26 January 2007

Time, Space, Math

Filed under: Culture, Ktismata, Language — ktismatics @ 4:14 pm

More from Crosby’s The Measure of Reality. Here are some of the quantitative advances that helped transform medieval Western Europe into the most powerful place on earth. (This post is my first use of my new computer – a MacBook. Hopefully it won’t crash a couple times a day like my old PC did.)

Clocks. People tend to think of time as a continuous flow, so the medievalists based their inventions for measuring time on flowing substances like water, sand and mercury. The Chinese invented a mechanical clock in the 10th century, but it wasn’t until late in the 13th century that some anonymous European rethought the basic metaphor, conceiving of time as a succession of discrete quanta. The key technological breakthrough was the “escapement” – a kind of oscillating notched gear that regularly interrupts the descent of a weight into thousands of small steps per day. Soon every big city in Europe, and then even the smaller ones, made sure it had at least one clock. It was the one piece of complicated machinery the average Joe encountered in everyday life. The clockwork mechanism itself became a metaphor for how God might operate the universe: not like an organism but like a machine.

Maps. The compass found its way from Asia to Europe in the 11th century. In the late 13th century the portolano was invented: a map of coastlines with compass courses drawn on them with a straightedge. But a really useful navigational map would show accurate distances as well as headings. Around 1400 a copy of Ptolemy’s Geographia arrived in Florence by way of Constantinople. In the 2nd century Ptolemy slapped a gridwork across the earth’s surface, even compensating mathematically for the earth’s curvature. By the time the Americas and the Pacific were discovered, mapmakers already had a way to represent them.

Astronomy. In the 14th century, the Frenchman Nicole Oresme wondered aloud why God would put the earth at the center of the universe. Why wouldn’t the center be somewhere in the heavens, where God lives? Nicholas of Cusa, a 15th century cardinal and philosopher, argued that God transcended the universe, so from God’s perspective there was no center. In the early 16th century Copernicus the Pole reintroduced an old Platonic and pagan idea: maybe the earth revolves around the sun. The heavens are huge: how could they circle the earth in a day? Far easier to set the earth in motion. Copernicus was a mathematician, and he performed exhaustive calculations to describe the heliocentric model quantitatively. The dimensions of the Aristotelian model were large; in Copernicus’s system they were nearly inconceivably vast – at least 400,000 times larger. Later in the 16th century Giordano Bruno, a monk, described the universe in terms that anticipated Newton, but that would also get him burned at the stake:

There is a single general space, a single vast immensity which we may freely call Void: in it are innumerable globes like this one on which we live and grow; this space we declare to be infinite, since neither reason, convenience, sense-perception nor nature assign to it a limit.

Mathematics. In The Geometries of Qualities and Motion, Oresme generalized the key principle that had made clocks possible:

For measuring continuous things it is necessary that points, lines, and surfaces, or their properties be imagined… Although indivisible points, or lines, are nonexistent, still it is necessary to feign them.

Roman numerals remained the dominant counting system in Europe until the 16th century. Arabic numerals greatly aided calculation, but only if you understood the place values of numbers lined up in columns, and if you could get your head around the mysterious number zero. Numbers are for counting things that are; zero is a sign for what is not. People started mixing the numbering systems together; e.g., IVOII meant 1502: I in the thousands column, V in the hundreds… The signs + and – didn’t show up in print until 1489. The 16th century witnessed the invention of the = sign, the decimal system for expressing fractions, and algebraic notation. Mathematical advances also stimulated mystical applications: astrology grew in popularity, and kabbalistic numerical interpretations of the Bible abounded.



  1. You got a Mac??? Good for you.

    I absolutely love the Mac/PC commercials. Have you seen the most recent one about the new Vista os? Very funny. PC is going to have to have major surgery to be prepared for the upgrade, and he doesn’t know if he is going to come out alive! Hillarious! And so true. Vista is a hog of resources, from what I have seen and heard.

    I’ve about had it with PCs. Next laptop will be a mac.


    Comment by Jonathan Erdman — 26 January 2007 @ 7:11 pm

  2. Ktsmatics,
    Was reading one of Paul Davies books about Cosmology recently. He mentioned how things would be in 20 billion years or so time with nothing left “burning” in the night sky and all matter starting to pull apart in I think the term is entropy? Anyway he discribed the total blackness of the universe and how with nothing “passing” time would be unable to be measured it would in effect cease. (Or something to this effect) I have been unable to stop thinking about this since I read it.



    Comment by Ivan — 28 January 2007 @ 11:30 pm

  3. I still haven’t bought Davies’ book. The strong anthropic principle says (I think) that a universe can’t exist unless it produces beings who can conceptualize the idea of a universe. Once those beings become extinct, does the universe they once inhabited go out of existence?


    Comment by ktismatics — 29 January 2007 @ 4:44 am

  4. Can you please stop toying with my brain.. seriously..



    Comment by Ivan — 29 January 2007 @ 7:29 am

  5. Some interesting thoughts about the conecption of time. We generally think of times as clokwork – one moment, then another moment, then another moment….But, as I understand it, the theory/ies of relativity debunk that thinking just a bit. Could we say that the pre-mechanical method of conceiving time (flowing like water/sand) might have been a better analogy???


    Comment by Jonathan Erdman — 29 January 2007 @ 7:02 pm

  6. Hmmm, good question. Not being right on top of relativity theory it’s tough to say, but I’d speculate this. A series of incremental changes in thinking probably took place, going from flow to quanta and then back to flow. The math available in the 13th century was more suitable for measuring discrete units, and since time moves at an even rate that was fine. But when the questions moved to things like acceleration, where the challenge was measuring rates of change, then the mathematicians had to break things down into smaller and smaller increments. Finally in the early 17th century Leibniz and Locke invented the calculus, which resmoothed the little increments into continua again. Time that moves at different speeds in different parts of the universe is kind of a calculus-type problem. Solving one level of problem seems to create higher levels of problems; inventing a mathematical technique makes it possible to imagine a slightly more complicated technique. I’m thinking about this sentence as a whole, but I break into into phrases and words, the words into letters, and the computer breaks it into binary bits. Then it all gets assembled back into the incredibly smooth prose you see before you. Amazing, isn’t it?


    Comment by ktismatics — 29 January 2007 @ 7:26 pm

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