George De Bothezat's Teaching on the Infinitesimal

Scientific Paper
Title	George De Bothezat's Teaching on the Infinitesimal
Read in full	Link to paper
Author(s)	Peter F Erickson
Keywords	infinitesimal, continuity, infinity, absolute space, philosophy of mathematics, Georges A de Bothezat
Published	2012
Journal	Proceedings of the NPA
Volume	9
No. of pages	3
Pages	146-148

Read the full paper here

Abstract

This is a review of George de Bothezat's account of the spatial infinitesimal in his work, Back To Newton: A Challenge to Einstein's Theory of Relativity.

Overview

This is a philosophy-of-mathematics paper rather than a physics one. Peter F Erickson reviews the treatment of the spatial infinitesimal in Back to Newton: A Challenge to Einstein's Theory of Relativity (1936), a book by Georges A de Bothezat — the Russian-American aeronautical engineer better known for building the US Army's first helicopter, the quadrotor "Flying Octopus", in 1922.

Erickson explains that he had not encountered de Bothezat while writing his own Absolute Space, Absolute Time, & Absolute Motion (2006), and that the book was brought to his attention afterwards — by Greg Volk, according to his references. He recognises de Bothezat as a precursor of his own work, and the review is structured accordingly: what de Bothezat got right, and where Erickson thinks he stopped short.

The governing question is stated in the conclusion: "Is there a stop to the division of space?" Erickson holds that there is, and that the stopping point is the spatial infinitesimal — the smallest possible element of space, having location but no parts, no centre and no edge, and incapable of motion. The alternative, he argues, is an "inner infinity" that dissolves matter into nothing.

The argument

What de Bothezat is credited with

Erickson credits de Bothezat with grasping that a point of literally zero dimension is self-contradictory, quoting him: "To consider points as having zero dimension and line and surfaces having no thickness is but a self-contradiction." On de Bothezat's account a straight line is made of "infinitesimally small dots touching one another".

Erickson stresses the importance of the line being completely filled — any gaps would admit an inner infinity at every point, a notion he attributes to Lenin, Russell, Cantor and "perhaps Einstein". He also credits de Bothezat's treatment of irrational numbers as quantities on a continuous segment that no finite division into equal parts can ever reach, adding that this exhibits the distinction between the finite and what he calls the sub-finite.

He departs from de Bothezat on the imagery: infinitesimals are not dots, because a dot has a shape, and the infinitesimal lies below the level at which shape exists — "it is precisely because they have no shape that all shapes are possible."

Phenomenalism and the doubling argument

Erickson locates de Bothezat's central weakness in an acceptance of Berkeley's "to be is to be perceived". The illustration is de Bothezat's endorsement of Poincaré's doubling thought-experiment: if everything in the world doubled overnight, including every measuring standard, nothing would have changed, because size is a ratio and all ratios are preserved.

Erickson rejects this. Size, he argues, is not a ratio but an intrinsic magnitude: "A pencil lost in a landfill still has its size, though we are unaware of its presence." An object twice as large would necessarily occupy twice as much space — that is, contain twice as many infinitesimals — whether or not any instrument could register the change.

Numbers, zero and infinity

On multiplication, de Bothezat held that the product of two unit-quantities is a quantity of a different kind: 3 feet times 2 feet is 6 square feet. Erickson accepts this but says it is not universal, proposing a "linear multiplication" in which the unit is unchanged — "2 feet times 3 feet is simply 6 feet along a line".

On zero, he objects to de Bothezat's account of it as "a negation expressed by an affirmation", and introduces a distinction between two zeros: the zero of neutrality, a point on the number line between +1 and −1, and "utter zero", a total absence of being. He regards this distinction as indispensable to his The Nature of Negative Numbers (2011).

On infinity, de Bothezat defined an absolute infinity as n/0. Erickson replies that if the divisor is utter zero no quotient exists at all, and if it is the neutral zero the result is merely very large — not boundless. He proposes instead an "immeasurable infinity", n/mε, a quantity not endlessly great but permanently beyond human ascertainment, of which the number of infinitesimals in a given length is his example.

