Geometrical algebra: a parable

The following parable illustrates the historiographical madness that I argue against in my recent paper in the Archive for History of Exact Sciences.

Imagine a time some two thousand years in the future, when almost all records of our civilisation have been lost. Imagine that the only surviving mathematical works are some 17th-century calculus texts and a concise, definition-theorem-proof treatise on Lebesgue integration from the 20th century.

A few mathematicians are the first to study these surviving documents. They feel that these works are strongly connected and ultimately concerned with the same line of thought. To be sure, there are many immediate differences between these two groups of texts. The early calculus works are concerned with calculating areas. They seek specific numerical answers to specific problems, often applied ones. They rely on figures and informal reasoning. The Lebesgue integration treatise, on the other hand, contains no numerical calculations, no figures, no applications, no mention of areas. It is formulated solely in terms of sets.

But despite these stark differences, the mathematicians reading these texts feel that they are very closely connected when one looks beneath the surface. They see Lebesgue integration as ultimately a culmination of the line of thought begun with the 17th century works on the calculus. Their main argument for this is that the Lebesgue integration treatise makes a lot of mathematical sense when read in this way. Of course one could read it as a study of the properties of sets for their own sake, but from this point of view it is quite a mystery why the author found the various propositions in it to be of any particular interest. If one reflects on the foundations of the integration methods of the older calculus treatises, on the other hand, one can easily imagine how the informal methods used in them could lead to foundational quandaries, and how the specific propositions in the Lebesgue integration treatise make perfect sense when read as a concerted effort at providing these techniques with a formal foundation. Of course it is regrettable that the sources do not contain explicit statements that this was the purpose of the treatise, but then again it is not so strange since one can easily imagine that this problem context was commonplace background knowledge at the time, which the readers of the treatise would have been aware of. After all, the basics of integration would have been well-known for hundreds of years by the time the Lebesgue integration treatise was written, so it would not be strange for this treatise, which is clearly very advanced, to assume these more elementary matters as common knowledge.

But when the mathematicians put forward this interpretation of the texts, historians object vehemently. The mathematicians’ interpretation is pure fantasy and speculation, they say. They ridicule the mathematicians’ interpretation as amateur history and bombastically pronounce that real historians would never engage in such groundless flights of fancy. A proper historian sticks to the actual historical sources, they say, not some hypothetical reconstruction that happens to sound reasonable to a modern mind. And the sources in this case are unequivocal: there is not a shred of direct evidence that the Lebesgue integration treatise has anything to do with the earlier texts. The mathematicians alleged that it was ultimately about the problem of area calculation, but the fact of the matter is that it simply never speaks of area calculations at all. Instead it is about sets, which separates it from the earlier works by a vast conceptual gulf. To feign connections between works that embody such completely different modes of thought is nothing less than the cardinal sin of the historian. Historical texts must be understood in their own right, which in this case means that the Lebesgue integration treatise must be read as being about sets and only sets, and any allusion to the notion that these sets somehow are supposed to correspond to areas must be condemned as historical blasphemy. The historians go on to replace the supposedly naive earlier studies with “more sensitive” studies of the Lebesgue integration treatise that never uses any notions or notations not found in the text itself. These new studies seem to their authors to be superior to the old ones almost by definition, for what could good historical method consist in if not close adherence to the explicit documentary record, which is the only evidence we have?

Of course in this case the mathematicians are right and the historians are wrong, and the ostensibly “more sensitive” studies produced by the latter are in fact far less sensitive and indeed pre-destined by their historiographical assumptions to fundamentally miss the point of the text they are studying.

Lebesgue’s theory of integrals is so refined and abstracted that it has ceased being about specific numerical cases. It is primarily about certain theoretical, foundational questions suggested by concrete, numerical integrals, rather than being about such specific integrals itself. Nevertheless its core motivations ultimately trace back to such concrete integrals. It’s certainly possible to study Lebesgue integration as a formal theory about sets without knowing anything about the hundreds of years of more concrete analysis preceding it. But anyone who does so robs himself of the opportunity to understand the motivations of the theory in the first place.

The geometrical algebra hypothesis effectively says that the relation of Euclid’s geometrical algebra to specific quadratic equations is of precisely the same type. Specific numerical cases are much too trivial to interest Euclid. He is concerned with much deeper, foundational questions. As mathematicians regularly do in such situations, he uses a refined, formal language, which doesn’t make the connection with the previous, more concrete tradition immediately apparent. Nevertheless it would be a grave mistake to assume on this basis that his work is completely discontinuous with this previous tradition.

While this cannot be proved conclusively for the case of the algebra of quadratic equations, there is an analog in the Elements that is completely unequivocal: Euclid doesn’t show how to compute the areas of specific rectangles and circles, though of course he knew how to do so (such things were of course well-known “especially in the taxation office,” as Grattan-Guinness puts it). What Euclid does instead is to prove in the abstract how (the areas of) rectangles (I.36, VI.23) and circles (XII.2) are related to their sides and radii, i.e., the results that constitute the theoretical foundations for practical area formulas. The geometrical algebra hypothesis makes the analogous claim for the case of quadratic equations.

The Elements is obviously a highly formalised work that synthesises diverse strands of previous thought, much of which was surely expressed less formally at a previous stage. That the Elements is a work of this character is recognised by all, but the opponents of geometrical algebra seem to ignore its implications. For it is evident that a work of this character will by necessity and design distort previous conceptions of the topics it expounds. Its goals are not to convey faithfully the intuitions, motivations, and modes of thought and expression of the traditions that it incorporates, but rather to show that the main results of this tradition can be subsumed within its particular formal system.

Other works of this type are for example Weierstrass’s arithmetisation of the calculus, Hilbert’s axiomatisation of geometry, or the Peano--von Neumann construction of the natural numbers. Of course the authors of such works write in a formalised way, using terms and constructs that are far removed from the underlying, informal modes of thought that these works are intended to formalise. Nevertheless these authors of course take for granted that the reader will have the background knowledge to understand that the purpose of the work can only be fully understood in this context. They would not want their readers to take it as absolutely forbidden to depart from the exact literal expression of their treatises in any way. Their particular contribution lies in their particular formalisation, to be sure, but to understand the purpose and context of the work one must keep a more open mind. To maintain that the only “sensitive” way of understanding such works is to stick slavishly and pedantically to every quirk of its formal language, and condemn any attempt at understanding its motivations in broader terms, would clearly guarantee from the outset that no real understanding could ever be reached. And yet this is the exact analog of the policy that the opponents of geometrical algebra so eagerly advocate in the case of Greek mathematics.