This is going to be another one of my difficult posts, so let's begin with a pretty picture:
This is from Oliver Byrne's color-coded The First Six Books of the Elements of Euclid, 1847, which you can read for free here. Ibn al Rabin (aka math professor Mathieu Baillif--I'm allowed to say that, right, Mathieu? I don't think it's very much of a secret in the comics community) recommended it in his fascinating post on mathematical proofs as abstract comics. Ibn writes, "I was inspired by an incredible book of 1847 recently re-edited by Taschen: The elements of Euclid by Oliver Byrne. Anyone interested in abstract comics and/or mathematics should at least have a look at it." I couldn't agree more. As a matter of fact, I have bought the Taschen reprint since, and it's absolutely gorgeous.
In the comments to Ibn's post, I wrote about his comics/visual proofs: "What I find fascinating about this is the re-visualization of geometry, so to speak--an argument that geometry can function visually only (or primarily). A large part of the history of math has been so anti-visual, reducing geometry to equations, functions, as if the visual had to be purified away. I think the ideological basis of that move really needs to be addressed by a philosopher of mathematics." I thought I would elaborate on this a bit.
To begin with, let me explain what I was referring to there. Here are a few quotes from my trusty Eves and Newsom, that is, Howard Eves and Carrol V. Newsom, An Introduction to the Foundations and Fundamental Concepts of Mathematics (New York: Holt, Rinehart and Winston, revised ed., 1965). You all remember Euclid and the axioms of geometry, right? The authors, at this point, are discussing the attempts by several late nineteenth-century mathematicians (Moritz Pasch in 1882, Giuseppe Peano in 1889, and David Hilbert in 1898-99) to provide new sets of axioms for the discipline, by "ferreting out" the implicit assumptions of Euclid, which supposedly made geometry a less than rigorous subject.
As for Peano:[Pasch] declared that the creation of a truly deductive science demands that all logical deductions must be independent of any meanings which might be attached to the various concepts. In fact, if it becomes necessary at any point in the construction of a proof to refer to certain interpretations of the basic terms, then that is sufficient evidence that the proof is logically inadequate... From this point of view, Euclidean geometry is essentially a symbolic system whose validity and possibility for further development do not depend upon any specific meanings given to the basic terms [such as "point", "line" or "plane"] employed in the postulates of the geometry; Euclidean geometry is reduced to a pure exercise in logical syntax. Where Euclid appears to have been guided by visual imagery [my italics], and thus subjected to the making of tacit assumptions, Pasch attempted to avoid this pitfall by deliberately considering geometry as a purely hypothetico-deductive system. (91-92)
A rejection of visual imagery as logically unsound, an attempt to purify it away from the proper body of rigorous mathematics: no matter the validity of such a move (and the so-called "formalist" tendency to which these attempts belonged was soon questioned by mathematicians calling themselves Intuitionists; nevertheless, Formalism is still, as far as I can tell, largely dominant, not the least through its influence on computer science), it seems to invite the kind of critique of an anti-ocular bias that Martin Jay, in his book Downcast Eyes, diagnosed in philosophy. (I should add that similar moves can be found in other branches of mathematics, for example in the transition, that occurred, in the work of Henri Poincaré, at about the same time as the developments I mentioned above, from early topology to algebraic topology, something about which Mathieu I'm sure would have a lot to say.) [edit, in response to an email from Mathieu: my phrasing, I think, was a bit confused here. In no way did I mean to claim Poincaré was a Formalist like Hilbert. Quite the opposite, philosophically he was clearly much closer to the Intuitionists. What I was trying, in a muddled way, to say, was that his move toward alegebraic topology, where spatial intuition is increasingly replaced by algebraic formulae, was a move similar in its anti-ocular bias to that of the Formalists. I was not trying to imply any further parallels beyond that.]From many points of view Peano's work is largely a translation of Pasch's treatise into the notation of a symbolic logic which Peano introduced to the mathematical world. In Peano's version no empiricism is found; his geometry is purely formalistic by virtue of the fact that it is constructed as a calculus of relations between variables. Here we have the mathematician's ultimate cloak of protection from the pitfall of over-familiarity with his subject matter. We have seen that Euclid, working with visual diagrams in a field of study with which he was very familiar, unconsciously made numerous hidden assumptions which were not guaranteed to him by his axioms and postulates. To protect himself from similar prejudice, Peano conceived the idea of symbolizing his primitive terms and his logical processes of thought. Clearly, if one is to say, "Two x's determine a y," instead of "Two points determine a straight line," one is not so likely to be biased by preconceived notions about "points" and "straight lines"... The derivation of theorems becomes an algebraic process in which only symbols and formulas are employed, and geometry is reduced to a strictly formal process which is entirely independent of any interpretations of the symbols involved. [all my italics, again]
What is more paradoxical, though, about this move is that it transforms the very goal of the discipline. Geometry began as (and has been since, for the largest part of its existence) an inquiry not into just proofs about two-dimensional, three-dimensional, or what have you, shapes, but into their very constructibility. And by this I mean, not only "constructibility" as in the possibility to construct an abstract mathematical object (such as, oh, Aleph One), but the much more empirical possibility of drawing a line or circle to exact specifications, using just a compass and ruler.
