“It is by logic that we prove, but by intuition that we discover.” (Henri Poincaré)

## Introduction

It is our day-to-day experience that rigid objects, e.g. two books, can be placed right next to each other, in a way that they touch but don’t overlap. However, if we think of space as composed of points, meaning that space is nothing more than a set of points, the concept of touching becomes different: namely, if two geometrical objects touch each other, they always overlap. For example, when a cube touches a sphere, they will have one point in common where they overlap. This is inconsistent with our intuition that a given piece of space, even a single point for that matter, cannot be occupied by multiple rigid objects at the same time.

## Veronese’s geometry

There is a geometry little known today, developed by G. Veronese in the 19th century, that avoids the aforementioned inconsistency by design. The idea is that the continuum, e.g. a straight line, or the 3-dimensional space, is not composed of points.(1) The points do belong to the continuum though, but are not part of it. Just like the graph of a computer network: it belongs to the network, but is physically not part of it.

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Fig. 1: Point of separation.

To illustrate the idea, imagine that Fig. 1 depicts a geometrical straight line. It consists of two disjoint, touching half-lines: one is of green color, the other blue. The point is the “thing” in the middle that marks where the two are separated. Clearly, the point belongs to both half-lines, since it marks where they end. But is it also part of them? If it was part of one, then, due to the symmetry of the situation, it would be part of the other too. With that, the green and the blue half-lines could not be disjoint. Thus, the point is not part of either half-line, and as such it’s not part of the line either.

Book 1      Book 2
Book 1      Book 2
Book 1      Book 2
Book 1      Book 2
Book 1      Book 2
Book 1      Book 2
Book 1      Book 2
Book 1      Book 2
Book 1      Book 2
Book 1      Book 2
Book 1      Book 2

Fig. 2: Two books touching each other.

In general, the contour of a geometrical object belongs to the object, but is not part of it. Fig. 2 shows another illustrative example, where two planar books touch each other. Again, the straight line segment in the middle (would be a rectangular surface in 3 dimensions) that marks the separation between the books is not part of either book, nor it is part of the plane in which the books “live”. But it does belong to them!

In Veronese’s geometry, the continuum in question is investigated by means of a superimposed system of all points. The underlying assumption is that differences among geometrical objects manifest themselves through differences in the respective collections of points that belong to each object. That is, a geometrical object is unambiguously determined, and thus can be defined, by its points.(2)

The superimposed system of all points must be dense enough, so that applying geometrical operations cannot lead out of it. For example, ℤ3 would not be appropriate in a 3-dimensional continuum because, among other shortcomings, the straight line that passes through (0, 0, 0) and (100, 1, 0) would not intersect in ℤ3 any of the straight lines that pass through (n, 0, 0) and (n, 1, 0), where n is an integer between 1 and 99.

The ordinary analytic form of Euclidean geometry would correspond to the case where the system of all points is ℝ3, which is already dense enough. However, Veronese goes beyond that and postulates a 3-dimensional (number) system that includes infinitesimally small as well as infinitely large quantities. He also provides a method by which similar systems can be constructed for continua of up to infinitely many dimensions.

The paradoxes start with the assumption that motion does exist. Then, an argument is presented that arrives at a contradiction, and the conclusion is drawn that motion cannot exist.

### Achilles and the tortoise

Problem
Eventually, Achilles catches up with the slower tortoise. His path will then necessarily include infinitely many (gradually decreasing) consecutive segments. Therefore, he would have to cover an infinite number of distances before he finally reaches the tortoise. But this is impossible, as no one can ever complete an infinite sequence of actions.

Solution
The seeming contradiction arises from the unspoken assumption that Achilles must become conscious of each individual segment during his motion. Intuition tells us that humans can only have a finite number of discrete conscious experiences behind them at any point in time. How is then Achilles’ motion possible? Well, being conscious of the whole (path) does not imply being conscious of its (infinitely many) parts.

Remark
One can eat a bowl of rice without becoming conscious of every single grain of it.

