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 Euclidappears 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 bookFrom 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 thatEuclid, 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]

*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 PoincarĂ©, 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 PoincarĂ© 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.]

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.

Another example of geometry functioning visually are the japanese Sangaku -- wooden tablets placed in shrines or temples that contain a geometric puzzle. I would think the spirit is similar. Both in Euclid's elements and in the sangakus there is abstract beauty, but also a hidden statement that you can only decipher when you speak the language.

ReplyDeleteMatthias

Thanks, Matthias! I'll have to look them up. The relationship between shrines and geometry is a little puzzling, I must say. So to speak... :)

ReplyDeleteAny thoughts on this notion of totally formalized, purely abstract geometry, versus visual intuition? I'd love to have more input from mathematicians on this one. (I'm still waiting to hear from Mathieu).

I have many disorganized thoughts on this.

ReplyDeleteAs for Euclidean geometry: I believe its origins lie in shear necessity: For large scale architecture (temples, pyramids), valid geometric theorems are essential. You need to be sure that your angle is a right angle. This certainty can only be warranted by reason. So, as philosophers used to do, the early geometers subjected themselves to "evident truths" and drew conclusions. It is hard to imagine what it must have felt like.

The other side (visual intuition) is much harder for me to understand. I would contest some of your claims. For instance, that interest in visuality. When I use my pictures in (math) talks, people (math people) invariably ask about the significance of the colors. That is an ironic question: They either know or suspect that the slightly sicklish green cannot have a mathematical meaning, and are eager to expose this. Artists have a much better understanding of this, and I think I know why.

There is a common ground among artists and mathematicians: Both subject themselves to certain conditions: For the mathematician, it's axions and similar assumed conditions; for the artists, there are choices of medium, mood, or topic. This is more evident in music than in visual arts.

The 12 tone music confines the composer to a chosen tone row, and the choice of a Raga confines the Indian musician to scale and certain further limitations in using the material. Both mathematician and artist use these confines to explore possibilities. This process of explorations often leads to discoveries of something unseen or unheard. For instance, if the mathematician decides to consider all polyhedra that have regular polygons as faces such that at any vertex the same number of polygons meet, such an exploration would lead to cube, tetrahedron, and maybe octahedron. But only careful exploration would reveal that there are two more stunning objects, dodecahedron and icosahedron. The surprise is one half of the artistic element.

The other half is pure choice, unknown (despised?) to the mathematician. If one were to draw a dodecahedron -- the mathematician would not care about its rendering, as long as it conveys its essence. The artist, however, would realize that there are remaining choices: color, perspective, shading, that affect our senses rather than just our comprehension. Only of they are in some sort of concordance with the phenomenon (the dodecahedron, here), the artist will be satisfied.

This becomes rather apparent when looking at older mathematical illustrations (done by artists) of geometric figures. They are clearly not as abundant as today, but usually more sophisticated. For instance, it Hilbert and Cohn-Vossen's book _Geometry and Imagination_ there are perspective renderings of line configurations where the thickness of the line decreases with the distances. Logical and natural, but not so easy in Adobe Illustrator. Over a 100 years ago the artists knew what was important to convey the meaning of an image. It hasn't become easier with computer graphics if we ignore that knowledge.

Much of this is far over my head (at least my little-kid-interrupted-sleep head) but I am very enthusiastic about your observation that when "normal" comics conventions are jettisoned a new lexicon can be created. I certainly have tried that. I see it in Cubist work as well - a vocabulary that is shifting for sure and so maybe not too useful for math, but still, a language in all that means with all its slipperiness.

ReplyDeleteMatthias, I did say that visual intuition is not totally absent (despite what the Formalists might have intended) from geometry as actually practiced, and that as a matter of fact there has been a rise in interest in it more recently--which is where I showed your own work! So I actually think we agree on that. I was only using that (attempted) separation as a way of wedging in, so to speak, the possibility of abstract art, and especially sequential art, also picking up, in a way, the Euclidian mantle.

ReplyDeleteThe constraints you refer to, in music and math, have been even further exploited and formalized by literary movements such as the OuLiPo (which consisted of writers AND mathematicians) and, in comics, the OuBaPo (the Ouvroir de Bande Dessinee Potentielle). The OuLiPo, for example, worked a lot with combinatorics. And, also, isn't there choice in math too? In set theory, for example, as far as I can tell people choose whether to work in ZFC or NBG or some other axiom system. Which can produce different results.