Infinitesimals of higher order

The decisive criticism concerns de Bothezat's infinitesimals of higher orders, in which each infinitesimal interval is itself divided into infinitesimals of the next order. Erickson argues that this reintroduces exactly what de Bothezat's best insight excluded: if the process is endless it produces an inner infinity, in which "every point would contain within it holes... Yet, even the rims of these holes would have holes", and space and its contents "amount to a fallacious sum — utter nothing".

He anticipates the reply that the orders might terminate, like a stack of cannon balls with chinks packed by smaller spheres, and answers that a final order fitting the chinks would have to have a shape — which contradicts the shapelessness that makes shape possible. He concludes that de Bothezat "was not far enough from Einstein's conception of curved space", and that for all his desire to return to Newton he "took a middle position".

Assessment

The review is a genuine review: it credits as well as criticises, cites de Bothezat's pages precisely, and rescues an obscure 1936 book by a figure of real accomplishment. Erickson is also explicit about his own commitments, which makes the argument easy to follow. Three difficulties should be recorded.

The central "evident contradiction" is not one. Erickson's argument against endless divisibility is that a body and a body twice its size would have the same quantity of potential parts, which he calls "an evident contradiction" and attributes to Russell and Cantor as an embraced absurdity. But a set that can be placed in one-to-one correspondence with a proper subset of itself is precisely Dedekind's definition of an infinite set, and the equinumerosity of the intervals [0,1] and [0,2] is a theorem of standard set theory, not an inconsistency in it. No contradiction has ever been derived from it. Erickson and de Bothezat are both entitled to reject Cantorian set theory in favour of a discrete alternative — that is a legitimate philosophical position with a long history — but rejecting a consistent theory because its consequences are counter-intuitive is a different act from exhibiting a contradiction within it, and the paper presents the second where it has done the first.

The correction on multiplication runs the wrong way. De Bothezat's rule — that multiplying two unit-quantities yields a quantity of a different kind — is ordinary dimensional analysis, and it is correct: length × length has the dimensions of area. Erickson's proposed "linear multiplication", in which 2 feet times 3 feet gives 6 feet, is not the multiplication of two lengths but scalar multiplication, 2 × (3 feet), in which one factor is a dimensionless number. The two operations are distinct and both are standard; de Bothezat had described the first correctly, and the amendment conflates it with the second. This is the one point in the paper where the correction offered is itself the error.

The doubling argument is a genuine dispute, not a mistake. The position Erickson attributes to de Bothezat's phenomenalism is Poincaré's, and in physics it is the mainstream one: measurable predictions depend on dimensionless ratios, and a global rescaling that changes every standard alike changes no dimensionless quantity and is therefore unobservable in principle. Erickson's reply — that the pencil in the landfill retains its size regardless — is a metaphysical assertion of intrinsic magnitude. It is a coherent position, essentially Newton's against Leibniz, and it is the one this wiki's absolute-space tradition generally takes; but it disagrees with de Bothezat about what size is rather than catching him in an error, and the paper's flat "He was wrong" understates that.

Finally, it is worth saying that the underlying question is not idle. Whether space is infinitely divisible or has a smallest element is live in contemporary physics, where discrete approaches such as loop quantum gravity and causal set theory take the second view, and the Planck length is routinely invoked as a scale below which classical geometry is not expected to hold. Erickson argues the case a priori, from the demand that a line be filled without gaps; the modern programmes attempt to make the same question empirical, by looking for observable consequences of discreteness. The two approaches do not meet, and the paper does not engage the physical literature — but the question it presses is a real one.

See also

• Georges A de Bothezat — helicopter pioneer and author of Back to Newton
• Peter F Erickson
• Absolute Space, Absolute Time, & Absolute Motion, The Nature of Negative Numbers
• Henri Poincaré
• Aether