Euclid writes randomness into his Elements Cropped from source (Garrett Coakley)

Euclid is, of course, the Greek mathematician, who is often referred to as the “Father of Geometry.”

Today Ken and I want to talk about an “error” that appears in the famous volumes written by Euclid a few years ago—about 2300 years ago.

The “error” is his use of the word ‘random’ when by modern standards he should be saying arbitrary. I find this surprising, since I think of random as a modern concept; and I find it also surprising, since the two notions are not in general equivalent.

It seems clear that Euclid said ‘random.’ The root-word he used, tuchaios, endures as the principal word for “random” in modern Greek and is different from words meaning “arbitrary” or “generic” or “haphazard” or even “stochastic.” The only meaning of tuchaios or Euclid’s exact phrase hos etuchen we’ve found that would make his statement remain strictly correct is, “it is unimportant which.” However, the way hos etuchen was put in the voice of Pope Clement I seems not to square with that meaning either.

## Euclid and Random

I never studied the Elements, Euclid’s famous collection of thirteen books on geometry. Not that long ago, many schools used the Elements as the textbook for the introduction of mathematics. Abraham Lincoln is said to have studied the Elements until he could recite it perfectly. I never looked at any part of it until I came across Book II while doing some research. And I was quite surprised to see the use of the notion of “random” there in the text.

Book II is focused on a geometric approach to identities, which are much easier to understand as algebraic identities today. The proposition that caught my eye is Proposition II.4.

Proposition 1 If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments.

Perhaps this is easy to understand as a geometric statement. Today we would write algebraically that it stands for the identity $\displaystyle (a+b)^{2} = a^{2}+b^{2}+2ab.$

Why did Euclid state it geometrically? Perhaps the main advantage was that it allowed him to reason directly about geometric objects. After all he was writing about geometry, so a square with sides of length ${a}$ stood in nicely for the term ${a^{2}}$. An even better reason might have been his lack of modern algebraic notation—the equals symbol was invented by Robert Recorde a few years after Euclid, in 1557.

Here is the proof as it appears in the Elements. The length compared to simply expanding ${(a+b)^2}$ shows the power of modern notation: ## The Issue

What surprised me was the exact statement of Proposition II.4. Note that it starts,

If a straight line be cut at random

In online Greek editions such as this the phrase hos etuchen meaning “at random” is set off with commas. Euclid reiterates this phrase in the first line of his proof. However, the result is actually true for any cut of the line, which is more than saying “at random.” So why does Euclid say “random”?

Euclid seems nowhere to define in any precise way what “random” means in this context. Recall that one of the great achievements of the Elements was its claim to be a precise and axiomatic approach to geometry. But using an undefined term like “random” seems to run overtly counter to that goal.

Looking for other usages doesn’t clearly let Euclid — or his main ancient editor, Theon of Alexandria — off the hook. Pope Clement I was St. Peter’s first, second, or third successor. A novelization of his acts and homilies has him using the same Greek phrase at the beginning of Book 1, chapter IV:

Our Peter has strictly and becomingly charged us concerning the establishing of the truth, that we should not communicate the books of his preachings, which have been sent to us, to any one at random, but to one who is good and religious, and who wishes to teach…

This aligns with the modern meaning: the writer was saying that most people would be unqualified to preach Peter’s sermons. It does not mean that all people would be bad or that it is unimportant who receives the books. There is support in other ancient examples for the reading, “to anyone you happen to meet,” but even then the inference stays one of “mostness” not “all.” In any event, Euclid’s proposition is correct with “all”—even if the line is “cut” at one of the endpoints.

The difference is not a quibble. It is easy to make statements in Euclidean geometry that are true for “random” but not for “all”: A random triple of points in the plane makes a triangle. A random line through a point outside a circle is not tangent to the circle.

However, there are also cases where holding for “random” is sufficient for holding for “all.” Equalities like Euclid’s ${(a+b)^{2} = a^{2}+b^{2}+2ab}$ have that property. So does the Schwartz-Zippel lemma: if ${p}$ is a polynomial and ${p(\vec{x}) = 0}$ for a random ${\vec{x}}$ over a large enough field then ${p}$ is the zero polynomial. In fact Euclid’s identity is a case of this—could we add Euclid as sharing credit for the lemma?