I love the discussion of perspective you mention! Here is an example I always show in class to discuss how comic artists can manipulate line thickness (and line inflection vs. non-inflection) to show depth: http://i381.photobucket.com/albums/oo253/A280/DSCN4513.jpg. And, if I may, here is a page from my current project where I did some of the same thing: http://4.bp.blogspot.com/-vq2qYh6nqAo/UI9tWr7KPgI/AAAAAAAABJc/02zJkZKoaWo/s1600/408677_4418786742778_965904097_n.jpg

Delete"And, if I may, here is a page from my current project where I did some of the same thing: http://4.bp.blogspot.com/-vq2qYh6nqAo/UI9tWr7KPgI/AAAAAAAABJc/02zJkZKoaWo/s1600/408677_4418786742778_965904097_n.jpg"Andrei, do you know this drawing by Iannis Xenakis (a sketch for Erikhton) : http://2.bp.blogspot.com/_apFpwnMI2hs/TSAgAHdHaNI/AAAAAAAADi8/QzTTfgaz9Jw/s400/Xenakis.jpg ; when I saw again this, last day, I thought of some of your comics.

Mattias--weird, I had not seen that before! I've seen some of Xenakis's scores over the years, may even have posted some here, but I've never seen one that looks like that. Cool.

DeleteWarren--thinking about it, and also about what Matthias said--yes, and I think these new principles of narration end up being very musical, by which I mean a kind of inevitability can be established in the same way that a piece of music builds up tension and release through cord changes, and reaches a satisfying conclusion via a return to the tonic. Which also reminds me of the arc of a mathematical proof, with the final tonic being akin to the finally proven theorem... There may be more than a simple metaphorical parallel there, given the strong numerical basis of music. In a way, a piece of music is a passage through a series of groups of numbers (an A chord, for example, being 440 Hz, about 600 for the E, 880Hz for the A above that, etc). So in a way an E7 chord requiring to be resolved into an A is a mathematical relationship between sets of numbers...

ReplyDeleteI will cut my speculation short, I'm tempted to go far afield now.

Sorry, I meant about 660 for the E.

ReplyDeleteI am quite happy that my "proofs as abstract comics" got you inspired, Andrei, and that you liked Byrne's version of Euclid. And your post contains many quite interesting suggestions. I'll try to give my opinion on some of the points you mention, although I am a little bit unconfortable to write about mathematics from a philosophical/historical point of view for the following reasons: 1) My knowledge of the subject is quite scarce, so what I'll say will be approximative thoughts based mostly on my intuition (for what it's worth), and 2) to answer properly much more than a short post would be needed, and it's easy to draw false conclusions because of too many simplifications or generalizations. (And also, my level in english is probably not good enough for a philosophical discussion.) (And I am not a math professor but rather a high school teacher.)

ReplyDeleteThat said, I would agree to some extent with Andrei's claim that mathematics severed the links with visual intuition at some point in its history, I would say at the start of the 20th century. But I don't think it is actually true: my impression is that, due to the birth of new branches of mathematics such as set theory and logic (although some logic did exist during the Greek times) which had lead to paradoxes in the mere fundations (or to-be-fundations) of the subject, there has been a great need for some really rigorous and syntactical ways of defining what are the objects and rules that mathematicians use.

This "fundational" work has mainly been done during the first 30 years of the 20th century, and as a result the mathematicians and logicians of that time constructed a system in which every aspect of mathematics could

in principlebe encoded. But I think that most mathematicians satisfy themselves with the fact that this work having been done, one can completely forget about it and continue to do mathematics, using as much visual intuition as needed, because the tedious checking whether there are no unwritten assumptions is already behind them. (I am simplifying much here.) (I was too long: the rest in the next answer.)But there was a side effect to this surge of formalism: the way mathematics are communicated (which is usually very different from the way it is done) is very formalistic. Let me cite a long extract of an answer of William Thurston to a question on the forum Mathoverflow:

Delete"Mathematical ideas, even simple ideas, are often hard to transplant from mind to mind. There are many ideas in mathematics that may be hard to get, but are easy once you get them. Because of this, mathematical understanding does not expand in a monotone direction. Our understanding frequently deteriorates as well. There are several obvious mechanisms of decay. The experts in a subject retire and die, or simply move on to other subjects and forget. Mathematics is commonly explained and recorded in symbolic and concrete forms that are easy to communicate, rather than in conceptual forms that are easy to understand once communicated. Translation in the direction conceptual -> concrete and symbolic is much easier than translation in the reverse direction, and symbolic forms often replaces the conceptual forms of understanding. And mathematical conventions and taken-for-granted knowledge change, so older texts may become hard to understand."

