An imaginative extension of Euclid’s parallel postulate into life, liberty, and the pursuit of happiness.
“The joy of existence must be asserted in each one, at every instant,” Simone de Beauvoir wrote in her paradigm-shifting treatise on how freedom demands that happiness become our moral obligation. A decade and a half later, the mathematician and writer Lillian R. Lieber (July 26, 1886–July 11, 1986) examined the subject from a refreshingly disparate yet kindred angle.
Einstein was an ardent fan of Lieber’s unusual, conceptual books — books discussing serious mathematics in a playful way that bridges science and philosophy, composed in a thoroughly singular style. Like Einstein himself, Lieber thrives at the intersection of science and humanism. Like Edwin Abbott and his classic Flatland, she draws on mathematics to invite a critical shift in perspective in the assumptions that keep our lives small and our world inequitable. Like Dr. Seuss, she wrests from simple verses and excitable punctuation deep, calm, serious wisdom about the most abiding questions of existence. She emphasized that her deliberate line breaks and emphatic styling were not free verse but a practicality aimed at facilitating rapid reading and easier comprehension of complex ideas. But Lieber’s stubborn insistence that her unexampled work is not poetry should be taken with the same grain of salt as Hannah Arendt’s stubborn insistence that her visionary, immensely influential political philosophy is not philosophy.
In her hundred years, Lieber composed seventeen such peculiar and profound books, illustrated with lovely ink drawings by her husband, the artist Hugh Lieber. Among them was the 1961 out-of-print gem Human Values and Science, Art and Mathematics (public library) — an inquiry into the limits and limitless possibilities of the human mind, beginning with the history of the greatest revolution in geometry and ending with the fundamental ideas and ideals of a functioning, fertile democracy.
to make these concepts less vague.
even, for example, about such matters as Life, Liberty, and the Pursuit of Happiness!
just a sordid struggle for mere existence!
happened at the beginning of the 19th century.
is still on the increase!
as it proved to be in mathematics.
as in a quiz program!
of the greatest value to the human race.
And this we certainly need more of!
How can THEY be “proved”?
from which to derive them!
from which to derive them.
This circularity of certainty permeates all of science. In fact, strangely enough, the more mathematical a science is, the more we consider it a “hard science,” implying unshakable solidity of logic. And yet the more mathematical a mode of thinking, the fuller it is of this circularity reliant upon assumption and abstraction. Euclid, of course, was well aware of this. He reconciled the internal contradiction of the system by considering his unproven postulates to be “self-evident truths.” His system was predicated on using logic to derive from these postulates certain consequences, or theorems. And yet among them was one particular postulate — the famous parallel postulate — which troubled Euclid.
The parallel postulate states that if you were to draw a line between two points, A and B, and then take a third point, C, not on that line, you can only draw one line through C that will be parallel to the line between A and B; and that however much you may extend the two parallel lines in space, they will never cross.
Euclid, however, wasn’t convinced this was a self-evident truth — he thought it ought to be mathematically proven, but he failed to prove it. Generations of mathematicians failed to prove it over the following thirteen centuries. And then, in the early nineteenth century, three mathematicians — Nikolai Lobachevsky in Russia, János Bolyai in Hungary, and Carl Friedrich Gauss in Germany — independently arrived at the same insight: The challenge of the parallel postulate lay not in the proof but, as Lieber puts it, in “the very ATTITUDE toward what postulates are” — rather than considering them to be “self-evident truths” about nature, they should be considered human-made assumptions about how nature works, which may or may not reflect the reality of how nature work.
This may sound like a confounding distinction, but it is a profound one — it allowed mathematicians to see the postulates not as sacred and inevitable but as fungible, pliable, tinkerable with. Leaving the rest of the Euclidean system intact, these imaginative nineteenth-century mathematicians changed the parallel postulate to posit that not one but two lines can be drawn through point C that would be parallel to the line between A and B, and the entire system would still be self-consistent. This resulted in a number of revolutionary theorems, including the notion that the sum of angles in a triangle could be different from 180 degrees — greater if the triangle is drawn on the surface of a sphere, for instance, or lesser if drawn on a concave surface.
It was a radical, thoroughly counterintuitive insight that simply cannot be fathomed, much less diagramed, in flat space. And yet it wasn’t a mere thought experiment, an amusing and suspicious mental diversion. It bust open the floodgates of creativity in mathematics and physics by giving rise to non-Euclidean geometry — an understanding of curved three-dimensional space which we now know is every bit as real as the geometry of flat surfaces, abounding in nature in everything from the blossom of a calla lily to the growth pattern of a coral reef to the fabric of spacetime of which everything that ever was and ever will be is woven. In fact, Einstein himself would not have been able to arrive at his relativity theory, nor bridge space and time into the revolutionary notion of spacetime, without non-Euclidean geometry.
The very dignity and ingenuity driving mathematics, Lieber points out in another lovely conceptual bridging of ideas, is also the motive force behind the central aspiration of human life, the one which Albert Camus saw as our moral obligation — the pursuit of happiness.
CANNOT SERVE as an instrument of thought!
just like any other postulate.
is a goal of human effort.
Man’s spirit and faith and even his life.
ACCEPTANCE of LIFE for the HUMAN RACE.
as in mathematics and sciences.
and is determined to conquer.
(e) FRIENDSHIP between K and K’!
(b) Differences in color of skin!
Complement Human Values and Science, Art and Mathematics with Carl Sagan on science and democracy, Robert Penn Warren on art and democracy, and Walt Whitman on literature and democracy, then revisit Lieber’s equally magnificent exploration of infinity and the meaning of freedom.