s definitions of point, line, and straightness allow a range of mathematical and philosophical interpretation. Historically, however, these definitions may not have been in the original text of the Elements at all. Regardless, the subtlety of defining fundamental concepts such as straightness is best seen by considering the geometry not only of a flat plane but also of curved surfaces. Transcript “A point is that which has no part.” What a bonkers way to start a book. But that’s Euclid for you. Let’s start the whole thing off with a negative, Euclid apparently told himself. He’s like: Let me tell you what a point is. Think of things that have parts. It’s not that. It’s the other stuff. Stuff that doesn’t have a part. Pretty weird that the first thing you introduce is actually defined by exclusion, in terms of what it is not. But anyway, never mind that. There are important interpretative issues at stake here. The first two lines of Euclid’s Elements are the most misunderstood. They define the concepts of point and line. “A point is that which has no part” and “a line is a length without breadth.” We might interpret this as saying that a line is 1-dimensional, and a point is 0-dimensional. Here’s how people misunderstand this. They say: Aha, told you! Geometry is not about physical things; it’s about objects in some ideal realm, just like Plato said. Because if you draw a line with a pen for example, it will always have some breadth, no matter how thin it may be. No physical object can ever be a “breathless length.” This proves that Euclid is not talking about physical space. But that is a terrible argument, which makes no sense. It is demonstrably false. Yet you hear it repeated again and again. Some ancient philosophers made this argument. Aristotle mentions it in the Metaphysics (998a). Still today, many modern scholars walk into this fallacy all the time. But don’t worry, I’m here to save you from this mistake. There is no inconsistency between Euclid’s definitions and a physicalist view of geometry. On the contrary, these kinds of idealisations are an essential part of any physical theory. Ptolemy, the astronomer, treats the moon as a point for the purposes of many of his demonstrations, for instance. Obviously no one would infer that he is therefore believes the moon is a mathematical point with no extension. The convention of treating the moon as a point is simply a common-sense idealisation that is the only sensible thing to do for many mathematical purposes, regardless of what one’s estimation of the actual body of the moon may be. It is the same for instance in Archimedes’s work on levers, where the lever arm is a weightless mathematical line and the weights are applied at mathematical points. Since such idealisations are unequivocally used all the time without further ado in applied mathematics, it makes no sense to take them to be inconsistent with a physicalist view of geometry. On the contrary, such idealisations are exactly the standard assumption one would expect in physicalist geometry, just as one invariably finds it any other mathematical theory pertaining to the real world. So if this argument is right, that Euclid’s definitions prove that his geometry is divorced from reality would, then it is equally true that the Greeks did not intend their astronomy or their statics to apply to the real world either, which is obviously absurd. So it’s madness to infer from Euclid’s definitions that he thinks geometry is non-physical. It is more plausible to read these definitions as specifications of idealisations made in geometry, rather than as claims about the ultimate nature of geometrical objects. Indeed you can find support for this in ancient sources. Heron, for example, clearly takes such a view. He writes: “Already in ordinary language use we have the notion of a line as something which has only length, but not at the same time width and thickness. For we say: a road of 50 stades, as we concern ourselves with the length only, but not at the same time its width.” Here the identification of geometry with everyday physical objects is evident. The allegedly Platonic or ontological aspects of the definitions is merely a common-sense matter of simplifying assumptions and directing attention only to the relevant aspects of the situation. Proclus makes the same point as Heron. He also uses the example of a road. And he attributes this view to “the followers of Apollonius.” In other words, Proclus puts this view right at the mainstream of Greek geometry at its peak. Apollonius is at the heart of the mathematical establishment. Heron was also a mathematical author. So mathematicians were the ones who thought a road was a good example of a line. Meanwhile, those who tried to use Euclid’s definition to drive a wedge between mathematics and physical reality were philosophers. It’s typical, of course, that philosophers focus on the first two lines of Euclid and try to dismiss the relevance or status of geometry on that basis. Perhaps they never made it past the first page of the Elements. How convenient that they immediately found an excuse to dismiss geometry based on the first two definitions. How convenient that their objective analysis just happend to justify ignoring all technical mathematics. Such a motivation is quite transparent in at least one of these philosophical authors, Sextus Empiricus. He gives probably the most extensive articulation of this idea that the first definitions of Euclid undermines the credibility of mathematics. The very title of his work is Against the Mathematicians. “The mathematicians talk idly,” he accuses, “for the straight line shown to us on the board has length and breadth, whereas the straight line conceived by them is ‘length without breadth’.” Gotcha, huh? You can decide for yourself if you think Sextus Empiricus is a razor-sharp philosophical mind who has outsmarted all the mathematicians, or whether he’s a guy who doesn’t like mathematics and wants to rationalize his own ignorance. Those of us who read Euclid beyond the first page quickly realize that there is a further compelling argument for why one must not make too much of the alleged ontological import of Euclid’s definitions of point and line. Namely, that these definitions are the most extraneous part of the Elements. The Elements is obviously a very carefully constructed logical theory, where almost every statement is carefully formulated to correspond precisely to the justification of specific inferences in deductive proofs. Obviously postulates and propositions are of this type, and so are many definitions, such as the definition of a circle which is used already in the very first proposition to infer that since two line segments are radii of the same circle, they must be equal. However, the definitions of point and line are not of this type. These definitions serve no direct role in the deductive structure of the theory. They are effectively ornamental. They are arguably the most inconsequential parts of the entire Elements, since they are never actually used in any proof. Yet these are the very lines always cited as virtually the only textual evidence in mathematical sources of alleged anti-physicalist tendencies in Greek geometry. Madness. In fact, these definitions may not even have been part of Euclid’s original text of the Elements at all. The version of the Elements we have has been edited, unfortunately. When Euclid wrote it, it was a sophisticated analysis of the foundations of geometry. It’s readers were high-level mathematicians. Later it became a textbook for schools. Editors interfered to make it more accessible. Possibly adding the first couple of definitions for example. This is especially clear with respect to Definition 4 of the Elements, the definition of a straight line. Here’s what it says: “A straight line is a line which lies evenly with the points on itself.” This definition is meaningless drivel. What does it even mean to “lie evenly with itself”? How can such a masterful work, which is clearly written by a top-quality mathematician, open with such junk? There’s a compelling answer to this conundrum, proposed by Lucio Russo. It goes as follows. Euclid didn’t define “straight line” at all. His focus was on the overall deductive structure of geometry, and for this purpose the definition of “straight line” is essentially irrelevant. Indeed, the utterly useless Definition 4 is never actually used anywhere in the Elements. Archimedes agreed with Euclid as far as the Elements were concerned, but in the course of his further researches he found himself needing the assumption that among all lines or curves with the same endpoints the straight line has the minimum length. He therefore stated this property of a straight line as a postulate in the context where it was needed. The Hellenistic era, which included Euclid and Archimedes, was one of superb intellectual quality. Unfortunately it did not last forever. Heron of Alexandria lived about 300 years later. This was a much dumber time. Fewer people were capable of appreciating the great accomplishments of the Hellenistic era. But Heron was one of the best of his generation. He could glimpse some of the greatness of the past and tried his best to revive it. To this end Heron tried to make Euclid’s Elements more accessible to a less sophisticated audience who didn’t have the background knowledge and understanding that Euclid’s original readers would have had. He therefore wrote commentaries on Euclid, trying to explain the meaning of the text. To these new, more ignorant readers of geometry, it was necessary to explain for example what a straight line is. Heron realised that Archimedes’s postulate about the line as the shortest distance captured well the essence of a straight line. However, it is not itself suitable as a definition, because a line should have the property of being the shortest distance between any two of its points, not be phrased in terms of only two fixed points, as Archimedes. Remember, Archimedes was not trying to define a straight line, only to make explicit an assumption about straight lines that was particularly relevant in a particular work of his. To adapt Archimedes’s idea into a definition, Heron therefore explained that, and now I quote him: “a straight line is [a line] which, uniformly in respect to [all] its points, lies upright and stretched to the utmost towards the ends, such that, given two points, it is the shortest of the lines having them as ends.” In this passage, the phrase “uniformly …” obviously refers to the universality of the shortest-distance property. The point of this phrase is to highlight that this property applies to any two points on the line. This is what later becomes Euclid’s phrase “evenly with the points on itself.” The original purpose of this phrase was to say that the distance-minimization property of the straight line holds for any pair of points on the line: that is to say, the property holds “uniformity” or “evenly” across the entire line. Not only for the endpoints. The definition in the Elements is a mutilated version of what Heron said. Heron’s point is that no matter which two points on the curve you pick, the straight line is always the shortest path between them. The mutilated version ignores the part about shortest distances, and distorts the part about it applying across all points into the vague phrase about evenness of all points. How did that happen? To understand this we need to fast-forward another 300 years. Intellectual quality has now plunged deeper still. Geometry is in the hands of rank fools. Euclid’s Elements, which was once written for connoisseurs of mathematical subtlety, is now used by schoolboys who rarely get past Book I and “learn” that only by mindless rote and memorisation. A time-tested way (still in widespread use today) to “teach” advanced material to students who do not have the capacity to actually understand anything is to have them blindly memorise a bunch of definitions of terms. In this context, therefore, there is a need for an edition of the Elements which includes many definitions of basic terms, which must be short and memorisable, and which don’t need to make mathematical sense. In this era of third-rate minds, some compiler set out to put together an edition of the Elements that would satisfy these conditions. Heron’s commentary on the Elements is appealing in this context since it affords opportunities to focus on trivial verbiage instead of hard mathematics. But Heron’s description of a straight line is still too complicated. It’s too long to memorise as a “soundbite” and the mathematical point it makes is moderately sophisticated. The compiler therefore makes the decision to simply cut off Heron’s description after the bit about “uniformly in respect to [all] its points.” This solves all his problems: the definition becomes shorter and easier. The only drawback is that the “definition” becomes utter and complete nonsense. But since the whole purpose of it is nothing but blind memorisation anyway this doesn’t matter anymore. This is how the ridiculous Definition 4 ended up in “Euclid’s” Elements. It’s a mutilated version of what was once a very good definition. According to Russo’s hypothesis, which is compelling. As Russo also observes, in the works of other great Greek mathematicians such as Archimedes and Apollonius (who “belong to the same scientific tradition” as Euclid) “there is nothing analogous to the pseudo-definitions of fundamental geometrical entities contained in the Elements. The introduction of terms implicitly defined through postulates is instead frequent.” So this supports the hypothesis that the Elements was corrupted due to its association with introductory teaching. While these more advanced works remained less tampered with. If we want a definition of a straight line consistent with Greek geometry, I would propose defining it as follows: a straight line is the path of a stretched string. In other words, a straight line is a curve that doesn’t change shape when you pull its endpoints. This is closely related to the notion of the shortest distance between two points. Related, but not equivalent. To get to the bottom of the notion of straightness it is useful to consider not only the usual plane but also other surfaces. Euclid’s geometry is the geometry of a flat plane, a flat piece of paper so to speak. Other surfaces have other geometries. A cylinder, for instance, like a Pringles can. It has its own geometry. Pringles lines, Pringles triangles. To appreciate the geometry of a surface we should forget for a moment that it is located in three-dimensional space. We should look at it through the eyes of a little bug who crawls around on it and thinks about its geometry but who cannot leave the surface and is unaware of any other space beyond this surface. Think of for example those little water striders that you see running across the surfaces of ponds. They know the surface of the pond ever so well. They can feel any little movement on it. But they are quite oblivious to the existence of a third dimension outside of their surface world. This makes the water strider an easy prey for a bird or a fish that strikes it without first upsetting the surface of the water. It is instructive to think about the intrinsic geometry of surfaces in this way. It forces us to realise that many things we take for granted as “obvious” objective truths in geometry are really a lot more specific to our mental constitution and unconscious assumptions than we realise. In some ways we are as ignorant of our own limitations as the water strider. Let’s transport ourselves into the cylinder world to practice seeing geometry from a different point of view. On a cylinder there are stretched-string curves that are not the shortest path between its two endpoints. Wrap a shoelace around a Pringles can. You can make various spirals that are stretched strings. Or a helix as it’s called, a corkscrew curve. So these are straight lines, according to my definition. But they are not the shortest distances between their endpoints. Even if you have to stay on the surface of the cylinder, you can still get from one endpoint to the other more directly than by a spiral that winds around and around an excessive number of times. So “stretched string lines” and “shortest distance lines” are not the same thing, as this example shows. It is arguably the stretched string that gets it right. It makes straightness a “local” property. We can alter the distance characterisation of straightness to be local too. Then we would say: a curve is a locally shortest path if, for any given point on the curve, there is a neighborhood around that point such that the distance along the curve between any two points on the curve in that neighborhood is the shortest possible distance between those points. This picks out the same straight lines as the stretched string definition. Being a stretched string is the same thing as being a locally shortest path: it’s the shortest path between points on the line when you zoom in, but not necessarily between points on the line that are far apart. Straight lines can also be defined as curves possessing half-turn symmetry about every point: a curve has half-turn symmetry if, for any given point P on the curve, there is a neighbourhood around that point such that when this neighbourhood is rotated about P by half the angle-measure around P then the curve ends up on top of itself. More loosely, a curve is straight if it always “cuts angles in half”; it “leaves the same amount of space on either side.” To test for this kind of straightness on surfaces one can use the “ribbon test”: if a ribbon or band can be laid flatly along the curve without creasing on either side, then the curve is straight. Try it on your Pringles can. You can use a measuring tape for instance, for instance those free paper ones you can get at hardware stores or furniture stores. That’s your “ribbon.” Try wrapping it around the Pringles can. Some ways of wrapping it makes it lay flat against the surface; those are straight lines. Other ways of wrapping it makes it crease up on one side or the other; those are not straight lines because they don’t leave the same amount of space on either side. Here’s a fun thing to investigate and think about. We have now defined straight lines on a Pringles can in two different ways: one in terms of a stretched string, like a shoelace, and one in terms of a flat ribbon, like a