Abstract
Wallis’s attempted proof of Euclid’s Parallel Postulate is an important but oft neglected event leading to the discovery of non-Euclidean geometries. Our aim here is to show Wallis’s own reliance on three non-constructive diagrammatic inferences that are not (fully) explicit in his own supplement to Euclid’s axioms. Namely, there is i- an implicit assumption concerning the possibility of motion; ii- an implicit assumption about the continuous nature of space and time; and iii- an explicit assumption about the existence of similar triangles which conceals an appeal to a combinatoric principle of reasoning that is tantamount to appealing to the Axiom of Choice.
This paper was inspired by the unpublished lecture notes of Michael Hallett as well as his interest in the proposed thesis of this paper. As such, we extend special thanks to Prof. Hallett and duly note that Sect. 3 and 4 rely heavily on shared insights.
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Notes
- 1.
John Wallis (1616–1703) was appointed to the Savilian Chair of geometry at the University of Oxford by Oliver Cromwell in 1649. It was Henry Savile, his predecessor, who had famously remarked that “On the most beautiful body of Geometry there are two moles, two blemishes, and, so far as I know, no more”, the chief blemish being the Parallel Postulate [3]. The incumbent of this chair was obligated to give a lecture every year on classical geometry. Wallis is most known for a decades long feud with Hobbes, as well as for making headlines in 1685 for calculating the square root of a 53 digit number entirely in his head during a bout of insomnia, and remembering the 27 digit result entirely from memory the next morning [11].
- 2.
Indeed, kinetic diagrammatic reasoning via the contiguously developping field of kinematics was indispensable to the development of the calculus. For our purposes, Isaac Barrow’s work on the infinitesimal derivation of tangent lines – disseminated in a series of lectures at Cambridge in the mid-1660s – is particularly relevant due to the limit ‘characteristic triangle’ similar to the triangle inscribed by the tangent, the x axis and any line perpendicular to the x axis [4].
- 3.
In fact, Wallis commissioned a translation into Latin and was the first to publish the work independently of Clavius’s Commentary on Euclid (1574) [2].
- 4.
See also I, 27 for an equivalent formulation. By contrast, the Fifth Postulate states that “if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angle” [6].
- 5.
For it is not at all given that parallel motion is possible on an idealized plane ostensibly frozen in both space and time.
- 6.
And, indeed, by Euclid himself, though he was somewhat more parsimonious in his deployment of this technique. It is tempting to analyze the few usages of motion in Euclid’s Elements as an unsavoury fail-safe when no constructive method could be ascertained to deliver the Proposition in a more agreeable manner.
- 7.
At least, not in the XVIIth century. Nowadays, it is quite easy to compile a kinematic diagram showing exactly this type of parallel motion on the idealized plane.
- 8.
“One should not object here that Euclid himself in his proofs never appears to have applied the movement of a straight line and never mentioned this in the postulates, since just as in the explanation of the sphere, he uses the movement of a circle, in the explanation of a cone, he uses the movement of a triangle, in the explanation of a cylinder, he uses the movement of a rectangle, he could have used, if necessary, the movement of a straight line in his proofs. From time to time Archimedes, Apollonius, and other geometers do this. Indeed, Euclid himself uses the movement of two straight lines where the angle between them does not alter, and indeed very close to the beginning, in that he proves Proposition 4 [i.e., I, 4] by a covering argument, and that assumption is necessary to the covering. And in my Lemma, I use the notion of movement in exactly the same way. In addition, the same is assumed in the third postulate (namely, to describe a circle with any given centre and radius), since one assumes (in the drawing of the circle) that the circular surface is described by the moving around of the radius (while one of its endpoints remains fixed at the centre). I mention this in order that I do not give the impression of having neglected the Euclidean rigour in proofs and that I have brought in new postulates (other than those admitted by Euclid himself)” [12].
- 9.