(William Thurston was one of the greatest mathematicians from the 70s up to very recently. He passed away in 2012.)

What I wanted to say here is that like any other field, maths is prone to fashion, and at some points in its history it was fashionable (for valid reasons) to try to reduce maths to a symbolic art form. But intuitions of any kind, and among them visual intuitions, never really disappeared, although it may not seem apparent in its written form.

Now, getting back to abstract comics, I tried to do these "proofs as abstract comics" most of all as an experiment to see whether the fact of putting carefully chosen images in a sequence was sufficiently powerful to convey mathematical "abstract" ideas without using any typographical symbols. Well, I cheat since the title tells half of the story, and I have also been interested in the purely esthetical sensation the final rendering could bring (above all after seing Byrne's book which is a perfect example of something having both an esthetical and an abstract content). I would not dare to say that my comics bring anything new to the way mathematics can be seen, explained, made or thought. I hope, though, that they somehow show something about the "power of the sequence", which is the feature of abstract comics that fascinates me the most.

I hope that what I wrote is more or less understandable, please do feel free to correct my english mistakes or shaky language if you like (anyway, can anyone other than me correct it ?)

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Delete(removed above because of typos) : these kinds of ruminations (above) are what make blogs like this so viable - the pleasure of (over)hearing the personal reactions/ responses / thoughts of (intelligent) others on an esoteric subject is so rare....

ReplyDeleteThanks, Rosaire!

DeleteThis post and the comments are very stimulating. A couple points which might be of interest:

ReplyDeleteOne motive for the move away from visualization to symbolic reasoning (and for the emphasis on hidden assumptions in Euclid) was the surely the failure to "prove" the parallel "postulate" and the discovery of non-Euclidean geometries.

Also, I think it is important to distinguish visualization in the literal sense from visualization in the metaphorical sense of using imagined structures to intuitively reason about mathematics. I would guess that even avowedly "formalist" mathematicians make use of the latter type of visualization. Only computers "discover" proofs purely through symbolic manipulation - deciding which symbols to manipulate is driven by intuition and "imagery" on some level for any human.

A reason to emphasize this distinction is this interesting fact (learned in a psychology talk on spatial reasoning): amongst blind mathematicians, the most popular subdiscipline is geometry. In order for this to be a meaningful data point, some use of "visualization" in the above metaphorical sense must be at stake, yet also this is clearly not visualization in the literal sense. Anecdotally, apparently some blind geometers have claimed they can reason intuitively about space more easily because they are not hampered by the visual bias.

(Can there be an abstract comics for the blind? A non-visual abstract comics?)

Anyway, fascinating considerations in the above article - thank you for posting it and for the rest of this great blog.

Just a note--I have a policy against allowing anonymous comments, but this one had substance, so I thought I would let it through. But I hope the commentator comes back and identifies himself or herself.

DeleteAs to the substance of the comment itself--I don't know if the two kinds of visualization are altogether separate. The notion of the blind geometers would seem to confirm Kant's idea of humans having a pre-programmed, so to speak, understanding of Euclidean space--which is to say, an intuition of it; and this understanding can clearly be filled in through touch as well as sight. This would only be of historical interest if Brouwer and some of the other intuitionists hadn't made a similar claim, based on Kant, for the number system too (if I remember correctly).

So "visualization" can function as a mental construction of space (in geometry), whether sight assists it or not.

As for formalists using visualization--yes, I agree, see what I wrote above about practicing mathematicians. But in terms of the rhetoric and ideology of formalism, clearly such intuition was meant to be thoroughly banned.

I'm not sure I follow the notion of the failure to prove the parallel postulate as leading away from visualization--Euclid had never intended it to be proved, since it was an axiom, and therefore failure to prove it cannot be seen as a failure of the Euclidean system. Furthermore, as an axiom, it is clearly based on a visual intuition (albeit only within Euclidean space), foregrounding the priority of such visualization to formal proofs. Interestingly, while the Lobachevskian space is hard (for me, at least, and despite Poincare's attempt) to visualize, Riemannian elliptic geometry is quite intuitive.