measuring tape. Are they the same? Are there some lines that are “shoelace-straight” but not “ribbon-straight” or the other way around? I’ll leave that to you to explore. So we have two notions of straightness, and both of them get at something very fundamental: The stretched string highlights the idea of straightness as minimization, or as a tight fit. This idea is reflected in many real-world occurrences of straightness. For instance, the path of a cross-Atlantic flight. You know that when you look at the path on a map, in the flight tracker, it looks curved. It looks like you’re flying from Paris up toward the North Pole, and then back down again to get to New York. Why not go “straight across” instead? Of course the path is in fact straight. It looks curved only because the map is an imperfect representation. If you have a globe you can stretch a string between Paris and New York and feel for yourself that the shortest path indeed goes “up” toward the North Pole. But that path is straight, according to the stretched string definition. But we also have the second idea of straightness: that of straightness meaning “the same amount of stuff on both sides.” This is also reflected in various familiar situations. For instance, when you fold a piece of paper, the edge is straight. Why is that? This doesn’t have to do with stretched strings or least distances. Instead it has to do with the sameness of both sides. To fold something you match up points on one side with points on the other. Folding is only possible if the two halves are precisely equal. There is also a kind of three-dimensional version of this. Namely the axis of rotation when a solid body is rotated. For example a döner spit at a Middle Eastern restaurant, or a basketball spinning on your finger tip. The axis of rotation is a straight line. Why? This is again because of sameness on all sides. The moving parts have to fit into each others’ space. So they have to be equal on either side. Here’s an example from engineering. Mirrors are made flat by rubbing two of them against each other face-to-face, with a fine sand or other polishing agent applied between them. This too embodies the idea of flatness or straightness as equivalent to sameness on both sides. Another example is rowing a boat. You go straight in a rowboat if you apply equal force to each oar. This is again symmetry-straightness, not stretched-string straightness. It’s not built into the very rowing process that this necessarily corresponds to the shortest distance between the endpoints of the journey. But it is built into the very act of rowing this way that you leave equal amounts of space on either side. Light rays are straight. But this is more like the stretched string again. Light “cares” about minimizing the time of travel, so to speak. Just like the airline. The airline stretched a string across the globe to find out how to fly from Paris to New York. They also tightened their purse strings, so to speak, with the same move, because the shortest path is also the cheapest path. Light is a bit of a penny-pincher too, it would seem; or it is impatient, perhaps. Because it chooses the quickest path. For instance, if it has to go from point A to point B via a flat mirror, then it chooses to bounce off the point on the mirror that makes the total distance as short as possible. You can reproduce this path with a stretched string. Suppose A and B are two points on a wooden table. Let’s hammer two nails into those points. One of the edges of the table we regard as the mirror. Take a vertical metal bar and put it against the edge of the table. Now wrap a string from A, around the metal bar at the end of the table, and then to B. Now pull the string as tight as you can. The metal bar forces the string to go to the edge of the table and back. But the bar can move along the edge of the table. When we pull the string we force the bar into a particular position, namely the position that minimizes the total distance. The path of the string is the same as the path of light between these points via a mirror at the edge of the table. You can try it out with a laser pointer if you don’t believe me. So light is like stretched strings. Indeed artists use this sometimes. The pull strings to simulate light rays in order to get vantage points and perspectives just right. I’m trying to emphasize with these examples how thinking about what straightness means is connected to many aspects of culture and experience. Isn’t it fascinating how the mathematical notion of straightness is a sort of root of all these diverse phenomena? Once you’ve read the Elements you see geometry everywhere. Flight paths, döner spits, spinning basketballs, light and mirrors, rowboats, Pringles cans––henceforth, anytime you encounter these things you will go: ah, of course, that reminds me of Euclid’s Definition 4! The idea of straightness as corresponding to stretched string also generalizes well to other surfaces that are not homogenous. So far we have mentioned the plane, the cylinder, and the sphere. These surfaces are all homogenous in that every point is alike. If you cut out a piece of the surface, it fits on top of any other part of the surface. Some surfaces are not like that. For example, the surface of the human face. It has regions of different curvatures, as we say. A flat piece of paper has zero curvature: it’s not curved at all. A ball has positive curvature: it curves the same way in all directions. A saddle has negative curvature: it curves in different ways in different directions. A saddle for riding a horse. It curves “upwards” along the spine of the horse, and “downwards” where your legs go. Opposite directions of curving. This is what makes the curvature negative. The human face has both negative and positive curvature. Some parts are like a saddle. For instance the side of the nose, or the area just below your mouth. If you put your finger there and run it top-to-bottom, then it curves one way. But if you ruin it side-to-side, it curves the others way. So those are regions of negative curvature. They are like a saddle. Other parts of the face have positive curvature, like a ball. For instance the chin and the cheeks. There the surface curves the same way no matter which direction you run your finger. Felix Klein, a 19th-century mathematician, thought this might be the key to a mathematical analysis of the elusive concept of human beauty. Since the face has regions of positive curvature and regions of negative curvature, there’s a diving line running between them. Between the cheek and the nose, between the lips and the chin, and up again on the other side. So Klein drew this line of zero curvature on a classical sculpture. You can google it, Felix Klein Apollo Belvedere, and you can see photos of this. Klein was hoping that a simple pattern would emerge that would “explain” the beauty of this face. But it didn’t work. No such pattern was discernible. Still it makes for a good story. It’s also a good piece of “first date mathematics.” You can explain this idea to your date over some glasses of wine. And of course slowly reach out and sensually trace these curves on their face and so on. Great stuff. But where were we? I wanted to discuss how the notion of straightness extends to these other surfaces. Surfaces with variable curvature. We can still say that straight lines are stretched strings. We often call them geodesics rather than straight lines in such cases. But the stretched-string idea is still the same. Here are some examples. Think of bandaging an injured limb. The bandage needs to be tightly wrapped. This means that it must follow a geodesic path, a stretched-string path. The bandage is a “straight line” in the sense that it is a stretched string. In other words, it always takes the locally shortest distance. Of course not the shortest distance overall, since it wraps around and around. But the shortest distance between any two nearby points on its path, because otherwise it would create slack which you would never do of course. Another example: The heart beats through the contraction of muscular threads across its surface. These muscular threads must be geodesics. They must be stretched-string paths. Because the heart beats by contracting these threads. If these muscular threads were not positioned along geodesic paths, then when they contracted they would just slide around on the surface of the heart instead of contracting it. The human heart is carefully designed with this geometry in mind. And if it wasn’t we would all die very quickly. So the stretched-string notion of straightness is truly a matter of life and death.
35 minutes | 2 months ago
What makes a good axiom?
How should axioms be justified? By appeal to intuition, or sensory perception? Or are axioms legitimated merely indirectly, by their logical consequences? Plato and Aristotle disagreed, and later Newton disagreed even more. Their philosophies can be seen as rival interpretations of Euclid’s Elements. Transcript What kinds of axioms do we want in our geometry? How do you tell a good axiom from a bad one? Should an axiom be intuitively obvious? Should it be empirical, physically testable? Should it be logically self-justifying, or are axioms logically arbitrary? The time has come to take a stand. As we have been reading Euclid backwards, we have seen how the Pythagorean Theorem can be reduced to a theorem on the areas of parallelograms, and how this theorem in turn can be reduced to triangle congruence. So now we have to prove triangle congruence somehow. If two triangles have the same side-angle-side, then they are the same triangle. How to prove such a thing? We can’t keep playing our game of reducing every theorem to a simpler one, because we’re running out of “simpler.” Maybe this theorem is as simple as it gets? Maybe it’s the rock bottom? How do you decide anyway what’s simpler than what? It’s becoming more philosophy than mathematics to answer these questions. This theorem—side-angle-side triangle congruence—is Euclid’s Proposition 4. Ok, so it’s a proposition, not an axiom. So apparently he has reduced it to something. But what? Let’s read the proof. So we have two triangles, and they have certain measurements in common. Two sides of one triangle are equal to two sides of the other, and also the angle between those sides are equal in both triangles. Euclid says he can prove that the other measurements are equal too. The remaining side, the remaining angles: it’s all equal. They’re the same triangle basically. And here’s how Euclid says you can prove this. Take one of the triangles and put it on top of the other. We know that they have side-angle-side in common, so those parts line up perfectly. These three attributes are enough to “lock” the entire triangle into one unique shape, in fact. Because suppose it wasn’t. Suppose the two triangles were different. Since they have side-angle-side in common, they lined up at least on those parts. This “locks” into position two of the sides and all three of the vertices. There is no way one of the triangles can stick out beyond the other in terms of these two sides or in terms of any one vertex. So the only way the triangles could be not equal would be if the third side somehow missed. This would mean that the endpoints of the third sides were the same for both triangles, but the line joining them would be different. Impossible! You can’t have multiple lines connecting the same two points. Or Euclid puts it: two straight lines cannot enclose a space. You can’t draw a straight line from A to B, and then another straight line from A to B, in such a way that these two lines miss each other and have some space in between them. Since this is impossible, the third sides of the triangles must line up on top of each other, and therefore the two triangles are identical, or congruent. That’s the proof. Once again the point of the proof is not to convince us that the theorem is true, but to reveal how its truth can be reduced to more basic truths. Euclid has now taken this as far as he can. We’re all the way down to the axioms: things that cannot be broken down any further. The proof of the triangle congruence theorem rests most prominently on two axioms. One, as we saw, is that “two lines cannot enclose a space.” Which is equivalent to saying that, for any two points, there is only one straight line between them. This corresponds to Euclid’s Postulate 1, which states as an axiomatic principle that we can “draw a straight line from any point to any point.” It is understood that this line is unique. That is to say, there’s only one way you draw that line. So that’s an axiom. You can’t reduce it any further. But there was another axiom involved as well in our proof of the triangle congruence theorem. Namely the assumption that we could put one triangle on top of the other. This corresponds to Euclid’s Common Notion 4: “things coinciding with one another are equal to one another.” This is basically a definition of equality. What does it mean for two things to be equal? Put one on top of the other. If neither sticks out beyond the other, then they are equal. That’s what equal means. In fancier words you could say: equality means alignment under superposition. So that’s another axiom that Euclid states at the beginning of his work, and which he cannot prove from more basic principles. What should we make of these two axioms? Since we can’t prove them from other things, they must be justified some other way. What way would that be? Euclid apparently thought these two principles were especially suited to be axioms. He could have done it differently. He could have chosen other axioms. For example, the triangle congruence theorem itself could have been taken as an axiom. That’s what Hilbert later did, in his modern and very authoritative axiomatisation of geometry. So from the point of view of modern mathematics it makes a lot of sense to take the triangle congruence principle as axiomatic. From a logical point of view that’s perhaps the best approach. Modern logicians don’t like Euclid’s proof one bit. Bertrand Russell called it “logically worthless.” If you want mathematics to be logic, then that makes sense. But what is “good” mathematics? That depends on your philosophy of mathematics. You must first decide what kind of thing mathematical knowledge is. What it should be. Only after you have made that philosophical decision do you have any basis for judging whether Euclid’s approach is better or worse than that of others. Euclid’s choice of superposition as an axiomatic principle is quite interesting in this regard. It seems almost physical or empirical. In the proof of the triangle congruence theorem, you are literally, physically picking up one of the triangles and placing it on top of the other triangle. This seems to assume that triangles are physical objects, like cardboard cutouts or some such thing. And the idea that equality means alignment under superposition also has a somewhat physical feel. The thing fits on top of the thing. It’s something you could test practically, in the real world. The modern authors I mentioned do not approve of these connotations. They don’t like it one bit that mathematics is so to speak contaminated by empirical considerations. They want mathematics to be pure reason. They don’t want it to depend on sense perception and physical experience. But Euclid’s use of superposition suggests that he was less dogmatic about this. It could be interpreted as a sign that he was open to the idea of geometry as ultimately physical. Of course geometry is still very theoretical. Obviously, to Euclid, you can’t justify things like the Pythagorean Theorem just by measuring things, the way you would verify a physical law by making a bunch of measurements in a lab. Of course geometry is not like that. But the fact remains that the axioms cannot be justified by the axiomatic-deductive process itself. What axioms are the “right” axioms, or the “best” axioms, is a question that cannot be answered by purely mathematical means. Some philosophical assumptions will necessarily be involved in such judgements. I wanted to use this as a bridge to discuss some Plato and Aristotle. I’m trying to emphasize how these things go together. Mathematics and philosophy. Reading Euclid leads naturally to philosophical questions. We reduced the Pythagorean Theorem down to superposition and uniqueness of lines. We faced the questions: Why stop there? Why these principles and not others? What kinds of foundations should geometrical knowledge be built upon? This is the right time to read philosophy, with these burning questions fresh in our minds. Mathematics itself does not answer these questions. As Aristotle says in the Posterior Analytics: “for the principles a geometer as geometer should not supply arguments.” So there is a kind of division of labor. Justifying the axioms is not the business of the geometer “as geometer.” But of course Aristotle didn’t mean by this that you should have mathematicians over there and philosophers over here and there’s no point for them to talk to each other. A better way to read it, I think, is this: geometers, as geometers, cannot justify their axioms, and therefore any geometer needs to be a philosopher as well. Aristotle discussed the axiomatic-deductive method at length in this treatise, the Posterior Analytics. Here’s a quote that sums up his view: “Demonstrative understanding must proceed from items which are true and primitive and immediate and more familiar than and prior to and explanatory of the conclusions.” Quite a list of demands! Axioms, such as those of geometry, should have all of those characteristics, according to Aristotle. Obviously this means that the axiomatic-deductive method is a whole lot more than merely logical deductions from arbitrary assumptions. Indeed Aristotle says as much: “There can be a deduction even if these conditions are not met, but there cannot be a demonstration––for it will not bring about understanding.” This places very significant restrictions on what could be a legitimate axiom in geometry. It must be “primitive and immediate and more familiar than and prior to and explanatory of the [theorems].” So axioms need to be self-evident, in other words, it seems. That’s more or less what Aristotle means by “immediate,” I suppose. And axioms must also be irreducible, not in turn derivable from some other principle. That seems to be the meaning of Aristotle’s demand that they