This led Ken and me to think about a problem: Can we make something out of ‘Euclidean’ randomness?

## Random vs Generic vs Arbitrary

There is a third concept lurking here: generic. Generic usually implies random but is more special and does not require probability or (Lebesgue) measure. In fact it basically means “not special.” Three collinear points are special; a line tangent to a circle is special.

The exact notion of “generic” is context-dependent, but at the interface of geometry and algebra we can pin it down: a set of elementary objects (points, lines, etc.) is special if it satisfies some finite set of simple arithmetical equations and is generic otherwise. Collinear points and tangent lines are clearly special in this sense. More formally, the special sets are those closed in the Zariski topology, apart from the whole (Euclidean) space. So now we ask:

Could Euclid have been in any sense aware of the idea of genericity?

If so, then Euclid could have been led into deep waters. Consider just a line segment going from 0 to 2. The midpoint ${x = 1}$ is special because it satisfies the equation ${x+x = 2}$. Similarly so are the points ${x = \frac{1}{2}}$ and ${x = \frac{3}{2}}$. It quickly follows that all rational numbers are special. Now so is ${x = \sqrt{2}}$ since it satisfies ${x*x = 2}$. And likewise ${\sqrt{2}}$, so all points for algebraic numbers are special too. Euclid would certainly have suspected that the cubic and higher points might not be constructible. So although he might have suspected that a “random” point was not constructible, he would have had a hard time realizing that the non-constructible points include the special subset of algebraic ones.

Of course, it took until the 1600s to articulate modern meanings of “random” and until the 1800s for Georg Cantor and the topological notions underlying genericity to arise. It still interests us what “hints” might have been perceived in the intervening centuries.

## A ‘Random’ Road to Geometry?

Alfred Tarski, the famous 20th-century logician, created formal axioms for geometry. His axioms modeled that part of geometry that is called “elementary.” This includes statements of plane geometry that can be stated in first-order logic and only refer to individual points and lines: arbitrary sets are not allowed. The above reference has the details of his axioms. They were built on two primitive notions:

• Betweenness: The sentence ${Bxyz}$ denotes that ${y}$ is “between” ${x}$ and ${z}$: that is ${y}$ lies on the line segment from ${x}$ to ${z}$.
• Congruence: The atomic sentence ${Cwxyz}$ means that the length of the line segment ${wx}$ is equal to the length of the line segment ${yz}$.

Tarski proved that this theory is decidable. And actually it has a remarkable property: any statement in the theory is equivalent to a sentence ${S}$ that is in universal-existential form, a special case of prenex normal form. In this form all universal quantifiers precede any existential quantifiers: $\displaystyle S = \forall u \forall v \ldots\exists a \exists b\ldots (\cdots)$

This form is close to just having equations, so it is tantalizing to ask, given any formula ${\phi}$, does either ${\phi(\vec{x})}$ or ${\neg\phi(\vec{x})}$ hold for generic ${\vec{x}}$? Or for some notion of “random” ${\vec{x}}$? The basic ${B}$ and ${C}$ formulas have this property: their negations hold generically—even though they do not hold for all ${\vec{x}}$.

However, the following seems to be a weighty counterexample to any kind of “zero-one law” holding here: Tarski’s system can define a formula ${Axyz}$ meaning that the angles at ${x}$ and ${z}$ are acute. Now fix ${x = (0,0)}$ and ${z = (2,0)}$ in the plane. Then any ${y = (a,b)}$ satisfies ${Axyz}$ if and only if ${0 < a < 2}$ and ${b \neq 0}$. Neither the set of such ${y}$ nor its complement is a nullset.

