WHAT MIGHT CAUSE THE CHESHIRE CAT TO VANISH of everything but its grin? An increasing abstraction in mathematical discipline would surely be to blame, naturally. To claim that each of us inhabit Wonderland will come as no surprise to some of you. Perhaps the far more practical question is—like Alice; are we trying to make sense of ourselves in proportion to the nonsensical world surrounding us, or like the creatures which tormented her tutoring; have we settled in as one of Wonderland’s curiouser and curiouser residents? Hold off on your response, because author Lewis Carroll drew a line of the mathematical nature in the sand, and which side we find ourselves on, based of course on the author’s own account of what may now be referred to as the Science delusion, particularly the math which makes its fabrications entirely possible, will conveniently answer the question as to where we should stand.
Charles Dodgson, pen name Lewis Carroll, was a stubbornly conservative mathematical lecturer and tutor at Christ Church College in the University of Oxford, tasked with preparing its students to pass examinations in the numbers department. For the record, it was the 19th century. His was a generation of swirling tempests, when new controversial concepts in mathematics, such as imaginary numbers (like the square root of a negative number), symbolic algebra, and projective geometry—characteristic expressions only found in allegorical formalism—were explored and ultimately exploited as groundwork to expand whatever perceived realities its master magicians could conjure.
In earlier centuries—for more than two-thousand years, in fact, Euclid’s Elements had been the personification of rationality. The very adjective “Euclidean” was unnecessary on the mere basis that no other sort of geometry had been conceived. His was grounded in a physical reality and backed by rigorous reasoning. Theorems which might fall under his axioms, once proven, were deemed an absolute truth. In such a world, mathematical objects were conceived as the ideal representation of their physical counterparts, both in universal applicability and the objects themselves. Masterfully applying the geometry of circles, quadrilaterals, parallel lines and trigonometry, Euclid could settle complex arguments using simple, logical steps. And it worked every time.
These were principles however which Dodgson’s contemporaries were openly straying from. The Euclidean presentation was worn fashion; old news; a tired code of moral conduct; because after all, it was the 19th century, and humanism reigned supreme. Rene Descartes was leading the charge of abstraction within the darkened catacombs of his skull. He introduced geometry of an analytical nature. Post-Cartesian mathematics gave its artists freedom to explore new unseen ideas, so long as these manipulated concepts found a consistent framework of operations. Such calculational techniques of modern mathematics weren’t simply the favored tool for esotericism. New Science had found its art too. Russian Nikolai Lobachevsky (1792-1856), Hungarian János Bolyai (1802-1860), Germans Bernhard Riemann (1826-1866) and Carl Friedrich Gauss (1777-1855) further advanced the post Euclidean abstractions once explored by Descartes, and in the intervening years, an entire fairyland of geometric worlds would explode to consciousness beyond that which Euclidean logic had been intended, mainly in N-dimensions, projective geometry, affine geometry, and finite geometry. And yet all of this would seemingly culminate with Albert Einstein’s theory of general relativity. Space itself, according to Einstein, is not Euclidean.
In short, Wonderland inhabited both inner-space and the fourth-dimension of time. Over a decade after Alice’s famous adventures—1879 to be exact, Dodgson would publish his mathematical treatise, “Euclid and his Modern Rivals,” where-as thirteen contemporary geometry textbooks were exhibited and shown to be of inferior quality or equally—if not better explained—by Euclid. According to Dodgson, post-Cartesian mathematics was a nonsensical land of the delusional mind, where his students were being guided to perversion and led away from an arithmetic which actually described the real world. For a true Euclidean, planes are flat and parallel lines never meet. Yet this is simply not so on a globe. The earth has multiple longitude lines that all meet at the North and South Poles, despite being parallel. Alice’s Adventures in Wonderland was written with an ink-dipped dagger, mocking such reckless concepts which opened up a slippery slope between what one could fathom through the language of algebra and that of geometry at the cost of concrete existence. Wonderland was a warning for the 20th century, particularly what it might become. Depending on what side of the line you’re standing on, you’ll agree with me. Mad results followed.
“I wonder if I shall fall right through the earth! How funny it’ll seem to come out among the people that walk with their heads downward!” – Alice
Indeed, we all inhabit Wonderland. And like Alice, falling down the rabbit hole means leaving the world of Euclidean geometry behind. Goodbye plane Earth. Hello globe logic.
More to come.