be “primitive” and so on. It gets pretty interesting when Aristotle elaborates further on what he means by some of these terms, because then he commits himself to the perhaps controversial stance that axioms are ultimately grounded in physical experience. Here’s what he says: “I call prior and more familiar in relation to us items which are nearer perception.” So immediate perception must be the ultimate foundations of “demonstrative understanding.” Not pure thought, but sensory perception. The axioms are generalized or idealized facts of experience. As Aristotle says: “We must get to know the primitives [that is to say, axioms] by induction; for this is the way in which perception instills universals.” For instance, for any two points there is a unique line connecting them. This is fact of experience, but of course generalized––“by induction,” as Aristotle says. That is to say, we have observed this in many examples. For this particular pair of points there’s a unique line, and for that pair, and so on. These are facts of perception. And then “perception instills universals by induction”: that is to say, we generalize from these examples to the general principle that the principle will work for any two points, not just the numerous examples we have witnessed. So Aristotle thinks the axioms of geometry ultimately come from concrete experience. The credibility of the axioms, the certainty of the axioms, derives from immediate sensory experience. This fits pretty well with the principles to which Euclid reduced everything. It is known through experience that there is a unique line from any point to any point. For instance by pulling a string between two points you can get a very direct sensory feeling for the existence and uniqueness of that line. And the principle of superposition, of putting one triangle on top of the other, can likewise be seen as an idealized version of a very immediate and basic physical experience. But not everyone agreed. Plato is the opposite of Aristotle. He has complete contempt for the physical world, and he loves mathematics precisely because it is something purer and higher than physical experience. Let me quote Proclus expressing this view. Proclus is a follower of Plato. He is keen to argue that mathematics stems from the soul, not sense experience. He addresses the Aristotelian view, and he sums it up like this: “Should we admit that [the objects of mathematics] are derived from sense objects, either by abstraction, as is commonly said, or by collection from particulars to one common definition?” That’s what Aristotle had argued, but Proclus says: No, we should not accept that. And here’s why. Geometry cannot be based on physical experience, Proclus says, because “The unchangeable, stable, and incontrovertible character of the propositions [of mathematics] shows that it is superior to the kinds of things that move about in matter. And how can we get the exactness of our precise and irrefutable concepts from things that are not precise? We must therefore posit the soul as the generatrix of mathematical forms and ideas,” not physical reality. Plato was quite obsessed with this idea that pure thought is the highest and most noble thing in human life. In the Timaeus he elaborates on this idea in a rather amusing and poetic way. To philosophise is the purpose of life. Human anatomy is merely an appendix to the soul and the mind. “The entire body” was created “as its vehicle,” Plato says. That is to say, the body exists only to make philosophising possible. For example, consider the intestines of the human digestive system. They are very long and winding, right? Like you roll up an extension cord when putting it away in a drawer; it looks like that in our insides. Food doesn’t go in a straight line from the mouth and out the other end. Instead the body passes it through the intestines that go back and forth, back and forth, a very long distance. Plato thinks he knows why. Here’s how he explains it: “The intestines are wound round in coils to prevent the nourishment from passing through so quickly that the body would of necessity require fresh nourishment just as quickly, there by rendering it insatiable. Such gluttony would make our whole race incapable of philosophy and the arts, and incapable of heeding the most divine part within us.” So the human body is just a means to an end. The only thing worth anything is philosophy. Eating doesn’t have any value in itself. The only purpose of eating is to put off the annoying needs of the body for a while, so as to give us time to think. Plato has a similar theory regarding eyesight. To Aristotle, the senses were a source of knowledge. The foundations of geometry rested on sensory experience. Of course Plato disagrees. The purpose of eyesight is just like that of the intestines: it’s just a physical crutch whose ultimate goal is to support pure philosophy. Here’s how Plato puts it: “Our ability to see the periods of day and night, of months and of years, of equinoxes and solstices, has led to the invention of number and has given us the idea of time and opened the path to inquiry into the nature of the universe. These pursuits have given us philosophy, a gift from the gods to the mortal race whose value neither has been nor ever will be surpassed. I’m quite prepared to declare this to be the supreme good our eyesight offers us.” So eyesight is not a good in itself, but merely a stepping-stone toward philosophy. It’s a kind of necessary evil, like the intestines. It would be better if we didn’t have to eat at all, but given that we live in this feeble physical world, the best we can do is to make the food take a long time to go through us so we have as much time as possible to think in between meals. In the same way, ideally, we wouldn’t need eyesight. Ideally, we would do pure philosophy, which transcends feeble physical reality. But we are stuck in physical form and with imperfect minds. So we need these support mechanisms to push us toward philosophy. Eyesight leads to astronomy which leads to mathematics and thus philosophy, and then we’re in business. It would have been better if we could have skipped those preliminary steps and gone straight to philosophy. Then eyesight would have been redundant. Eyesight isn’t actually needed for true philosophy. We only need it because of our imperfections. We need this little push to get us started on philosophy, but once we’re up and running with philosophy we can pretty much poke our eyes out because they’re not needed anymore. In this passage, Plato was talking about astronomy but he could just as well have said the same thing for geometry. This is how we must think about the role of geometrical diagrams and sensory perceptions in Plato’s philosophy of mathematics. True mathematics is independent of all that physical stuff, according to Plato. Geometry is not based on physical and sensory experiences with moving figures, drawing lines, and so on, as Aristotle claimed. Diagrams and reliance on the senses are only a stepping stone to true geometry. We need this crutch because our minds and bodies are feeble and imperfect. But once we’ve reached the philosophical level of doing geometry, we can kick away this ladder because then it serves no purpose anymore. Here’s another colorful image Plato has for this. He’s explaining why birds exist. “[Birds] descended from simpleminded men––men who studied the heavenly bodies but in their naiveté believed that the most reliable proofs concerning them could be based upon visual observation.” And conversely, “land animals came from men who had no tincture of philosophy and who made no study of the heavens whatsoever. As a consequence they carried their forelimbs and their heads dragging toward the ground.” So the philosophising human is the perfect balance between these poles: not focused on worldly gratification like the beasts, but also not making the mistake of trying to understand thing by looking. The birds thought that the best way to understand the stars was to get as close as possible to get a good look. But humans know better. We understand that the best way to understand the stars is by thinking, by philosophising, not looking. Once again the same can be said for geometry. Too much looking and not enough thinking: that is the cardinal sin that we must avoid not only in astronomy but in geometry as well. This also fits well with another work by Plato, the Meno. In this work, Plato shows how an ordinary uneducated slave boy can be led to recognize geometric truths, such as a special case of the Pythagorean Theorem. Socrates draws a simple diagram and asks some simple questions, and step by step the boy fills in the reasoning and arrives at the theorem. Plato interprets this as a sort of awakening. Learning is a form of recollection, he claims. That is to say, the boy did not reach this geometric insight through instruction, or through empirical investigation dependent on the senses. Rather, the boy realized that he knew something that he didn’t know that he knew, so to speak. His inner philosopher was awakened. External input was the trigger for this awakening, but the knowledge had really been there all along. The senses are just a trigger for reawakening this knowledge, not an actual basis for that knowledge. This story sums up the role of the senses in geometry, according to Plato. So what does this mean for the axioms of Euclid? What kinds of things do the axioms of geometry need to be to conform with Plato’s vision of geometry as this kind of pure philosophy, a work purely of the mind? I think it comes down to a kind of innateness theory of axioms. The axioms of geometry need to be essentially pre-programmed into our minds. This fits with the idea that learning is recollection, and that mathematics is merely making the mind conscious of things it didn’t know that it knew. There is no external source of this knowledge, according to Plato. The mind just knows it, within itself. So axioms should be intuitive, instinctive. You should read them and you should go: of course! They should feel like the most natural and undoubtable thing in the world. That’s what Plato’s theory suggests. Proclus of course agrees. He’s Plato’s mouthpiece, and here’s what he says about axioms: “axioms take for granted things that are immediately evident to our knowledge and easily grasped by our untaught understanding”; “[axioms] must always be superior to their consequences in being simpler, indemonstrable, and evident in themselves.” That’s almost exactly what Aristotle said. So Plato and Aristotle arrive at the same view of axioms despite their very different outlooks. They disagree on the ultimate origin and foundation of this knowledge: whether it comes from sensory experience and the external world, or whether it comes purely from within our philosophical faculties. This opposition is famously captured in the the iconic fresco The School of Athens painted by Raphael. Plato is pointing to the sky, Aristotle is pointing straight ahead. They are basically pointing to where they think knowledge comes from. Aristotle thinks the source of knowledge is the world before our noses. Plato thinks knowledge resides in a higher realm, above the physical. But despite this orthogonal disagreement, Plato and Aristotle agree on the properties that axioms must have. Axioms need to be the simplest and most obvious first truths. Do you agree with them? No, you don’t. You don’t think axioms need to be obvious and intuitive. Either that, or else you think Newtonian physics is a hoax. Newtonian physics is an example of an axiomatic theory where the axioms are completely non-intuitive. In fact, they are very strongly counter-intuitive. The basic axiom of Newtonian physics is the law of universal gravitation. Any rock is pulling on any other rock, even if they are separated by thousands of miles of empty space. That’s just sheer witchcraft. In fact, you yourself is in a direct bond with all the universe through this mysterious force. It’s like something straight out of science fiction or new age spirituality. Every last one of the thousand stars in the night sky is actively and directly exerting a force on you at any given moment. That’s crazier than any occult astrology you’ve ever heard. Yet that’s Newtonian physics, the most successful scientific theory of all time. In fact, this example of Newtonian physics corresponds precisely to a kind of blind spot that we should have seen coming in our discussion of axiomatic philosophy. On the one hand we said axioms should be obvious, simple truths, but on the other hand we said axioms are what you are left with after you start with theorems like the Pythagorean Theorem and reduce and reduce and reduce. Those are two different ideals. And they are not necessarily compatible. The idea of reducing complex theorems into smaller part does not entail that the axioms you end up with are obvious truths. Axioms are just whatever results when you reduce many theorems to a few core principles. This process could be seen as agnostic as to the nature of the axioms. We just follow the reductive process where it takes us. Just like a chemist cannot decide in advance what kinds of elements he wants the period table to contain, so also the mathematician reducing geometry to its building blocks has to keep and open mind and follow the reductive process where it takes him. At least that’s how Newton interpreted the geometrical method. He’s very clear about this. He’s very explicit about this reductive process being the same in physics as in geometry. Geometry starts with things like the Pythagorean Theorem; physics starts with things like the speeds of the planets and so on. These are the “phenomena” as Newton calls them. And from the phenomena you reason backwards to the underlying causes or unifying principles. That’s what you do in geometry when you show how many theorems can be reduced to a few key principles, and that’s what you do in physics when you show that lots of astronomical data can be derived from a few laws. Newton is adamant that these two things are the same. “As in mathematics, so in natural philosophy,” he says. “Natural philosophy” means physics. The two are the same, in terms of methodology. That’s how Newton justifies his radical physics. By saying that it’s nothing but what the geometers had been doing all along. To make this shoe fit, Newton has to sacrifice the idea that axioms are obvious truths, as Aristotle and Plato had claimed. But his interpretation is not crazy. You could read Euclid that way. You could say: Euclid doesn’t care whether the axioms are obvious or not. He just follows the reductive process where it leads. He’s agnostic or open-minded about what kinds of axioms will be the outcome of this process. Of course that clashes with what Plato and Aristotle said, but they are philosophers so it doesn’t really matter. The important thing is what the mathematicians thought, and their texts are ambiguous enough to allow for the possibility of Newton’s interpretation. So Newton interprets Euclid a certain way in order to justify his own methodology. Newton’s interpretation is hardly very likely, but it’s also not provably wrong exactly. He’s a clever guy, Newton. He knows his physics is crazy and occult, so he massages an interpretation of the Euclidean tradition to legitimate it. I don’t think Newton was right in the way he interpreted Euclid. But his perspective is very illuminating nonetheless. For one thing it’s striking that Euclid’s geometry was so authoritative still 2000 years after it was written that cutting-edge modern science was justified on the grounds that its method was the same as that of Euclid. There was no more solid pillar of respectability than Euclid, to anchor your theory to. Even then, 2000 years after the Elements was written. Euclid’s city, Alexandria, had burned any number of times, and seen several new religions come and go. But the impact of the geometrical method was above such transient circumstances. But even just for understanding Greek philosophy of geometry in itself the Newtonian example is useful. Greek philosophers seem to have been blissfully unaware of the possibility of such a theory, where the reductive process leads to non-obvious axioms. In fact, in Aristotle’s Posterior Analytics there is a phrase that pretty much sums this up. Here’s what Aristotle says: ”I call the same things principles and primitives.” Principles are the logical starting points of a deductive system, and primitives are the immediately given truths grounded in perception. Aristotle thinks you might as well regard these as synonyms, apparently. He does not serious consider the possibility of viable scientific theory in which these two concepts would not align. But Newtonian physics is such a theory. It has principles that are not primitives. That is to say, it has axioms obtained by reducing the phenomena down to their smallest parts, but those axioms are not obvious and not intuitive and not known by direct experience. So the Greeks could have their cake and eat it too. They could have the idea of “reasoning backwards”–of reducing geometry to a few core principles–and at the same time maintain that these core principles should conform to various predetermined philosophical requirements as well, such as being obvious. Newtonian physics shows that you can’t always have it both ways. At a certain point, you have to pick sides. So you have to decide which of the two you’d rather sacrifice. Newton picked the brave side, I think. The path less travelled. He sacrificed the idea that axioms should be intuitive. A huge sacrifice, almost unthinkable. It’s like a military general sacrificing 90% of his troops in an audacious manoeuvre. Few people would have dared to even contemplate such a move. But it worked. Even though it was a huge sacrifice, it got Newton into such a strong position that he won the war anyway. Many people at the time thought Newton was crazy for making this sacrifice. He got a lot of pushback for this. Reduction to non-obvious axioms?! It’s such a radical idea. It goes against everything Plato and Aristotle said. But in a way Newton’s idea is already contained in Euclid. It’s the idea of reading Euclid backwards. Newton’s perspective may not have been Euclid’s exactly, but it’s useful to keep the example of Newtonian physics in mind to highlight what’s at stake in this tension between the “backwards” and “forwards” directions of reading Euclid.
