I’m teaching our intro to proofs course (well, one of them) and one of the classic illustrations of an overturned “axiom” is the Greek axiom of commensurability, which stated that all segments are commensurable (“any two lengths have a third length which divides both an integer number of times”), which would mean in modern terms that all real numbers are rational. The fact that the diagonal of a square stands in irrational ratio to the square’s sides overturned this “axiom” and forced a redevelopment of a chunk of geometry.
That story I can get in a lot of places. What I’m interested in is an example of an actual proof using the axiom of commensurability. How is the idea that everything is commensurable useful in a geometry proof?
The example can be either a true theorem that had to be reproved, or a false claim that followed from the axiom — preferably, in that case, not one that is obviously related to rationality itself. I specifically would like to see the argument that involved commensurability.
To define a ratio of two geometric segments early Pythagoreans assumed commensurability, subdivided them into smaller subsegments of equal lengths, and took the ratio of the numbers of the subsegments as the ratio of the segnments. So all theorems that used ratios of segments, about similar triangles for example, had to rely on it and were put in doubt when Hippasus (allegedly) discovered that the side and the diagonal of a unit square had no common measure. A more complex and potentially infinite process of laying segments off of each other, similar to constructing continued fractions of irrationals, was used to define ratios after the incommensurables were discovered, see Fowler.
Eventually Eudoxus of Cnidus developed a new theory of proportion based on a new definition for comparing geometric ratios, where he implicitly treated ratios as Dedekind cuts of rationals, and old proofs of the ratio theorems could be repaired by replacing exact matches with squeeze estimates from above and below. This was an early application of the method of exhaustion that served Greeks as a surrogate for the theory of limits.
One example of using commensurability is in the proof of a theorem attributed to Thales that parallel lines cut opposite sides of an angle in the same ratio. Assuming commensurability we can subdivide both segments on one side into small subsegments of common measure, each of the two segments then contains a whole number of them. Drawing parallel lines through the endpoints of the subsegments subdivides the opposite side of the angle into subsegments too. The segments cut by the original lines contain exactly the same numbers of subsegments, hence the same ratio. Another example is analysed in detail in Lecture 6 of Eves’ Great Moments in Mathematics (pp.53-56): the areas of triangles of the same height are to each other as their bases. He sketches a proof based on commensurability and then shows how it has to be reworked with Eudoxian definition. Fowler gives an example from Aristotle for areas of parallelograms of the same height.
By the way, one can probably construct models where the axiom of commensurability holds, it’s just inconsistent with some other Euclid’s axioms, like the axiom of parallels that leads to the Pythagorean theorem. In essence, Eudoxus replaced the “axiom of commensurability” with a weaker but consistent axiom now called the “axiom of Archimedes”.
I agree with André’s comment about the “incorrectness” of the reference to a Greek axiom of commensurability.
- Craig Smorynski, History of mathematics : A supplement (2008)
we can find some refernce to a “commensurability assumption” [page 50] and to an “axiom of commensurability” [page 51].
I strongly support the first locution; the problem of incommensurability was prior to (and the main source of) Euclid’s axiomatization; thus, can be historically mesleading to speak of “axiom”.
Of course, it was an (implicit) assumption common to all “archaic” Greek math: given two magnitudes, e.g. to segments of lenght $a$ and $b$, according to the assumption it is always possible to find a segment of “unit lenght” $u$ such that it measure both, i.e. such that [using modern algebraic formulae which are totally foreign to Greek math] :
$a = n \times u$ and $b = m \times u$, for $n,m \in \mathbb N$.
From the above instance of the assumption, it follows that :
$a/b = (n \times u) / (m \times u) = n/m$.
The assumption amounts to saying that the ratio between two magnitudes is always a ratio between numbers (i.e. in modern terms : a rational number; but note that for Greek math the only numbers are the natural ones and they are distinguished from magnitudes : a segment, a square, … which are “measured” by numbers).
The well-known discovery of the existence of irrational magnitudes, through the proof that the case where $a$ is the side of the square and $b$ its diagonal is not expressible as a ratio between (natural) numbers, leads Greek math to the withdrawal of the “commensurability assumption” and to the axiomatization of geometry.