In Euclid’s Third Postulate, the construction of a circle of any center and radius is allowed, since any such circle may be drawn by moving the radius around the centre. The length and curvature of the radius are assumed to not incur any distortions through this motion. However, Euclid falls slightly short of postulating the actual existence of infinitely many circles of arbitrary size. The Third Postulate remains a construction postulate, not an existence postulate. As we shall see, Wallis will also attempt to justify the explicit assumption at the center of the second part of his proof through appeal to this Postulate.
- 10.
Pace Wallis, the argument there could be achieved by a different kind of motion: lifting the figure from the plane and repositioning it, or by folding the plane, etc.
- 11.
For instance, like the depiction of space and time presented in Zeno’s fourth paradox of motion, the Stadium argument. Our only source for Zeno’s Stadium argument is due to Aristotle. Due to the difficulty in interpretating this paradox, it is usually presented alongside two simple diagrams. It is perhaps notable that, while these diagrams are genuine heuristic aides to understanding the paradox, so difficult is the task of ‘visualizing’ the diagrams discretely (that is, of ‘visualizing’ these diagrams outside the assumption that space and time are both continuous) that seldom does the paradox strike us on our first, second, or even third time encountering it.
- 12.
Consider a Poincaré disk model where line CD cuts straight through the center. As the line \(\alpha \beta \) moves closer to the center and approaches some point \(\pi \) where it might cross CD, one or both of its nodes would appear to get ‘left behind’ due to the potentially infinite distances they must travel compared to the segment of the line closest to the center (where the deviation from the Euclidean planar model is lesser). Such a space is continuous, but it is not uniform: here, the first part of Wallis’s argument is not just fallacious, it is simply not possible. Yet, even when looking at a diagram of a non-Euclidean space represented on a Euclidean plane, it is very difficult for us to see how the hyperbolic parallel motion of line \(\alpha \beta \) behaves. In our world, as in our diagrams, objects moving along the shortest path from one point to another move smoothly, without morphing, disappearing and reappearing, etc. The intellectual knowledge that our experience of the world may not be as it appears does little to lift the nearly insurmountable obstacle presented by our senses.
- 13.
Curiously, the first part of Wallis’s proof can be entirely dispensed with in favour of a Euclidean construction. As such, we have not belaboured this point. Nevertheless, this explicit kinetic reasoning is highly informative as to the implicit kinetic reasoning we conjecture is at play in the second part of his proof.
- 14.
See note 9. “Indeed, since magnitudes can always be divided and multiplied without restriction, this seems to follow from the nature of relationships between magnitudes, namely that every figure can always be reduced or increased without restriction while retaining its own form. In fact all geometers have made this assumption (without expressing it or even remarking on it), among them Euclid. For when he requires that a circle with given centre and radius can always be described, he assumes that there is a circle of arbitrary size or with arbitrary radius, and when he assumes that something is possible, then he requires that one can carry this out. To be sure, it would be no automatic requirement that, without the necessary knowledge being set out, one should be able to draw a similar figure to a given one according to a given measure. But [given this demand] one can just as well assume that this can be carried out for an arbitrary figure as for a circle” [12].
- 15.
Yet Wallis’s explicit assumption entails more than does Euclid’s Third Postulate. For all circles are by definition similar in the desired respect. The ability to construct a given circle entails the ability to construct a similar circle of arbitrary size. It follows from the ability to construct any circle – barring, of course, certain physical constraints placed on the construction of compasses. On the other hand, triangles vary widely in the desired respect. The ability to construct a given triangle does not immediately entail the ability to construct a similar triangle of arbitrary size. Euclid shows how to construct specific triangles only twice: 1- in I, 1, Euclid shows how to construct equilateral triangles of arbitrary size by inscribing them within constructed circles; and 2- in I, XXII, Euclid shows how to construct a triangle out of three given lengths – yet his method again relies on the ability to construct arbitrary circles, given here that we are also given the exact length of all three sides. It is then far from clear that Euclid believed the construction of similar triangles of arbitrary size to be anything close to a fundamental principle.
- 16.