36 minutes | 3 months ago
Consequentia mirabilis: the dream of reduction to logic
Euclid’s Elements, read backwards, reduces complex truths to simpler ones, such as the Pythagorean Theorem to the parallelogram area theorem, and that in turn to triangle congruence. How far can this reductive process be taken, and what should be its ultimate goals? Some have advocated that the axiomatic-deductive program in mathematics is best seen in purely logical terms, but this perspective leaves some fundamental challenges unresolved. Transcript Here’s a way to think about one of the key ideas involved in Euclid’s proof of the Pythagorean Theorem. Picture a stack of books sitting on your desk. It has the shape of a rectangle. Let’s say you’re looking at the side with the spines of the books; they make a rectangle. Now, give the stack of books a whack with your hand. So the pile is knocked askew. The shape of the stack is now a parallelogram instead of a rectangle. But the area is the same. I mean the area of the side facing toward you, the side with the spines of the books. It’s obviously the same area because it’s made up of the same books as before. You just moved the books around. You moved the same amount of area into a new configuration. Also the height is the same: the height from the desk to the top of the pile. This is still equal to the sum of the thicknesses of each book. This illustrates the geometrical theorem that the area of a parallelogram is equal to the area of a rectangle with the same base and height. This is Euclid’s Proposition 35. This is a key ingredient in Euclid’s proof of the Pythagorean Theorem. To prove the Pythagorean Theorem we need to show that the area of the squares on the sides is equal to the area of the square on the hypothenuse. We do this by starting with one of the small squares on the sides and showing that its area can be remolded and made to fit into the big square in such a way the theorem becomes clear. So the idea of Euclid’s proof is to transform one area into another. Its shape is transformed but the area remains the same. And the transformation he uses is basically this one with the stack of books knocked over into a parallelogram shape. Euclid starts with a stack of books corresponding to one of the small squares. He knocks it over into a parallelogram shape. He rotates the parallelogram by 90 degrees so it’s now aligned with the big square instead. And he straightens the parallelogram back out again, just like you would straighten out a stack of books. This is how he shows the equality of areas that the Pythagorean Theorem asserts. The book analogy is not perfect because Euclid so to speak slices his stack of books two different ways. If we want to think of his first step, transforming a square into a parallelogram, in terms of a book stack, then we must visualise the spines to go a particular way. Then when Euclid is straightening the parallelogram back out later, if we want to visualise that in terms of books, we need to picture the spines of the books differently, sitting in another direction. It’s a different stack of books, so to speak. Different but equal. If you have Euclid’s text in front of you, you can draw this into the diagram, how the books need to be oriented for each step to work, and you will see clearly that you have to change perspective halfway through. Euclid is talking about triangles instead of rectangles and parallelograms but that doesn’t matter, the principle is the same. So we are continuing our adventure of reading Euclid backwards. We reduced the Pythagorean Theorem to a more basic proposition, the book stack proposition, 35. What does that in turn depend on? Remember that we are trying to boil everything down to its molecular components. How does Euclid prove Proposition 35? That is to say, how does he reduce this this proposition to more basic ones? I should clarify that Euclid doesn’t do anything like this stuff with the books. I explained this theorem with this analogy to a stack of books, but certainly Euclid’s logic doesn’t depend on anything like that. That would be much too informal. The books need to be “infinitely thin” for the argument to work perfectly, and that’s a whole can of worms foundationally that Euclid certainly doesn’t want to go in to. Instead he offers a purely finitistic proof. Euclid’s proof of Proposition 35 is very clear and satisfying. Euclid proves that one area is equal to another by adding and subtracting pieces in a clever way. So he decomposes it into a couple of puzzle pieces that fit just right with each other. Even though the two areas as wholes have entirely different shape, Euclid shows that there is a clever way of cutting the situation into puzzle pieces that are equally suited to each area. The two areas are two parallelograms of different shape; they’re like two different languages so to speak. You would have thought that they couldn’t communicate very easily. But these puzzle pieces establish a common understanding; something that is equally natural and understandable in either language. So these puzzle pieces, this universal language, can be used to translate one area into the other. If we think in terms of reducing the truth of the theorem to more basic facts, this means that, with the puzzle pieces, we have basically reduced the equality of the entire areas to the equality of the each corresponding puzzle piece separately. The puzzle pieces are all triangles, and the fact that corresponding ones are equal comes down to triangle congruence theorems. That is to say: Under what conditions are two triangles the same? For example, they are the same if the have side-angle-side in common. That turns out to be the next step down if we keep reducing the Pythagorean Theorem. Like a French chef simmers a sauce to make it thicker, so we keep boiling the Pythagorean Theorem, and now we’re down to this. Triangle congruence, and some stuff about parallels as well. We have to keep reading Euclid to find out what happens if you keep cooking it. But before we keep wilting down the Pythagorean Theorem on the Bunsen burner to see what it’s made of, let’s take a moment to reflect on this theorem about the stack of books, or the areas of parallelograms. Proclus has an entertaining remark about this theorem in his ancient commentary on the Elements. He points out that it shows that the same area can have many different perimeters. The stack of books, if you make it more askew you will increase the perimeter while keeping the area the same. A very stretched-out parallelogram has a lot of perimeter but not a lot of area. According to Proclus, military commanders in antiquity did not understand this, with detrimental consequences. Suppose an enemy army is advancing toward your borders. You want to know how many they are. So you send a spy in the cover of darkness at night to scout the situation. The spy sneaks up on the enemy’s night camp and stealthily walks around it, counting the number of step. He then rides back and reports this number. So the number of steps around the camp is taken to be a measure of its size. For cities as well you could do this: How big is the city? Just walk around the city walls and count the steps. It’s so-and-so many steps big. Of course this is a mathematical mistake, because it measures the perimeter when you really wanted to know the area. And the stack-of-books theorem shows that they are not at all the same. Anyway, that’s just a fun story. All the propositions of Euclid have some cultural significance like this. It’s like you see sometimes the period table of chemistry and for each element they’ve added a little example of some familiar real-world thing where this element occurs. “You know kids, lithium isn’t just some weird science thing, you use it every day!” It’s in whatever, toothpaste or something. So you can do that with Euclid’s Elements as well. A little story for each theorem to lighten the mood and make things a bit more culturally relevant. But that’s just for kicks and giggles. Let’s get back to the more scientific purpose: the systematic reduction of all geometrical knowledge to some sort of ultimate minimum foundation. We are just a few steps in to this process and it’s already starting to raise some philosophical conundrums. It was natural enough to take apart the Pythagorean Theorem into more basic results, like the one about areas of parallelograms. Then that in turn could be reduced to triangle congruence. But this can’t go on forever. And we’re already down to such basic facts that it’s becoming very difficult to see how there could be anything “more basic” to reduce them to. This path of reduction, it looked so natural when we set out on it. Starting from the Pythagorean Theorem, this seemed like an obvious way to go. But our clear path through the woods is now becoming darker and thornier. It’s no longer clear where to go from here. Instead of blindly forging ahead in the same direction, we need to take a step back and think about where it is we want to go. What kinds of things should the foundations of geometry be? There are in fact a number of possible answers to this that are very different and completely incompatible with each other, yet each of them is quite plausible in their own right. Let’s have a look at some of the main ones. I mean philosophical views of the status of axioms, or starting points, in mathematics. Or what pretty much comes to the same thing: philosophical interpretation of the ultimate nature of mathematical reasoning and the source of its credibility. Do you think mathematics is ultimately empirical, like physics? Is geometry just the science of physical space? If so, that suggests that the axioms of geometry should be the most fundamental and testable things from an empirical point of view. Geometry should start from things you can check in the field or in a lab. Measuring things with rulers, for instance. That should be the starting point of geometry if you think the certainty of geometrical reasoning ultimately derives from sensory experience and data collected from the world around you. Or do you think mathematics is ultimately pure reason? Then the axioms don’t need to be physically testable but rather mentally fundamental. That suggests that goal of the reductive process is to boil theorems down to the most obvious or intuitively undoubtable starting points. This divide between empiricism and pure reason is mirrored in Aristotle and Plato, one might argue. We will look into that in more depth another time. Let’s focus now on yet another point of view: That of logic. There are two ways you can say mathematics is pure reason: One associates reason with the human mind. Intuition, aha-moments. Those are mental experiences, maybe to some extent subjective experiences. Another characterisation of pure reason is logic. This envisions the laws of reason as detached from human considerations, such as the mind and its subjective experiences. Instead it tries to give a purely objective account of reasoning. Suppose we try to argue that mathematics is basically logic. So it’s not based on anything contaminated by humanness, such as the senses or the mind. Instead mathematical truths are simply necessary truths in some absolute sense. Their truth follow from absolute laws of reason that are some kind of abstract truths more fundamental than human experience or physical reality. This point of view doesn’t really impose any evident restrictions on what kinds of things the axioms of mathematics should be. The starting points of mathematics do not need to be physically measurable, nor intuitively obvious, and so on. Logic does not imply such prescriptions, like the other views did. Mathematicians just deduce consequences of definitions and axioms. Mathematics doesn’t care what the axioms are. From this point of view, mathematics doesn’t make any claim to establishing absolute truths. All of mathematics is just “if ... then ...” statements. If these axioms are true, then these theorems follow. The axioms themselves, then, can be pretty much arbitrary for all the mathematician cares. This is a very modern view. Modern mathematicians pretty much accept this. It’s certainly a very convenient view for the mathematician. It’s a sort of abdication of responsibility. What is a philosophy of mathematics supposed to do? What is it for? Surely it should explain the obvious facts about mathematical reasoning, such as that it somehow establishes seemingly absolute truths. When we read a proof such as Euclid’s proof of the Pythagorean Theorem or the parallelogram area theorem, the proof is so compelling. It gives us complete conviction that the theorem must be true. It’s unlike anything we ever see in other domains. There are no such absolutely compelling and irrefutable proofs in politics or ethics. Why not? What’s so special about mathematics? History reinforces the point. Every last one of Euclid’s theorems are as true today as they were when they were written well over two thousand years ago. Every civilisation accepts these universal truths. Why does this happen only in mathematics? A philosophy of mathematics should answer these questions. But the logic interpretation of mathematics does not. It doesn’t pinpoint any particular characteristic of geometrical reasoning that explains why it should be so unique in these regards. It doesn’t explain why the particular axioms of geometry that Euclid investigated were universally accepted in so many contexts, and turned out to be so uniquely suited to describe the physical world in all kinds of scientific advances that the Greeks had not even dream of yet. So in this way the logic philosophy of mathematics is perhaps a kind of coward’s philosophy. It’s a non-philosophy, as far as many key questions are concerned. It just doesn’t have any kind of answer to the major questions that other philosophies of mathematics sees it as their duty to address. There’s a famous essay called “The unreasonable effectiveness of mathematics in the natural sciences.” Famous physicist Eugene Wigner said this in 1960. Everybody cites it all the time. But ask yourself: Why did no one say this until 1960? Did the effectiveness of mathematics somehow become unreasonable only then? Of course not. The effectiveness of mathematics in the natural sciences had been around forever. Including the effectiveness of ideas