Regarding the evolution of Greek mathematics, you can see :
- Wilbur Knorr, The Evolution of the Euclidean Elements : Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry (1975)
- Arpad Szabo, The Beginnings of Greek Mathematics (1978).
See Knorr, page 131 :
Within such a tradition [the pythagorean one] the commensurability of all geometric magnitudes must at first have been an implicit axiom. As we have seen, this axiom met its direct refutation in the discovery of the incommensurability of the side and diameter of the square, a discovery made and disseminated sometime within the last third of the fifth century B.C.
For Pythagoas and Pythagoreanism I’m referring to SEP‘s entries :
In the modern world Pythagoras is most of all famous as a mathematician, because of the theorem named after him, and secondarily as a cosmologist, because of the striking view of a universe ascribed to him in the later tradition, in which the heavenly bodies produce “the music of the spheres” by their movements. It should be clear from the discussion above that, while the early evidence shows that Pythagoras was indeed one of the most famous early Greek thinkers, there is no indication in that evidence that his fame was primarily based on mathematics or cosmology.
The interpretative issues are complex.
My “feeling” is that is a little bit “un-historical” to speak of axioms regarding Pythagoas’ math because the idea of math as a science based on axioms and proofs of theorems dates from Aristotle and Euclid.
The extant remains of P’s math are very few and second (or third) hand :
First, Pythagoras himself wrote nothing, so our knowledge of Pythagoras’ views is entirely derived from the reports of others. Second, there was no extensive or authoritative contemporary account of Pythagoras. No one did for Pythagoras what Plato and Xenophon did for Socrates. Third, only fragments of the first detailed accounts of Pythagoras, written about 150 years after his death, have survived. Fourth, it is clear that these accounts disagreed with one another on significant points.
My point of view is that math for P was more like zoology or elementary chemistry : it was the study of the properties of shapes and numbers (of course, with a “perception” of the peculiarity of those kind of “objects” compared with animals or stones).
We must “read” P’s math as a “science” (in a time when science and philosophy were not differentiated) and its “basic discovery” was the possibility of accounting for (describing, explaining, …) natural facts in terms of numbers.
The paradigmatic example of this discovery is the possibility of “translating” musical notes into mathematical relations.
If we see this discovery as a (primitive) scientific law, we can understand the source of the postulate regarding commensurability : assuming that the only numbers “available” in ancient Greek math was the natural ones, the above postulate is implicit in the search (and discovery) of mathematical relations describing natural facts.
This is quite far from an axiom used in a mathematical proof.
Some further quotes form SEP‘s entry :
There is, moreover, no talk of mathematical proof or a deductive system in the passage from Aristoxenus […]. Pythagoras is known for the honor he gives to number and for removing it from the practical realm of trade and instead pointing to correspondences between the behavior of number and the behavior of things. Such correspondences were highlighted in Aristotle’s book on the Pythagoreans, e.g., the female is likened to the number two and the male to the number three and their sum, five, is likened to marriage.
Proclus does not ascribe a proof of the theorem to Pythagoras but rather goes on to contrast Pythagoras as one of those “knowing the truth of the theorem” with Euclid who not only gave the proof found in Elements I.47 but also a more general proof in VI.31. Although a number of modern scholars have speculated on what sort of proof Pythagoras might have used (e.g., Heath 1956, 352 ff.), it is important to note that there is not a jot of evidence for a proof by Pythagoras; what we know of the history of Greek geometry makes such a proof by Pythagoras improbable, since the first work on the elements of geometry, upon which a rigorous proof would be based, is not attested until Hippocrates of Chios, who was active after Pythagoras in the latter part of the fifth century (Proclus, A Commentary on the First Book of Euclid’s Elements, 66). All that this tradition ascribes to Pythagoras, then, is discovery of the truth contained in the theorem. The truth may not have been in general form but rather focused on the simplest such triangle (with sides 3, 4 and 5), pointing out that such a triangle and all others like it will have a right angle.