This is curious because if one regards Euclid-type diagrams as finite structures like graphs, then the first-order zero-one law proved by four Soviet mathematicians and independently a little later by Ronald Fagin comes into play: as ${n\rightarrow\infty}$ the proportion of size- ${n}$ structures that satisfy a given first-order sentence (pure: no parity or counting) goes either to 0 or to 1. Still, we can ask two questions:

• Are there interesting classes of Tarski geometric formulas ${\phi(\vec{x})}$ such that either ${\phi(\vec{x})}$ or ${\neg\phi(\vec{x})}$ holds for generic ${\vec{x}}$, or such that ${\phi}$ obeys a zero-one law in measure?
• Which classes of ${\phi}$ are such that ${\phi(\vec{x})}$ holding for a random (or generic) ${\vec{x}}$ implies that ${\phi(\vec{x})}$ holds for all ${\vec{x}}$?

So what we are asking is, exactly when does Euclid’s use of “random” for “arbitrary” remain correct? Which geometric statements are guaranteed either to hold or to fail for “random” arguments?

## Open Problems

Do our questions have nice and simple answers? Are we the first to wonder how Euclid’s words fare when given a modern mathematical interpretation?

[fixed definition of ${Axyz}$]

1. January 12, 2016 10:36 pm

Compare C.S. Peirce’s concept of “tychism”. For some reason I always thought the English-French word “random” had something to do with horse races.

2. January 13, 2016 4:30 am

If Axyz means that x,y,z form an acute triangle and their coordinates are as given, then the absolute value of b must be greater than 1, not just nonzero. Also Axyz most certainly does satisfy the first-order zero-one law; if D(r) is the disc of radius r centered at (1,0) then the area of points in D(r) that satisfy Axyz divided by the area of points in D(r) that don’t satisfy Axyz goes to 0 as r goes to infinity.

• January 13, 2016 8:04 am

Thanks—I changed the formula instead. Great observation about the disks. Ad chorasimilarity, RAM is Μνήμη Τυχαίας Προσπέλασης. But one is right to say that the meaning is more of “arbitrary” there.

3. January 13, 2016 5:20 am
4. January 13, 2016 7:20 am

The distinction between “genuine” or “generic” relations and “degenerate” relations is a fundamental question in Peirce’s logic of relatives, which is critical to his semiotics or theory of triadic sign relations. He often uses the example of conic sections to illustrate the various types of genericity or degeneracy.

Random Readings On Relations, Well, Not Really

5. January 14, 2016 7:11 am

Nice post. It reminds me of a perhaps very stupid question which I have since I started reading geometry a year or so ago: Was Euclid really not familiar with curves? Anyone who could give me a hint is super highly appreciated. Thanks; Thomas

• January 15, 2016 8:48 am

In the following link, I believe you may find some good reasoning on why in ancient Greece the geometers were only interested in circles, spheres and straight lines and didn’t care about any other kind of curve.

http://bertie.ccsu.edu/naturesci/Cosmology/Cosmo1Background.html

• January 15, 2016 3:41 pm

Thank you very much, Pantelis. I’ll check it out tomorrow and, if I may, may get back to you.

• January 16, 2016 7:14 am

Thanks again, Pantelis. This is familiar Greek cosmology: All curves, circles & spheres there as well. Could it be that, for Euclid, they were so self-evident that he didn’t care bothering about them? And may he have thought, or may he have just presupposed, that they can be dealt with within his scheme?
I came to wonder about all this when pondering all the fuss about non-Euclidean Geometry: There must have been curved paths in Alexandria (& American cities were not invented yet). And since non-Euclidean Geometry came up largely for the 5th Postulate (at least so far I understand), I was wondering why the name ›non‹-Euclidean in the first place, when Euclid’s other Postulates, Axioms & Propositions basically still hold. To me, the question then is: Can we deal with curves & all that within the Euclidean scheme, plus maybe the algebra invented by Graßmann? Please forgive me these perhaps again stupid questions/remarks. I’m reading geometry for only a short time…