that were first developed for purely mathematical reasons but later proved to have hugely important and completely unforeseen scientific applications. For instance, the Greeks studies ellipses in great mathematical detail, and then two thousand years later it turned out, completely unexpectedly, that planetary orbits are ellipses. So this purely geometric topic became hugely important in science, which no one had predicted. Why didn’t people say then: the effectiveness of mathematics is unreasonable? Why would it take all the way to 1960 before anyone drew this obvious conclusion? I’ll tell you why. Because the conclusion that the effectiveness of mathematics is unreasonable only follows if one assumes the logic interpretation of mathematics. If mathematics is nothing but logical inferences from arbitrary axioms, then sure enough it’s a complete mystery, it’s completely unreasonable that mathematics can work so well. But what people used to conclude from this is that it is the logic conception of mathematics that must be unreasonable. It is unreasonable to think that mathematics is nothing but logical deductions. Because that completely fails to explain so much of what we know about mathematics. In 1960 the logic conception of mathematics had become the modern dogma that it remain to this day. It had become so ingrained in the mathematical psyche that mathematicians could no longer even conceive of rejecting it. Then they had no choice but to declare the effectiveness of mathematics in physics to be unreasonable. That’s why Wigner’s famous phrase is from 1960 and not 450 BC. It’s not a fact that effectiveness of mathematics is unreasonable. Rather, one of two things is unreasonable: either the effectiveness of mathematics is unreasonable, or the conception of mathematics as nothing but logic is unreasonable. For thousands of years people preferred to conclude from this that there must be more to mathematics than just logic. Euclid is not just “the axiomatic-deductive method.” This can’t be the whole picture. The axioms must be somehow more than arbitrary. What makes the axioms true? Logic itself doesn’t care and cannot help us with this question. So we need something more than logic in our philosophy of mathematics. So I claim that only in very modern times did the logic conception become the norm. Maybe in some future episode I will discuss what circumstances made that come about. The important thing for our present purposes, as we read Euclid, is to understand that with the reduction process that we have begun, that consists of breaking down theorems into smaller and smaller pieces, the end pieces, the ultimate rock-bottom pieces, need to have some sort of claim to credibility. They cannot simply be whatever you’re left with when you keep reducing and reducing. Or can they? I say everyone rejected that view, but I could play devil’s advocate. Listen for example to this fragment from Eudemus’ Physics: “As for the principles they talk about, mathematicians do not attempt to demonstrate them, they even claim that it is not their business to consider them, but, having reached agreement about them, they prove what follows from them.” This is a bit of a disturbing quote, in my opinion. It seems to almost assert that logic view that I said was regarded as unacceptable at that time. Mathematicians only prove what follows from axioms, and they claim that “it is not their business” to worry about the status or truth of those axioms. Sounds strangely modern, just the view I assigned to the 20th century. I think that’s not really what the quote says for various reasons. In part what Eudemus is saying is that the justification of the “principles” (that is to say the axioms) shouldn’t be regarded as part of mathematics but rather part of some other field, some more philosophical domain. But whatever, that’s just putting labels on things. That still means that the axioms are to be justified some way. So they are not arbitrary. The justification is “philosophy” rather than “mathematics”—sure, whatever, call it what you want, but it’s in any case very different from not justifying or being concerned with the nature of the axioms at all. The quote also said, if you noticed, that the mathematicians don’t care about the axioms, “having reached agreement about them.” What does that entail? On what basis did mathematicians reach such an “agreement”? This opens the door for all kinds of considerations of the status and nature of the axioms within mathematics, even according to this quote, the devil’s advocate quote. So I think it’s safe to say that the logic view by itself was not satisfactory. The starting points, or axioms, of mathematics need to have some kind of justification. In fact, there is one way in which logic itself can provide such a justification. So the problem we need to solve is this. We started with the Pythagorean Theorem, we reduced it to more basic statements, then those to more basic ones, and so on. Where do we stop this process? Do we stop when we just don’t see how to go any further? This is what I just criticised as untenable. Because this would mean declaring whatever we’re left with to be axioms, without convincing criteria of justification for which kind of things should be allowed to be axioms and which not. The axioms can’t just be arbitrary because then we can’t explain the successes of mathematics. One hope of some logicians has been that everything could be reduced to definitions. There are no axioms! Everything is at bottom just definitions. The meaning of words. Mathematics is about drawing out consequences contained in the definitions of concepts, without any assumptions being made. That would be great for the logician and some people have tried to fit geometry into such a mold. But it doesn’t work. Geometry needs assumptions, genuine axioms. You can’t get away with only definitions. You can’t reduce mathematics to a purely linguistic game. And besides, even if you could, what would be the guarantee that the definition corresponded to anything? That the entities defined actually exist? And that the definitions are not self-contradictory or inconsistent? Definitions alone cannot carry this burden of justification. You need something more. But there’s one more ace up the logician’s sleeve, and it’s a pretty clever one. There are statements that are logically self-justifying. Statements such that, if you try to deny them, you have actually committed yourself to accepting them. An example is the famous statement by Descartes: I think, therefore I am. How could you deny such a thing? What would you say if you wanted to deny it? “No, I don’t think that.” Or: “I think that’s wrong.” As you can hear, you walked right into the trap. By trying to deny that you are a thinking being, you made statements that actually presuppose that you are thinking being. The denial is self-defeating. You can’t deny the statement without actually implicitly conceding it. Such statements are justified by “consequentia mirabilis,” as it’s called. There’s an argument of this form already in an Aristotelian fragment. Aristotle uses it to prove the proposition: We ought to philosophise. Try to deny it. So you say: No, we should not philosophise. Well, in that case, it would be important to reach the conclusion that we should not philosophise. Reasoning our way to this conclusion would spare us from the mistake of philosophising. Then we could do more important things with our time instead of philosophising. But now we are caught in a trap again. We wanted to establish that we shouldn’t philosophise, but in trying to argue this we actually committed ourselves to the position that we should philosophise, namely we should philosophise in order to establish the conclusion that we shouldn’t philosophise. So once again the attempted rejection of the proposition actually implies acceptance of the proposition. Could it be that all the axioms of mathematics could be of this type? That would be a logician’s dream. That would be a great way of justifying ending the chain of reductions of theorems to lower and lower constituent parts. We have to keep reducing until you’re left with nothing but logically self-justifying statements. Consequentia mirabilis axioms only, which must be accepted as true because it is logically incoherent to try to deny them. This view had its adherents. Clavius was fond of the consequentia mirabilis. Clavius was influential in discussions of Euclid around 1600; he was the editor of the standard Latin version of Euclid that everybody used. Even Saccheri, who did some very sophisticated work on the foundations of geometry in the 18th century, was keen on trying to reduce the foundations of geometry to consequentia mirabilis. So this idea was clearly seen as very attractive. People really tried to make it work. But ultimately it failed. It was an approach based more on what the logician wanted than on what mathematics is really like and how mathematics wants to be understood. So altogether, the reduction of mathematics to logic is an idea that has had great appeal to many. Several times in history, a complete reduction of mathematics to logic has seemed within reach, only for the quest to end in bitter disappointment. This is also what happened with Frege and Russell, Hilbert and Gödel, and so on, centuries later. Bertrand Russell put it in interesting terms. Here’s what he says in his autobiography: “I wanted certainty in the kind of way in which people want religious faith.” He’s talking about his early career, around 1900. At this time he worked on an enormously ambitious project to reduce all of mathematics to logic. It didn’t work. As Russell himself says: “After some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.” Russell’s case is quite typical, one might argue. Others have had the same experience when they have tried to achieve the same goal. It’s a great temptation: one logic to rule them all; “my precious.” Many have been seduced by that idea, and spent twenty years obsessed with it only to fail, as Russell did. Let’s look at the most famous of the problems Russell ran in to: the so-called Russell’s Paradox. A popularised version of Russell’s Paradox goes like this. A barber shaves everyone who do not shave themselves. Who shaves the barber? There is no coherent answer. The barber cannot shave himself because he only shaves those who do not shave themselves. But he also could not not shave himself. Because if he didn’t shave himself he would by definition be one of the people he does shave, which is everybody who do not shave themselves. So either way leads to a contradiction. Mathematicians unknowingly allowed this type of paradox to enter their logical systems. This stuff about the barber is just a translation into everyday terms of something that first occurred within mathematics itself. Russell thought this problem was fixable. But others thought it was a comeuppance for logic that was both deserved and bound to happen. Consider for example the reply by Brouwer, an influential but eccentric mathematician in the early 20th century. Here’s what he says: “Exactly because Russell’s logic is no more than a linguistic system, there is no reason why no contradictions would appear.” That is to say, since logic is divorced from meaning, divorced from the real world, why wouldn’t it be inconsistent and self-contradictory? History shows that inconsistencies can very easily creep into formal axiomatic systems, against the best efforts of even top mathematicians devoted specifically to building rigorous and coherent foundations. A long list of leading logicians have published systems of logic which turned out to be inconsistent. According to Brouwer: “The language of Euclidean geometry is reliable only because the mathematical systems and relations, which are symbolized by the words of that language as conventional signs, have been constructed beforehand independently of that language.” That is to say, it is precisely because it is not merely logic that Euclidean geometry is so reliable. It is anchored in the real world, and the physical world has a much better track record of being consistent than the thought-constructs of logicians. Emil Post was another rebel at that time who likewise called for “a reversal of the entire axiomatic trend of the late 19th and early 20th centuries, with a return to meaning,” as he put it. Logic had gone too far. Some formalisation and logic are powerful tools in mathematics. But you can take it so far that mathematical theories lose all bond with reality and meaning. Then there is no grounding anymore to protect you from contradiction and inconsistency. Logic is “the hygiene which the mathematician practices to keep his ideas healthy and strong,” said Hermann Weyl, another contemporary of these guys. But, like hygiene, you can overdo it. Some hygiene is much better than none, of course, but obsessive hygiene can undermine the natural state of the body and the immune system. Maybe logic is like that. It’s like cleaning everything away with bleach all the time. It’s good to clean, but if you overdo it you eventually clean away the very thing you were trying to protect. There were big debates about such questions in the early 20th century; the people I quoted were all part of those heated debates about logic. But that’s a story for another day. For our purposes, we are interested specifically in logic-centric attempts at interpreting Euclid, and accounting for the success of Greek geometry. Indeed, such logic-centered interpretations have been sought eagerly. They are very agreeable for some purposes; they have an almost religious appeal, as Russell said. But ultimately there are severe limitations inherent in such views, which have meant that most people from antiquity to early modern times have felt that some additional ingredient, beyond mere logic, is needed for a successful philosophy of mathematics. And as we read Euclid backwards, the closer we get to the beginning, the more essential it becomes for us to make up our minds about our philosophy of mathematics. Any moment now we have reached all the way down to the axioms and then push comes to shove. We’re going to have to take a stand and say: this is why we stop at these particular axioms and why you should believe them. Let’s keep reading Euclid and see how we can answer this challenge.