This can be seen in Euclid from the very beginning of his book. Proposition I asks how, in Byrne's translation, "on a given finite straight line to describe an equilateral triangle":
And Proposition II shows how, "from a given point, to draw a straight line equal to a given finite straight line":
That is, Euclid's geometry is not just about proofs and deductions: it's about space, it's about constructions. Spatial notions are not just "visual imagery," errors in waiting, always prone to lead into error the pure mathematician and to impurify his (or "her," but mostly "his"--have you ever set foot in a math department?) "hypothetico-deductive system." They are the very essence of Euclid's science. The moment geometry was transformed into a purely logical or algebraic formal system, it became something else, something else still designated by the same name.
Now, of course, working geometers might not care quite so much about the rigorous philosophies I mentioned; I doubt that an actual visual intuition is totally absent in the work of most of them (though I don't know about some topologists). And furthermore, with the rise of computer imagery, we have seen a new interest in visuality, for example through fractals, which are graphs of iterated functions of such visual complexity that they largely could not be plotted before computers; or for that matter through so called "minimal surfaces," as for example studied by my IU colleague (and neighbor!) Matthias Weber:
Nevertheless--and I guess this is my point--mathematics having at some point severed its connection to visual intuition (or at least having attempted to do so), maybe visual artists can claim squatters' rights to that part of Euclid's legacy? I don't want to make too large claims here, so take this with a grain of salt, or imagine me with my tongue in my cheek, or choose the idiom (or the cliche) you wish. But in its existence as a manual of visual constructions and proofs, The Elements can be seen as lying equally at the source of mathematics as at that of abstract art (not to mention architecture or design). As a book of which each proposition is a sequence of spatial moves or constructions, it can also be seen (as Mathieu argued) as an early form of abstract sequential imagery, of abstract comics.
Furthermore. In a good number of abstract comics (just skimming through the anthology, those of Warren Craghead, Mark Gonyea and Greg Shaw, and I hope mine too), representational, illustrated narrative having been abandoned, new sequential principles are clearly established. To my mind, more often than not, such principles, in attempting to build a formal narrative arc and reach a conclusion, are (at least intuitively) closer to the logical sequencing, the building up of tensions, and the resolution through drawing surprising connections, that characterize a geometric (and even other kinds of mathematical) proof. Mathieu's abstract-comic proofs make this eminently, beautifully clear:
One more thought, as a P.S. If unspoken assumptions are supposed to have been built into images, couldn't one say that they have similarly been built into the language in which the new axioms are phrased? And if that is the case, couldn't a picture of a line, or of a plane, as an act of sheer presentation, with no further assumptions, function just equally successfully as axiom, as a phrasing of such? Or as a premise? Assume there is a square, a proposition could begin. Or better yet, don't assume anything. Look. Here is a square. But don't read what I just wrote. Let the picture do the talking. Or rather, let it do the existing.
From here, we could go to a different kind of logic--say, perhaps, to a kind of box logic (yet one where the shapes of the boxes matter, where the forms are not just symbols...). But that is a discussion for another time.