IF WONDERLAND HAD A MAXIM, IT MIGHT READ: *reductio ad absurdum*, which, when translated from the Latin, means “reduction to absurdity.” In philosophical logic as well as mathematics, it is a form of argument quite familiar with Euclid’s proofs and which attempts to discredit a statement by establishing contention between both parties and showing the absurdity of its denial, which will, if its course is not properly redirected, inevitably lead all participating members to a ridiculous and most impractical conclusion. With that in mind, let me now introduce to you one of Wonderland’s most esteemed residents; the caterpillar whom Alice, being only 3-inches tall at the time, had the displeasure of taking advice from.

He is a 19^{th}-century London math professor, and his name, according to Oxford PhD graduate Melanie Bayley, is none other than Augustus De Morgan. De Morgan was the first British mathematician to lay out a consistent set of rules for symbolic algebra. Morgan’s book, *Trigonometry and Double Algebra*, first published in 1849, “explained the departure from universal arithmetic,” wrote Bayley, “where algebraic symbols stand for specific numbers rooted in a physical quantity–to that of symbolic algebra, where any “absurd” operations involving negative and impossible solutions are allowed, provided they follow an internal logic.” Bayley added, “De Morgan wanted to lose even this loose association with measurement, and proposed instead that symbolic algebra should be considered as a system of grammar. ‘*Reduce* algebra from a universal arithmetic to a series of logical but purely symbolic operations,’ he said, ‘and you will eventually be able to *restore* a more profound meaning to the system.’” Much like the hookah-pipe which the caterpillar smoked, De Morgan employed algebra’s original Arabic translation in his footnotes, “*al jebr e al mokabala*,” which means, “restoration and reduction;” but more on that in a moment.

If Dodgson, aka Lewis Carroll, took personal displeasure with symbolic algebra, it’s partly because of the added aggravation he saw in his own students, who had to *unlearn* a perfectly logical system of real world practicality for something which might easily become perverse in application, as Alice would soon come to find out.

Firstly, consider Alice’s baffling stabs at reality. Already she had grown far too behemoth in size to enter through the Lilliputian door that led directly into the rose garden. To quote Alice, “Now I’m opening out like the largest telescope that ever was! Good-bye, feet!” After indiscreetly indulging in the mysterious bottle that read: DRINK ME, she was then too small to reach the key on the table. An oddly placed cake on the floor made her enormous; the White Rabbit’s fan shrunk her down to size again. After nearly drowning in her own tears and surviving the political campaign-trail of a Dodo-based Caucus race, another unmarked bottle caused her to swell up again in the Rabbit’s house while, quite conveniently, pebbles thrown in the window turned to cakes and, upon ingestion, was thoroughly waned again.

“Being so many different sizes in a day is very confusing,” she confessed to the caterpillar, having been asked, rather impolitely, *who* she was, “…at least I know who I was when I got up this morning, but I think I must have been changed several times since then.”

Certainly she has changed—quite inconsolably too; fluctuating anywhere between 9 feet and now 3 inches. Alice functioned best in a rational world, where her multiplication tables and something as simple as poetic grammar might be rightfully recited. Not so here. “Alice,” writes Bayley, “bound by conventional arithmetic where a quantity such as size should be constant, finds this troubling.” The post-Cartesian rules which regulated Wonderland were very confusing indeed, *especially* for the traditional Euclidean sort such as Carroll. But for the caterpillar, who behaves as erratically as the imaginatively skewed world he inhabits—for you see, he has seemingly mushroomed up from nowhere; responds with the very apathy which might lend someone to believe the little girl, and her mocking narrator, are both siding with the wrong side of history. Regarding any concluding notion that Alice is dutifully confused or ebbed of mind, he quips, “It isn’t.”

Lewis Carroll’s views on the madness of this new symbolic algebra might best be explained—*reductio ad absurdum*—by the mushroom which the hookah-smoking caterpillar instructed her to partake of, seeing as how “one side will make you grow taller, and the other side will make you grow shorter.” Esoteric references aside, this advice was dreadfully confusing for Alice, “trying to make out which were the two sides of it; and as it was perfectly round, she found this a very difficult question.” Perhaps this perplexing circle of rationality relates to Dodgson’s many tutoring sessions. Unfortunately for Alice, she calculated the wrong balance between *tall* and *short*, because within a moment her chin collided with her foot.

What happens when Alice eats the other side of the mushroom? Carroll writes: “…she found that her shoulders were nowhere to be found: all she could see, when she looked down, was an immense length of neck, which seemed to rise like a stalk out of a sea of green leaves that lay far below her.”

Suffice to say, Alice might have interpreted the Caterpillars further promulgation of apathy, particularly his advice, to “Keep your temper,” quite differently than Lewis Carroll’s own contemporaries. Among Oxford’s educated, the word “temper” still retained its original definition, essentially meaning: “the proportion in which qualities are mingled.” One might consider *tempered metal*. And here, when faced with Alice’s difficulty formulating a tempered resolve with the mushroom, we can once more discern Carroll’s own frustration with symbolic algebra. It is a dreadfully poor substitute to the grounded realities of Euclidean geometry.

More to come.