42 minutes | 4 months ago
Read Euclid backwards: history and purpose of Pythagorean Theorem
The Pythagorean Theorem might have been used in antiquity to build the pyramids, dig tunnels through mountains, and predict eclipse durations, it has been said. But maybe the main interest in the theorem was always more theoretical. Euclid’s proof of the Pythagorean Theorem is perhaps best thought of not as establishing the truth of the theorem but as breaking the truth of the theorem apart into its constituent parts to analyse what makes it tick. Euclid’s Elements as a whole can be read in this way, as a project of epistemological analysis. Transcript Let’s read Euclid together. Euclid’s Elements, one of the most important and influential works in human history, who wouldn’t want to read that? “Euclid alone has looked on beauty bare,” as the poets say. Let’s do some episodes on this where we go through Euclid’s Elements Book I. And here’s the first twist: Let’s read it backwards. Well, not quite. But it’s a good idea to start at the end. Book I of the Elements ends with the Pythagorean Theorem and its converse. It’s not a murder mystery, it won’t spoil the fun to know the ending. I will explain why I think this is a good idea. This has to do with appreciating the refined goals of the Elements. It’s a very subtle work, in ways that are easy to miss. So I will use this idea of starting at the end as a way of highlighting some things to keep in mind in that regard, so that we approach the text with appreciation of these subtleties. It might be a bit dry to do only that, so I will also mix it up with some lighter things. Some stories related to the Pythagorean Theorem. Did the Egyptians use the Pythagorean Theorem to build the pyraminds, for example? Is that how they got the angles just right? We will discuss that soon. And I will also play a clip of RoboCop. I will try to do this for the Elements as a whole: a serious discussion of its finer points, as well as some entertaining tangents exploring the many cultural links of the various parts of the Elements. So here we go. My first goal is to outline the mindset with which we must approach Euclid’s text. If you’re a young person, you may look at Euclid’s Elements and say: yeah yeah, triangles and stuff, I saw all of that in high school too; our textbook had proofs just like this thing by Euclid; it’s pretty much the same thing. No, no, no. That’s like listening to Mozart and saying: yeah yeah, big deal, music is music. Forget it. There’s a world of difference. Euclid is on a whole other level of sophistication than some crappy high school textbook. You wouldn’t know it just by looking at the text though. The text looks the same as any other geometry text. Triangle ABC blah blah blah. It’s the same with musical scores, isn’t it? They all look the same when you just glance at the pages. You can’t tell Mozart from some hack. We must look deeper to appreciate the subtlety and genius of Euclid. The text itself doesn’t spell that out, just as a Mozart quartet doesn’t have a narrator telling you what’s great about it. But great works reward reflection. The more you study Euclid, the more you interrogate the text, the more you puzzle over its oddities, the more you come to appreciate the mastery that went into crafting everything just right. Euclid knew exactly what he was doing. His work is orders of magnitude more sophisticated than other superficially similar works in the same genre. The exercise of reading backwards is one angle we can use to start getting a handle on this. If we read Euclid from cover to cover, in the order it’s written, we get a strictly “bottom-up” perspective: we start with the most basic things and gradually get to higher and higher levels of sophistication. That’s how mathematics is typically written down. And with good reason. But the way mathematics comes into being is much more bidirectional. Mathematics grows like a tree: as the branches extend, so do the roots. Starting our Euclid adventure with the Pythagorean Theorem is a way of making us think about this. Of course when we read Euclid’s proof of the Pythagorean Theorem we find that it is based on earlier results. So you might say: Obviously you have to read those first before you can understand this proof. But that’s a bit simplistic. You could also say: Actually you need to look at the Pythagorean Theorem first because only then can you understand what the purpose is of those earlier propositions. From a purely logical perspective you have to read it linearly from start to finish, but to understand the meaning and purpose of these logical constructions you have to take a step back and interrogate the text from other angles as well. For a dogmatic understanding, it is enough to read it linearly, and parse the logical steps like a machine. But for a critical, independent understanding you want to not only verify the logic but also see how one could arrive at such logical constructions organically. That goes for any formal mathematics text, still to this day. Or maybe even more so today than ever. The definitions and axioms are the starting points of the way mathematics is written, but often they are almost the end product of the actual creative thought process. Only after you have figured out the hard parts of your theory do you know what the starting points need to be. Or at least there’s an interaction, a back-and-forth negotiation between the top and the bottom of the theory. Each is adapted to the other. So that’s one reason to read Euclid backwards. It’s a reason that applies to any formal mathematical theory, because they all have this element of bidirectionality. Actually geometry might be among the more unidirectional formal mathematical theories in how it was conceived, because the results of geometry were known in great detail, long before they were formalised. The tree came before the roots, so to speak. Here’s another way of visualising it. Think of the Pythagorean Theorem as the apex of a pyramid. The proof reveals which lower, more foundational stones it rests on. Those stones in turn rest on other stones, and so on. Something has to be the bedrock that is considered solid enough not to need any further support beneath it. Euclid’s Elements can be read in two directions: as a way of building up a more and more elaborate structure on top of solid foundations, or as a way of reducing advanced results to their basic components. So when we read the proof of the Pythagorean Theorem, one of the perspectives we should use is to think of it as “boiling down” this somewhat advanced result to more basic ones. This will help us appreciate the purpose and achievement of the more fundamental parts of the Elements when we get to those. Indeed, by the time Euclid wrote the Elements, the theorems themselves—such as the Pythagorean Theorem—had been known for hundreds or even thousands of years. Even proving the theorem wasn’t all that new. There were plenty of proofs. I bet Euclid knew two dozen proofs of the Pythagorean Theorem. We shouldn’t think of Euclid as saying: Hey guys, I discovered some things about triangles and stuff; check out this book where I explain how I came up with these theorems. No, no, no. That’s not at all what Euclid is doing. We must understand, when we read the Elements, that we’re way beyond that. If you just wanted to convince a random person that the Pythagorean Theorem is true, then there are much better proofs than Euclid’s. Simpler ones. More intuitive, based on simple diagrams. If all you want is a psychologically compelling argument that the Pythagorean Theorem is true then there are better options than Euclid. Euclid knew all of that, and he chose his proof very deliberately. Because it’s the best proof for his purposes. Namely the purpose of carefully analysing how the truth of the Pythagorean Theorem can be broken down into smaller truths. And more generally to do the same thing for all the truths of geometry in a comprehensive and systematic manner. So the proof of the Pythagorean Theorem isn’t so much about showing that the theorem is true. It’s more about showing what its ultimate foundations are. Here’s another metaphor for this. Think of a mathematical theorem as a dish that you cook. The Pythagorean Theorem is like a soup, let’s say. You can whip it up very quickly with store-bought ingredients like stock cubes or just microwaving something from a can. But Euclid doesn’t do store-bought. He’s going to do everything from scratch. And I mean really from scratch. If there’s going to be carrots in there, then Euclid is going to grow his own carrots. In fact you might say that Euclid is not so interested in cooking at all, even though a proof is like a recipe. Euclid is like a cookbook author who doesn’t like cooking and has no interest in feeding anyone. Instead he’s more like a chemist who is analyzing the molecular composition of foods. His recipes are not meant as a practical cooking guide but as an analysis of what the core ingredients of the dish are if you deconstruct the recipe as far as you possibly can. Here we have the idea of reading backwards again: Euclid isn’t really interested in making Pythagorean Theorem soup, but in starting with Pythagorean Theorem soup and taking it apart in the lab. Put it on the Bunsen burner. Different ingredients have different boiling points and so on, so you can carefully separate them out again. There was already plenty of geometry before Euclid. If theorems are food, everyone was already well fed, so to speak. Everyone already had their favourite dishes and neither they nor Euclid were looking to replace the traditional menus. What Euclid is bringing to the table is not new food but a refined theoretical perspective that stands apart from actual cooking. The idea of reading Euclid backwards is also related to a famous anecdote recorded about Thomas Hobbes, the 17th-century philosopher. Here
40 minutes | 5 months ago
Singing Euclid: the oral character of Greek geometry
Greek geometry is written in a style adapted to oral teaching. Mathematicians memorised theorems the way bards memorised poems. Several oddities about how Euclid’s Elements is written can be explained this way. Transcript Greek geometry is oral geometry. Mathematicians memorised theorems the way bards memorised poems. Euclid’s Elements was almost like a song book or the script of a play: it was something the connoisseur was meant to memorise and internalise word for word. Actually we can see this most clearly in purely technical texts, believe it or not. It is the mathematical details of Euclid's proofs that testify to this cultural practice. That sounds almost paradoxical, but I’m sure I will convince you. The surviving documentation about ancient Greek geometry consists almost entirely of formal treatises. Very stilted and dry texts. Definition, theorem, proof. Pedantically written. Highly standardised, formalised. Completely void of any kind of personality. Where is the flesh and blood, the hopes and dreams, the lived experience of the ancient geometer? It’s as if they were determined to erase any traces of all of those things, and leave only a logical skeleton. But it’s not as hopeless as it seems. At first glance it looks as if these texts have been scrubbed of all humanity. But, in fact, if we read between the lines we can extract quite a bit of information. There are implicit clues in these texts that reveal more than the authors intended. That’s our topic for today: How these seemingly purely logical texts actually say quite a lot about the social context in which they were produced. One thing we learn this way is that we should think of the Greek geometrical tradition as spoken geometry, not written geometry. Today we think of written texts as the primary manifestation of mathematics. When mathematicians disseminate their ideas, the published article is the official, definitive, primary expression of those ideas. The mathematician crafts a written document with the expectation that reading the text on paper is going to be the primary way in which people will access this material. Not so in antiquity. Oral transmission was considered the primary mode of explaining mathematics. Written documents were a last resort when personal contact was not possible. And the written document was not meant to be a primary exposition in its own right. Writing was merely the oral explanation put down on paper (or papyrus, rather). At least it must have been like that in the early days. Many conventions of Greek mathematical writing only make sense from this point of view. They must have been formed in an oral mathematical culture. Probably in later antiquity the situation was not so clear cut. Writing probably gradually became more of a thing in its own right, rather than merely a record of oral exposition. But even then, the conventions of written mathematics remained largely fixed. Greek mathematics never liberated itself from these conventions that had been set in an oral culture. They lived on. Perhaps in part due to tradition and conservatism, but probably also because the oral element remained a significant part of mathematical culture, perhaps especially in teaching. Here’s an example of this, which I have taken from Reviel Netz’s book The Shaping of Deduction in Greek Mathematics. Consider the equation A+B=C+D. Here’s how the Greeks expressed this in writing: THEAANDTHEBTAKENTOGETHERAREEQUALTOTHECANDTHED. This is written as one single string of all-caps letters. No punctuation, no spacing, no indication of where one word stops and the next one begins. A Greek text is basically a tape recording. It records the sounds being spoken. There is a letter of the alphabet for each sound one makes when speaking. The scribe just stenographically puts them down one after the other. From this point of view there is no distinction between upper or lower case letters: a letter just stands for a sound and that’s it. And there is no punctuation or separation of words, because those are not spoken sounds. And of course no mathematical symbols such as plus or equal signs, because that also does not exist in spoken discourse. The only way to understand a text like that is to read it out loud. You have to read it like a child who is just learning to read: you sound it out letter by letter, and then interpret the sounds, rather than interpret the writing directly. So the Greeks had a very limited conception of writing. They thought of writing only as a way of recording speech. They completely missed the opportunities that writing provides when embraced as a primary medium in its own right. Writing is a better way of representing equations, for example, than speech. But the Greeks completely missed that opportunity because they were stuck with the limited notion of writing as merely recorded sounds. I like to compare this with early movies. Think of those classic movies from, say, the 1950s or so. They are basically recorded stage plays. There are limitations inherent in the medium of theatre. The actors have to speak quite loudly, articulately, to be heard by the audience in the back of the theatre. And the scenery on stage cannot easily be changed or moved. In a play you better stick to one or two sets, such as the interior of a room. That you can set up carefully with furniture and all kind of stuff on the walls and so on. But because you can’t change it easily, you have to have to have large parts of the play take place in that single setting. These technical limitations constrain the artistic freedom of the playwright. You have to come up with a story where all the various characters have some reason or other to come and go into a single room, and once there to have loud conversations that drive the plot. All emotional depth and so on must be conveyed in this particular form. These things became second nature to writers. So when film came around they kept doing the same thing even though that was no longer necessary. Many treated film as simply a way of recording plays. So in early movies you still have a lot of these static scenes with a fixed camera at one end of a room, and characters coming and going, having loud conversations. Film affords new artistic possibilities. You are no longer limited to a static camera showing a fixed set, the way the audience of a theatre would be looking through the “fourth wall” of a room. You have many more options to convey things visually, instead of being limited to strongly articulated stage dialogs as the only driver of the plot. But many early movies didn’t take advantage of that. They just kept doing what they had always been doing at the theatre and just recorded that. They saw the new medium of film merely as a way of “bottling” existing practice. It’s just a storage medium. They didn’t consider that the new medium was in some ways better than the old one and enabled you to do completely new things. It was the same with writing in antiquity. Writing was merely for storing speech. They failed to take advantage of the ways in which writing could not only preserve existing cognitive practice but in fact transform it and improve it. Such as working with equations symbolically. Here is another consequence of this: the absence of cross-referencing. If a mathematical text is like a tape recording, you can’t easily access a particular place in the tape. The only way to make sense of the text is to “hit play,” so to speak, and translate it back into sounds. Only then can it be understood. You can “fast forward” and “rewind”—that is to say, start reading at any point in the manuscript. But you can’t turn to a particular place, such as Theorem 8. Modern editions of Euclid’s Elements are full of cross-references. Each step of a proof is justified by a parenthetical reference to a previous theorem or definition or postulate. But that’s inserted by later editors. There is no such thing in the original text. Because it’s a tape recording of a spoken explanation. Referring back to “Theorem 8” is only useful if the audience has a written document in front of them. If they are merely listening to a long lecture, or a tape recording of a lecture, then there is no use referring back to “Theorem 8”, because the audience has no way of going back specifically to that particular place in the exposition. For this reason, oral mathematics involves committing a lot of material to memory. In the arts, people memorise poems and song lyrics. Actors memorise the dialogues of plays. Ancient mathematics was like that as well. You would learn to recite theorems the same way you learn to sing along to your favourite song. This aspect of the oral culture thoroughly shaped the way ancient mathematical texts are written. Euclid’s Elements and many other texts follow a certain stylistic template that at first sight seems quite irrational, but which starts to make sense once we consider the oral context. Consider for example Proposition 4 of Euclid’s Elements. This is the side-angle-side triangle congruence theorem. It’s completely typical, I’m just picking a theorem at random. Let’s look at the text of this proposition. First we have the statement of the theorem in purely verbal terms. It goes like this: “If two triangles have two sides equal to two sides, respectively, and have the angle enclosed by the equal straight lines equal, then they will also have the base equal to the base, and the triangle will be equal to the triangle, and the remaining angles subtended by the equal sides will be equal to the corresponding remaining angles.” Ok, so: two triangles have side-angle-side equal, the it follows that they also have all the other things equal. Namely the remaining side, the remaining angles, and the area. “The triangle will be equal to the triangle,” says Euclid: this is his way of saying that they have equal area. After Euclid has st