The way that Proposition VIII is set up, Wallis does not need to invoke motion in the second part of his argument. But consider again Fig. 3. It differs from Wallis’s own Figure 7 solely by the extension of line \(\alpha \beta \) to \(\alpha \theta \). It is immediately clear that the only way that \(\triangle PAC\) could be constructed in a manner that would still concord with at least Wallis’s own construal of Euclidean methodology, is through motion. In doing so, it is also immediately clear how a dense infinity of similar triangles to \(\triangle \pi \alpha C\) appears to be given by the diagram. Nevertheless, Wallis’s argument still requires him to ‘choose’ the desired triangle from the uncountably infinite collection of all the similar triangles to \(\triangle \alpha C \pi \). Positing that line \(\alpha \theta \) ‘scans’ the space and halts when it coincides with line AB does the trick nicely.
- 17.
Wallis’s existence postulate was furthermore deeply rooted in his theory of magnitudes and proportions which he derived partly from Aristotle’s Categories.
- 18.
That is, that quadrilaterals with two sides perpendicular to the other two are rectangles (c.f., Euclid’s Definition X), itself equivalent to the Parallel Postulate [6].
- 19.
It is even the very basis of the Euclidean method which consists in proving a generalization by choosing an arbitrary but definite object, and then showing that the argument holds for that object [9].
- 20.
Indeed, Zermelo himself stated that “[t]his logical principle cannot, to be sure, be reduced to a still simpler one, but is applied without hesitation everywhere in mathematical deduction” [13].
- 21.
If the set S is finite (or consists of a countable infinity of positive integers), the existence of such a choice function follows through induction from the precepts of basic logic and of set formation; it is the case of an infinite set S (whether countable like \(\mathbb {Z}\) or \(\mathbb {Q}\), or uncountable like \(\mathbb {R}\)) where the rule for f(A) may be un-determinable – thus rendering the Axiom of Choice necessary [1, 9].
- 22.
Here, the nature of the geometrical continuum hides the uncountable infinity of similar triangles that must be scanned before arriving at the end point. This is a seeming paradox, which traces its roots back to Zeno’s Dichotomy paradox of motion (or, Achilles and the Tortoise). Thus, this is only a problem if we require that the line scan all the points contained in the line segment \(\pi P^{\prime \prime }\) and halt exactly at the first available moment. But Achilles is not asked to plant flags at all the half-way points, he is merely asked to cross the finish line. Any similar \(\triangle \ge \triangle PAC\) will do.
- 23.
Finite similar triangles may indefinitely be produced, but none of those are our litmus case. The litmus case is a similar triangle potentially infinite in area – the biggest possible one. As such, what we are primarily concerned with is the inability to halt on the triangle that lies ‘beyond’ all finite iterations of similar triangles and thus define even an arbitrary choice function f for this litmus case. For instance, suppose that the universe is continuous, finitely wide, but infinitely tall. Suppose that \(\angle \pi \alpha C\) is arbitrarily close to \(180^\circ \), and that base AC is the open interval bounded by the limits of the width of the universe. Does the line \(\alpha \theta \) always halt as it approaches the limit? Here, flags need to be planted, for there is no finish line to cross. After any flagged similar triangle there is an uncountable infinity of larger similar triangles. The choice function f for \(\triangle PAC\) cannot simply be defined retroactively.
- 24.
A classic illustration of this cognitive problem is Russell’s example of an infinite set of pairs of shoes and an infinite set of pairs of socks: while constructing a rule behind a choice function f on the (countably) infinite set of pairs of shoes is unproblematic (choose the right one), establishing any kind of a possible rule behind the ‘choice’ function f on the (uncountably) infinite set of pairs of socks is completely arbitrary.
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Therrien, V.L. (2020). A Diagram of Choice: The Curious Case of Wallis’s Attempted Proof of the Parallel Postulate and the Axiom of Choice. In: Pietarinen, AV., Chapman, P., Bosveld-de Smet, L., Giardino, V., Corter, J., Linker, S. (eds) Diagrammatic Representation and Inference. Diagrams 2020. Lecture Notes in Computer Science(), vol 12169. Springer, Cham. https://doi.org/10.1007/978-3-030-54249-8_7
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