• January 16, 2016 7:26 am

For the context of my questions, see my discussion with Jonah Miller

6. January 14, 2016 9:51 am

It is true that the way Euclid considers randomness is different than our current perception of randomness. But he is not making an error. In Greek the word “τυχαίος” which is translated as random in English, refers to an event that is the outcome of some unpredictable (random) procedure. It is not the event itself that has random properties, it is the way it is generates that has random characteristics. The term “ὡς ἔτυχεν” should probably be translated “as a cause of a random action”.
In school textbooks of geometry, at least in Greece, the word “τυχαίος” or else random is used as Euclid used to use it. And this is how teachers teach it. A random point, is any randomly chosen point among many others. A random cut is any cut created at a randomly selected point in a line. A random element on this sense is not an element that is not special, it is an element generated by a procedure which is not special. Whether or not the element that is generated during some random procedure has random description and properties is another issue.
If Euclid does not define the meaning of the term random it is probably because the interpretation of the word random on this way was common knowledge in Greek at that times.

• January 15, 2016 12:36 pm

Thanks for a very well-explained answer. The idea of “arbitrary outcome of a random process” also works for the Pope Clement usage.

7. January 14, 2016 12:05 pm

Is P equal to NP?

We pretend to show the answer of the P versus NP problem. We start assuming that P = NP. We prove a Theorem that states when P = NP, then the problem SUCCINCT HAMILTON PATH would be in P. But, we already know if P = NP, then EXP = NEXP. Since SUCCINCT HAMILTON PATH is in NEXP–complete, then it would be in EXP–complete, because the completeness of both classes uses the polynomial-time reduction. But, if some EXP–complete problem is in P, then P should be equal to EXP, because P and EXP are closed under reductions and P is a subset of EXP. However, as result of the Hierarchy Theorem the class P cannot be equal to EXP. To sum up, we obtain a contradiction under the assumption that P = NP, and thus, we can claim that P is not equal to NP as a direct consequence of applying the Reductio ad absurdum rule.

See more in

https://hal.archives-ouvertes.fr/hal-01249826

8. January 18, 2016 8:08 am

These are the forms of time,
which imitates eternity and
revolves according to a law
of number.

Plato • Timaeus • 38 A
Benjamin Jowett (trans.)

It is clear from Aristotle and even Plato in places that the good of reasoning from fair samples and freely chosen examples was bound up with notions of probability, which in the Greek idiom meant likeness, likelihood, and likely stories, in effect, how much the passing image could tell us of the original idea.

9. January 21, 2016 2:31 pm

The discussion here seems related to the “Data”, another book attributed to Euclid. It is about reasoning when geometrical information is “knowable”, more or less. So a line segment is said to be “given in magnitude” if it is possible to construct a line segment of equal magnitude; a circle is given in magnitude if its radius is given in magnitude, and so on. The propositions are all about deducing which information is “given”, if some other information is “given”. We could regard the random point as being given to us (somehow), which sidesteps the issue of how to choose it, by transferring responsibility – like the “for all… there exists…” pattern above.

The English translation by Robert Simson notes that (for example) an angle may be “known” in the sense that it is uniquely determined by other givens, but we still might not be able to construct it explicitly using the rules of Euclidean geometry. So he amends a definition that geometric objects “given in position” are those which “have always the same situation, and which are either actually exhibited, or can be found.” There seems to be something going on here which is a precursor of reasoning about constructibility or computability. In the Data, given-ness is sometimes considered in terms of the ability to construct a copy (e.g., a line of the same length).

Interestingly, Tarski’s axioms include such constructively doubtful examples as “(Bxyz and Bxyu and x != y) implies (Bxuz or Bxzu)”. This says that if x, y, z and u are collinear then one of z and u is closer to x than the other one is. But constructively, we may not be able to tell which case holds (see http://www.sciencedirect.com/science/article/pii/S0168007298000177).

• January 21, 2016 5:18 pm

Thank you very much for the detailed reply and links; this builds depth on the comment by Pantelis Rodis which I/we acknowledged above. Regarding your last point about Tarski, that will fall under themes in the next post after one on chess that will appear later today.

• January 22, 2016 1:20 pm

Galois used that same epistemological idiom of what can be “rationally known” in discussing solvability by radicals.