42 minutes | 6 months ago
First proofs: Thales and the beginnings of geometry
Proof-oriented geometry began with Thales. The theorems attributed to him encapsulate two modes of doing mathematics, suggesting that the idea of proof could have come from either of two sources: attention to patterns and relations that emerge from explorative construction and play, or the realisation that “obvious” things can be demonstrated using formal definitions and proof by contradiction. Transcript How did proofs begin? It’s like a chicken-or-the-egg conundrum. Why would anyone sit down and say to themselves “I’m gonna prove some theorems today” when nobody had ever done such a thing before? How could that idea enter someone’s mind out of the blue like that? In fact, we kind of know the answer. The Greek tradition tells us who had this lightbulb moment: Thales. Around the year -600 or so. Hundreds of years before we have any direct historical sources for Greek geometry. But we still sort of know what Thales proved, more or less. Later sources tell us about Thales. History is perhaps mixed with legend in those kinds of accounts, but key aspects are likely to be quite reliable. More fact than fiction. Let’s analyse that question, the credibility question, in a bit more depth later, but first let’s take the stories at face value and see how we can relive the creation of deductive geometry as it is conveyed in these Greek histories. So, here we go: What was the first theorem ever proved? What was the spark that started the wildfire of axiomatic-deductive mathematics? The best guess, based on historical evidence, goes like this. That love-at-first-sight moment, that theorem that opened our eyes to the power of mathematical proof, was: That a diameter cuts a circle in half. Pretty disappointing, isn’t it? What a lame theorem. It’s barely even a theorem at all. How can you fall in love with geometry by proving something so trivial and obvious? But don’t despair. It is nice, actually. It’s not about the theorem, it’s about the proof. Here’s how you prove it. Suppose not. This is going to be a proof by contradiction. Suppose the diameter does not divide the circle into two equal halves. Very well, so we have a line going through the midpoint of a circle, and it’s cut into two pieces. And we suppose that those two pieces are not the same. Take one of the pieces and flip it onto the other. Like you fold an omelet or a crepe. The pieces were not equal, we assumed, so when you flip one on top of the other they don’t match up. So there must be some place where one of the two pieces is sticking out beyond the other. Now, draw a radius in that direction, from the midpoint of the circle to the place on the perimeter where the two halves don’t match up. Then one radius is longer than the other. But this means that the thing wasn’t a circle to start with. A circle is a figure that’s equally far away from the midpoint in all directions. That’s what being a circle means. So we have proved that two things are incompatible with one another: You can’t be both a circle, and have mis-matched halves. Because if you have mis-matched halves you also have “unequal radii” and that means you’re not a circle. So a circle must have equal halves. Bam. Theorem. It’s a boring result but a gorgeous proof. Or a suggestive proof. It’s a proof that hints at a new world. Thales must have felt like a wizard who just discovered he had superpowers. “Woah, you can do that?!” By pure reasoning, by drawing out consequences of a definition, one can prove beyond any shadow of a doubt that certain statements could not possibly be wrong? That’s a thing? That’s something one can do? Wow. Let’s do that to everything! Right? So that’s how Thales discovered proof. As best as we can guess. A few other theorems are attributed to Thales as well. I want to bring up one in particular that I think is also a kind of archetype of what mathematics is all about. The theorem we just saw, about the diameter bisecting the circle, perfectly embodies one prototypical mode of mathematical reasoning. The pure mathematics paradigm, you might call it. Logical consequences of definitions, proofs by contradiction. That kind of thing. Thales’s proof really hits the nail on the head with that whole aesthetic. We’ve been doing the same thing over and over ever since. A modern course in, say, group theory, for example, is just Thales’s proof idea applied five hundred times over, basically. Now I want to take another one of the results attributed to Thales, and I want to argue that it is emblematic of another mode of mathematical thought. It’s a second road to proof. This second way is based more on play, exploration, discovery, rather than logic and definitions. The example I want to use to make this point is what is indeed often called simply “Thales’s Theorem.” Which states that any triangle raised on the diameter of a circle has a right angle. So, in other words, picture a circle. Cut it in half with a diameter. Now raise a triangle, using this diameter as one of its sides, and the third vertex of the triangle is on the circle somewhere. So it looks like a kind of tent, sticking up from the diameter. And it could be an asymmetrical tent that is pointed more to one side or the other. No matter how you pitch this tent, as long as the tip of it is any point on the circle, then the angle between the two walls of the tent at that point, at the tip, is going to be a right angle, 90 degrees. That’s Thales’s Theorem. How might Thales have proved this theorem? We don’t really know that based on historical evidence unfortunately. But let’s consider one hypothesis that makes sense contextually. We must imagine that Thales would have stumbled upon the proof somehow. We are not trying to explain how someone might think of a proof of this theorem per se. That’s the wrong perspective because it takes for granted that in mathematics one tries to prove things. What we need to explain is where this vision to prove everything in geometry came from in the first place. How could someone have struck upon Thales’s Theorem unintentionally, as it were, and through that accident become aware of the idea of deductive geometry? Indeed Thales’s Theorem is not terribly interesting or important in itself. If you had this vision of subjecting all of geometry to systematic proofs, why would you start with this theorem, or make this theorem such a center piece, as Thales supposedly did? You wouldn’t. The interesting thing about Thales’s Theorem is not that is was one of the first results to which mathematicians applied deductive proof. Rather, the interesting thing about it is that it was the occasion for mathematicians to stumble upon the very idea of proof itself, unintentionally. There’s a story about Thales falling into a well because he got so caught up in astronomical reasoning that he forgot his surroundings. It’s recorded in Plato: “While he was studying the stars and looking upwards, he fell into a pit. Because he was so eager to know the things in the sky, he could not see what was before him at his very feet.” A legend maybe, but the discovery of Thales’s Theorem must have been a little bit like that too. Discovering mathematical proof must have been like falling into a pit. You are looking in one direction, and boom, suddenly you find yourself having accidentally smashed face first into this completely unrelated new thing that you didn’t know existed. How could Thales’s Theorem be like that? Among all the world’s theorems, what makes Thales’s Theorem particularly conducive to this kind of fortuitous discovery of proof? Here’s my hypothesis. In this age of innocence, before anyone knew anything about proof, people still liked shapes. The had ruler and compass. They used these tools for measuring fields and whatnot, but they also liked the aesthetic of it. They were playing around with ruler and compass. Playing with shapes. After five minutes of playing with a compass you discover how to draw a regular hexagon. Remember? You probably did this as a kid. Draw a circle, and then, without changing the compass opening, run the compass along the circumference. It fits exactly six times. A very pleasing shape. We know for a fact that people did this before Thales. There are hexagonal tiling patterns in Mesopotamian mosaics from as early as about -700. Dodecahedra are another one of those things. The dodecahedron is like those twelve-side dice that you use in Dungeons and Dragons and stuff like that. Do-deca-hedron, it’s literally: two-ten-sided. So twelve-sided, in other words. Twelve faces, each of which is a regular pentagon. These things are in the archeological record. People made them of stone and bronze. A couple of dozen of dodecahedra from antiquity have been found, the oldest ones even predating Thales. They were used perhaps for oracular purposes, like tarot cards or something. Or maybe for board games, who knows? In any case, my point is that people were interested in geometrical designs for various purposes: artistic, cultural, and so on. Not just measuring fields for tax purposes. And they were clearly working with instruments such as ruler and compass to make these things. It’s easy to arrive at Thales’s Theorem by just playing around with ruler and compass, trying to draw pretty things. Start with a rectangle. Draw its diagonals. Put the needle of a compass where they cross, right in the midpoint of the rectangle. Set the pen of the compass to one of the corners of the rectangle. Now spin it. You get a circle that fits perfectly, snugly, around the rectangle. But look what emerged. A diagonal of the rectangle becomes a diameter of the circle. And the rectangle pieces sticking out from it are precisely those kind of “tent” triangles that Thales’s Theorem is talking about. This suddenly makes the theorem obvious. Why is Thales’s Theorem true? Why does any of those
36 minutes | 8 months ago
Societal role of geometry in early civilisations
In ancient Mesopotamia and Egypt, mathematics meant law and order. Specialised mathematical technocrats were deployed to settle conflicts regarding taxes, trade contracts, and inheritance. Mathematics enabled states to develop civil branches of government instead of relying on force and violence. Mathematics enabled complex economies in which people could count on technically competent administration and an objective justice system. Transcript How did geometry start? Who was doing it, and why, in early civilisations? The Greeks invented theorem and proof, but long before them there was geometry in Egypt and Mesopotamia. So that’s practical geometry, applied geometry. Or is it? Actually even the oldest sources have lots of pseudo-applications in them. Such as: Find the sides of a rectangular field if you know the perimeter and the diagonal. Or: I have two fields, and I know how much grain each field produces per unit area, and I know the total grain produced by both of them, and I know the difference between their areas, now tell me how to find the area of each field. Not the kind of situations you find yourself in every day exactly. You can judge for yourself if that deserves to be called applied mathematics. Given obscure and convoluted information, find something that should have been much easier to measure directly than this artificial data you somehow had access to. In any case, geometry like that, whatever you want to call it, was highly developed almost four thousand years ago. Why? What made people do this? Let’s try to find out. Early mathematics emerged where there was fertile soil. Rivers that made this possible. Agricultural abundance meant resources enough to expend some people specialising in mathematics instead of having all hands on the ploughs. Look at a modern population density map of Egypt. You will find that virtually the entire population is concentrated along the Nile; all the rest is pretty much desert. That’s still today. Even with the assistance of modern technologies the river area is by far the most liveable. Even more so back then when geometry started, thousands of years ago. It was the same in Mesopotamia, present-day Iraq. Also a river civilisation with very good agricultural conditions. They had legendary gardens that were praised in ancient sources. Google it: The Hanging Gardens of Babylon. You will see some nice pictures of what these luxurious gardens might have looked like. That’s a nice visual for this idea that it was agricultural abundance that made a specialised pursuit like mathematics possible in those societies. So that explains why they had the resources to support mathematics. But why would they want to? What did they stand to gain from geometry? Basically, mathematics was for a long time about commerce and taxes; bureaucratic management of workers and produce; inheritance law. Those kinds of things. Eleanor Robson’s book is very illuminating about this. “Mathematics in Ancient Iraq: A Social History”, the book is called. She emphasises especially that mathematics was very strongly associated with justice. A society without a functioning justice system is hampered by constant disputes about land, taxes, inheritance. Everybody is fighting with everybody. Like the old American West, you board yourself up and mind your own business and if there’s a disagreement, well, that’s what guns are for, isn’t it? Mathematics is the way out of this primitive state. Mathematics is objective. It can settle these disputes in a fair way. If everybody is wasting a huge amount of effort and resources on petty disputes in a lawless no-man’s land, who you gonna call? The mathematicians, that’s who. That’s how it went in ancient Iraq. A specialised, highly trained mathematician would come in and delineate all the plots of land, compute all the taxes owed, and distribute every inheritance. All according to exact calculations. This stuff used to be ruled by emotions, personal animosity, and the law of the jungle. But now, thanks to mathematics, that is replaced by objective rules. Who can argue with a calculation? Mathematics takes the worst sides of human nature out of the equation. When society is run by fair, universal rules, people no longer have to constantly look over their shoulder and fear that some lawless eruption of force could destroy everything they have at any moment. A functioning justice system enables people to work for the collective good and to plan for the long term. It is the authority of mathematics that makes this possible. These skilled mathematical technocrats had great credibility because people recognised that they were above the subjective and the emotional. They were bound by dispassionate calculation. Mathematics compelled them to be fair and rational. Indeed they explicitly said so themselves. As one mathematical scribe put it: “When I go to divide a plot, I can divide it; So that when wronged men have a quarrel I soothe their hearts. Brother will be at peace with brother.” That’s a quote by one of those mathematical technocrats, explaining what geometry accomplishes. Note that it has both of those elements I emphasised. Mathematics is the opposite of emotional disputes. It soothes heated hearts, it creates peace between warring brothers. And the quote also highlights that this happens because of the expertise of the mathematician: I know how to do this kind of thing, the technocrat is saying. It takes special training. The quote is from Eleanor Robson’s book. Here’s another thing she points out that is yet more evidence of the importance of mathematics in this context. The Sumerian word for justice literally means straightness, equality, squareness. Also in Akkadian: justice is the “means of making straight.” Again, another major indicator of this: “the royal regalia of justice were the measuring rod and rope.” Think of those Lady Justice statues that you see sometimes. She’s blindfolded because that shows that she’s unbiased, and she has these scales, showing that she’s considering both sides and weighing them carefully and fairly. That’s the symbol of justice in our society. But, in ancient Babylon, the symbols