Problem
A flying arrow occupies exactly one position at any moment. Thus, at every instant of time, the arrow is at rest. But then the motion of the arrow is impossible, since time is composed of such (motionless) instants. In other words, something that is always at rest cannot be in motion.

Solution
The paradox is baffling at the very least, until Veronese’s conception of the continuum comes to the rescue. Although instants (points) of time do exist, with exactly one position of the arrow belonging to each of them, time itself is not composed of instants. Time can only be decomposed into intervals of non-zero length, and in none of those is the arrow at rest.

Remark
This is a purely mathematical resolution of the paradox. No need to resort to physics.

## A note on time

Imagine people living in an infinitesimally small world. Their chosen unit of length (i.e. their “1 meter”) must be comparable to their physical sizes, which is infinitesimally small from our perspective. Analogously, since time resembles in many aspects the spatial dimensions, their chosen unit of time (i.e. their “1 second”) may also be infinitesimally small from our perspective.

The corollary would be that by the time we notice the tiniest change in our world, in an infinitesimally small world literally eternities would have elapsed.

This would open up interesting possibilities that seem to fall into the sci-fi category. To mention one, if we could mandate somebody in an infinitesimally small world to execute an extremely time-consuming algorithm for us, we would get back the result from them (or from one of their descendants) within the blink of an eye.

## Achilles’ arrow

Applying Veronese’s theory to time entails the existence of both infinitesimally small and infinitely large durations. This alone can be a rich source of riddles and paradoxes, especially when one attempts to reconcile it with human consciousness.

As an example, let’s assume that Achilles shoots an arrow whose velocity is infinitesimally small. Will it ever reach its (standing) target? Algebraically, yes, in an actual infinite amount of time. But what does it really mean? Will an immortal observer see it if they wait long enough?

G. Fisher, P. Ehrlich [ed.] (1994), Veronese’s Non-Archimedean Linear Continuum
J.L. Heiberg [ed.] (1883-1885), R. Fitzpatrick [trans.] (2008), Euclid’s Elements of Geometry
D. Hilbert (1898-1899), E.J. Townsend [trans.] (1902), The Foundations of Geometry
L. Keele (2008), Theories of Continuity and Infinitesimals: Four Philosophers of the Nineteenth Century
H.J. Keisler (2011), Foundations of Infinitesimal Calculus
P. Lynds (2003), Zeno’s Paradoxes: A Timely Solution
H. Poincaré (1902), E.V. Huntington [trans.] (1903), Review of Hilbert’s Foundations of Geometry
W.M. Strong (1898), Is continuity of space necessary to Euclid’s geometry?
G. Veronese (1891), A. Schepp [trans.] (1894), Grundzüge der Geometrie
G. Veronese (1908), P. Ehrlich [ed.], M. Marion [trans.] (1994), On Non-Archimedean Geometry

## FAQ

(1) Composed of points or not? Who said what?
Hilbert
The fundamental elements of Hilbert’s geometry are points, straight lines, and planes. Any other geometrical object is defined in terms of these. For example, the segment (of a straight line) is defined as a “system” of two (end-)points, and the circle as a “totality” of points. Obviously, Hilbert’s goal was logical soundness rather than coherent storytelling.
Poincaré
“There is supposed to be, between the elements of the continuum, a sort of intimate bond which makes a whole of them, in which the point is not prior to the line, but the line to the point.”
Euclid
In its original form, Euclid’s geometry presents a more appealing story than that of Hilbert, albeit logically less sound. The segment and the circle are both defined as “lines”, which is in accordance with intuition. The question whether lines, surfaces, and solids are composed of points is left open.

(2) What is a geometrical object and when does a point belong to it?
A geometrical object is part of either the continuum in question or a lower-dimensional sub-continuum thereof. A point that belongs to it marks where the object or a part of it may touch another geometrical object. To be able to define geometrical objects by their points, it is important to postulate criteria as to when a collection of points belongs exactly to a part of a (sub-)continuum and when it does not.