of justice were not a blindfold and a set of scales. Instead, Lady Justice was a geometer. She held her land-measuring tools. Those were the instruments of justice in ancient society. Maybe it’s pretty much the same today, four thousand years later. Back then, the trustworthiness of mathematics was a cornerstone of society. If people didn’t trust mathematics, there could be no law and order, no state bureaucracy, no complex economy, no civilisation. Today, that link is perhaps less evident. But perhaps no less crucial. We have added many layers of complexity to our society, but perhaps looking back at historical societies is the same thing as looking into the inner essence of our own. Maybe without faith in mathematics the entire fabric of our society would unravel. Maybe without mathematicians mediating their disputes, “brother would be at war with brother” as that ancient scribe feared. It is interesting also that this role of mathematics that I have outlined is really as much psychological as it is scientific. What makes this whole system work is not only that mathematics can give useful answers to certain technical problems. The psychological side is equally essential: mathematics has a kind of aura of objectivity, of trustworthiness, of professional expertise. That goes well beyond merely calculating the taxation rate of some field, or how many goats you can buy for a silver shekel. The system rests on a more nebulous trust in the mathematician class by the population at large. The idea of mathematics, the image of mathematics, is more important than the sum of its actual applications. That’s an important conclusion because it explains that striking feature of ancient mathematics: namely that many of the problems the ancients texts solve are super fake. They are pseudo-applications. For instance: Find the two sides of a rectangle, given that the sum of the length and the width is 24, and that the area plus the length minus the width = 120. So in other words, you basically have two equations in x and y, and if you solve for y in one and plug it into the other you have a quadratic equation in x. Lots and lots and lots of problems like that in Babylonian mathematics. Obviously nobody would ever face a problem like that in any real-word situation. It’s very often like this: you are looking for something simple, like the sides of a rectangle x and y, and you are given some super weird, like some convoluted combination of x and y is three eights of some other convoluted combination of x and y. Here’s another actual one: The width of a rectangle is a quarter less than the length. The diagonal is 40. What are the length and the width? In what real-world scenario can you realistically end up knowing the diagonal of a rectangle, and the difference between the sides, but not the sides themselves? And why couldn’t you just measure the sides? Someone did measure the diagonal, apparently, so why not the sides? Sometimes these texts hardly even try to hide how fake they are. One problem goes: I found a stone, but did not weigh it. I cut away one-seventh and then one-thirteenth, and then it weighed so-and-so much. What was the original weight of the stone? Who among us has not “found” whatever random stone, then chipped away an extremely exact ratio of it, and then suffered some kind of stone-cutter’s remorse I guess, and tried to reconstruct the original weight of the stone for some reason. Very relatable, isn’t it? Actually it kind of is. Not because we are sitting around cutting one-thirteenth out of random stones, or because we are running around measuring the diagonals of various fields and then later wish we had measured the sides instead. That never happens to any sane person in the real world. But it does happen in math books. Still today, we torture our students with such questions, one more artificial and unrealistic than the other. Some people think that kind of thing is modern pedagogy run amok. They see these kinds of problems in modern textbooks and they think: How silly modern pedagogy has become! These naive educators are bending over backwards to make math “relevant” to kids, but the just end up with silly fake problems. History offers a different perspective. The problems may be silly, but the cause is not a misguided obsession with real-world relevance among modern educators. Fake problems are as old as written mathematics itself. For as long as there has been mathematics education, students have been forced to go through page after page after page of pseudo-problems that only superficially, or linguistically, appear to be talking about real-world things, while actually corresponding to absurd scenarios that would never happen. In a way one might argue that history vindicates these problems. They are not so silly after all, if we consider them in the light of the role of mathematics in ancient Babylonian society. Mathematics doesn’t support the economy merely by keeping the account books. It’s more than that. Mathematics is what instills confidence in monetary law and order, without which any kind of complex economy would be impossible in the first place. For this system to work, there needs to be a specialised class of number-crunching technocrats. These people need to embody logic and reason and objectivity. They need to be math machines, detached from politics and emotion. A long schooling in artificial pseudo-problems makes some sense as a means of creating this class. From this point of view, it is even a strength that these problems are artificially divorced from real-world problems, because the mathematical technocrat is supposed to be detached from such concerns anyway. Mathematicians are valuable to society precisely because they are so disinterested in the needs of people of flesh and blood. It is this disinterestedness that makes people willing to trust the mathematicians to be the arbiters of disputes. The sheer volume of training in pointless problems also has its point. It is not enough that people at large know some mathematics: they could use mathematics as a tool for evil, as just one more incidental weapon in a society still ruled by greed and conflict. For a complex economy to take off, there needs to be faith that the law and the state administrative bureaucracy are fair and consistent. This faith comes from the credibility of mathematics. The mathematical technocrats need to be proper experts to justify the confidence placed in them. They need to embody mathematics; they need to single-mindedly look at any situation or conflict and see only the mathematics in it. Society needs the mathematicians to not only get the right answer, but to have great authority as proper experts. And it needs them to be “nerds,” so to speak, who are so one-sidedly developed that they can only see mathematics anywhere they look, and not let emotions or politics influence their work. A long and rigorous training in fake applied problems is not a bad recipe for bringing this about. Arguably, we pretty much still use the same recipe to the same end today, thousands of years later. So that’s the Babylonian tradition. We know quite a bit about it because they wrote on clay which is pretty durable. In Egypt, mathematics was recorded on papyrus, which isn’t going to survive for thousands of years normally. So we only have two or three or maybe four papyri that beat the odds and were conserved. But it seems the Egyptian situation may very well have been quite similar to the Mesopotamian one in terms of the role of the mathematicians. “Geometry” means “earth-measurement.” That’s from the Greek: geo metria. The ancient Egyptians had the same idea but their word for it was more concrete: a geometer was literally a “rope-stretcher.” A land surveyor stretches ropes to measure distances and delineate fields. A rope is pretty much equivalent to a ruler and compass. Pull the ends of the rope and you have a straight line. Hold one end fixed and move the other one while keeping the rope stretched: now you have a circle. Euclid explains how to make a square with ruler and compass. That’s Proposition 46 of the Elements. The Egyptians would have done that long before with their stretched ropes. Try it for yourself, it’s fun: go out into a field with a friend and try to make a perfect square using nothing but a piece of string. You will see why geometers were called rope-stretchers. Do you think you could make a square? Do you think anyone could? Back in the day, this skill could have given you a leg up in life. Suppose you make one square field, and then a rectangular field with the same perimeter. The square field will have greater area. But you could trick those less knowledgeable in mathematics. You could say: you get that field and I get this one, fair and square. Just try it for yourself, you would say, let’s walk around the fields and count the number of steps. 400 steps around my field, 400 steps around yours: aha, our fields are the same size. That’s what you tell the other guy, who isn’t such a math person. But you know that of course 100*100 is way more that 50*150. So later you get a much greater harvest. But of course you would pretend that that’s because you worked so hard while the other guy was lazy. Maybe that’s another way in which ancient society is like ours: privileged people use their privilege to rig the game in their favour, and then pretend it was all due to merit. According to Proclus, this kind of mathematical deceit did indeed happened: “The participants in a division of land have sometimes misled their partners. Having acquired a lot with a longer periphery, they later exchanged it for lands with a shorter boundary and so, while getting more than their fellow colonists, have gained a reputation for superior honesty.” Here’s how Thomas Heath paraphrases this in his History of Greek Mathematics: “Proclus mentions certain members of communistic societies who cheated their fellow members by giving them land of greater perimeter but less area than the plots which they took themselves, so that, while they got a reputation for greater honesty, they in fact took more than their share of the produce.” A dubious paraphrase, in my opinion. Can you spot the suspicious part of it? Good old Heath put something in there that was not in the original source. Hint: turn to the title page of Heath’s book. There are some clues there. The book was published by Oxford University Press in 1921. Heath’s name comes with some bells and whistles: it’s Sir Thomas Heath, in fact, and then K.C.B, K.C.V.O. That’s Knight Commander of the Royal Victorian Order etc. Titles upon titles. It’s an establishment guy, this Sir Thomas. A gentleman scholar, who was a civil servant as his day job at the Treasury. What part of Sir Thomas’s paraphrase of the ancient mathematical land deceit reflects his own social context more than that of the ancients he is trying to describe? I’m thinking of his phrase that these were “communistic societies.” The original source says nothing at all about this having anything to do with communism. But you can understand how Sir Thomas would have been concerned about communism at this time. The Russian Revolution started in 1917, Heath’s book is published in 1921. While writing the book, Heath was a secretary at the British Treasury. He would have read all about Lenin and Bolsheviks in The Times while having his afternoon tea. And those worries would have been at the top of his mind when he sat down in his study to do his scholarly work in the evening. It didn’t take much provocation, one imagines, for him to have a swing at how “communistic societies” were dreadful and corrupt. We must always read historical sources this way. Context matters. Now, the “original” in this case was Proclus. But that’s not much of an “original” to speak of. Proclus is nobody. He’s not particularly trustworthy. He was writing in the year 450 or so, thousands of years after the historical events he is talking about. So it’s anybody’s guess how much truth there is in what he is saying. And in any case, like so many other mediocre writers, both ancient and modern, Proclus is just copying what others had said. Let’s illustrate this point. Let’s see what we can learn by looking at Proclus’s account of the origins of geometry in Egypt. Here’s what Proclus says: “Geometry was first discovered by the Egyptians and originated in the remeasuring of their lands. This was necessary for them because the Nile overflows and obliterates the boundary lines between their properties. It is not surprising that the discovery of this and the other sciences had its origin in necessity, since everything in the world of generation proceeds from imperfection to perfection. Thus they would naturally pass from sense-perception to calculation and from calculation to reason. Just as among the Phoenicians the necessities of trade gave the impetus to the accurate study of number, so also among the Egyptians the invention of geometry came about from the cause mentioned.” Ok, sounds pretty plausible. But it’s worth running Proclus through a plagiarism checker, just as we do with modern student essays these days. Cutting-and-pasting from Wikipedia is nothing new. Proclus had many Wikipedia equivalents available to him. Perhaps he stole the whole thing for example from the Geography of Strabo, which was written more than 400 years before. Here’s what Strabo says: “An exact and minute division of the country was required by the frequent confusion of boundaries occasioned at the time of the rise of the Nile, which takes away, adds, and alters the various shapes of the bounds, and obliterates other marks by which the property of one person is distinguished from that of another. It was consequently necessary to measure the land repeatedly. Hence it is said geometry originated here, as the art of keeping accounts and arithmetic originated with the Phoenicians, in consequence of their commerce.” Basically a dead ringer for the Proclus passage. Plagiarism detected, SafeAssign™ would say. Actually Proclus has added something that is not in Strabo, namely the claim that this historical episode illustrates how human though passes from the world of the senses to the higher realm of reason. This is card-carrying Platonism. Proclus is a sycophantic follower of Plato. He sees everything through Plato-coloured glasses. Which is not helpful if we want to use him as a source of historical information. As Heath had his anti-communism, so Proclus has his Platonic axe to grind and it infects everything he says. Actually we can go back even earlier than Strabo. Let’s take an equal jump back in time again: another 450 years still. From Roman Strabo to classical Greek Herodotus. He too speaks of the origins of geometry in Egypt. Let’s listen to his account: “This king [Sesostris] also (they said) divided the country among all the Egyptians by giving each an equal parcel of land, and made this his source of revenue, assessing the payment of a yearly tax. And any man who was robbed by the river of part of his land could come to Sesostris and declare what had happened; then the king would send men to look into it and calculate the part by which the land was diminished, so that thereafter it should pay in proportion to the tax originally imposed. From this, in my opinion, the Greeks learned the art of measuring land.” Ok, I have to admit that this makes Thomas Heath look a bit better. “The king gave to each an equal parcel of land”: That is a bit more like communism. Heath said he was paraphrasing Proclus where there is no such phrase about equality. But Herodotus, the better source, kind of vindicates him a bit. You could imagine, in the scenario that Herodotus describes, that certain administrators in charge of implementing the king’s decree might secure a nice big square plot for themselves and trick the mathematically illiterate into a smaller plot with the perimeter trick. Perhaps not entirely unlike how corrupt middle-managers in the Soviet bureaucracy might manipulate the system for personal gain. But be that as it may. I think there’s another interesting thing about Herodotus’s description compared to Strabo’s. Strabo and Proclus give a cleaner and simpler account: the flooding of the Nile obliterates everything and you have to start afresh each year with the drawing of boundaries. Herodotus’s account is much less dramatic: some parts of properties might become damaged by the floods, and the task of the mathematician is not to redraw the entire agricultural map each year but rather to calculate what proportion of area has been lost in each case for taxation purposes. One can easily imagine how a desire to simplify and tell a clear and dramatic story might have led authors like Strabo and Proclus to prefer their version. The older source is a bit more “boring” but perhaps that makes it more credible. Indeed, Herodotus’s account fits better with what we said about the role of mathematics in Mesopotamian society. In Herodotus’s version, the mathematicians task is more technical, more specialised, more bureaucratic. Note his phrase: “the king would send men” to do the calculations. You have to send mathematicians. They are a small, specialised class of technocratic experts that are dispatched to solve disputes with authority and objectivity. That’s precisely the main point I have made today, so let us end there.
53 minutes | a year ago
More things Galileo didn’t do first
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Galileo and the Church
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Phases of Venus
Telescopic observations of Venus provided evidence for the Copernican view of the solar system. But was Galileo the first to see this, as he claims? Or did he steal the idea from a colleague and lie about having made the observations months before? Transcript Galileo and the phases of Venus: it’s a plot that mirrors … Continue reading Phases of Venus
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