Four Ways of Thinking About a Line

You don’t hear a lot of hot takes on straight lines. They’re lines; they’re straight; and that’s pretty much the full police report. If Hollywood ever options this into a screenplay, you’ll know it’s only because a truly A-list actor made it their passion project.

Image (9)

That’s how I know philosopher W.V. Quine is a Jedi master: because in his adorably named Quiddities, he manages to make me feel confused and breathless about the concept of straightness.

Specifically, he highlights four physical ways to test a line’s straightness, each quite distinct:

1. Stretch a piece of string along the line. If it matches the tight string, then it’s straight. Thus, straightness is a matter of tautness.

“This test recalls indeed the origin of our word line, Latin linea,” explains Quine; “it is related to linen and lint.”

Image (10)

2. Sight along the line, like an astronomer peering through a telescope. Your eye will detect deviations from straightness.

“We have here,” says Quine, “a notable quirk of nature: the light ray, which is our line of sight, matches the taut string…. Between yawns, try to recapture the fresh sense of naïve wonder that two such simple and disparate phenomena… should line up so nicely.”

Image (11)

3. Fold a piece of stiff cardboard. This will be naturally straight, so you can hold it against your candidate line to test its straightness.

Image (12)

4. Slide an edge along the line. For example, take a card’s edge, and slide it from one endpoint of the possible line to the other. If you can “preserve full contact” throughout the process, then you’ve got a straight line.

Image (13)

Now, what do taut string, sniper scopes, stiff cardboard, and sliding edges have in common? In practical terms, nothing; in geometric and conceptual terms, everything. Each is a different doorway into the same underground chamber, a different perspective on the same surprisingly rich idea.

This brings us to my favorite part of blogging: the Indefensible Generalization Game. Ready? I’ll go first:

Mathematics is the art of perfect synonyms.

I know that’s not usually how we define math. We call it “the study of logical systems” or “the queen of the sciences” or “the class between social studies and lunch.” But chew on the idea of synonymy, and you’ll be surprised how nutritious you find it.

Image (15).jpg

The greatest insights in mathematics are equivalences. You can call them equations, or isomorphisms, or symmetries, or “Hey, there are octahedrons in my cubes, and cubes in my octahedrons!” The point is that math is about connections, and a connection is when you can describe the same thing two ways.

Just listen to the theorist Simon Kochen:

Mathematicians don’t talk a lot about analogy. Not because it isn’t there, but just the opposite. It permeates all mathematics.

Or Joseph Fourier:

Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them.

Or Andre Weil:

Nothing is more fruitful—all mathematicians know it—than those obscure analogies, those disturbing reflections of one theory on another; those furtive caresses, those inexplicable discords; nothing also gives more pleasure to the researcher.

Okay, they’re talking more about “analogy” than “synonymy.” But my point, I guess, is that mathematical analogies establish two concepts as synonyms.

They’re ideas that rhyme.

Now, in case I’ve accidentally persuaded you to go buy Quine’s book, let me caveat. Quine is an analytic philosopher, which is to say that he makes a fetish of logic. I’m a big logic fan myself—it’s how I solve the Sherlock Holmes-style mysteries of where I’ve left my phone charger—but Quine takes logic to extremes that don’t always interest me.

Take this word-game on the meaning of “identity”:

Identity seems like a relation, but it does not relate things pairwise as a relation should; things are identical only to themselves. How then does identity differ from a mere property? Moreover, it applies to everything. How then does it differ from the mere property of existence…?

To me, logic is a cleansing agent, a clarifier. It weeds out certain pathogens in our thinking—inconsistencies, hypocrisies, circularities. Analytic philosophy like Quine’s, though, sometimes feels to me like an attempt to cleanse the cleansing agent itself. It’s like an autoimmune disorder of the intellect: logic’s antibodies attacking themselves.

Image (14).jpg

That’s why Quine’s discussion of lines jumped out at me: it’s concrete, physical. Its logic turns not inward, but outward, juxtaposing four tactile experiences and prodding us to inspect the commonality, the shared conceptual structure.

This, I think, is what math is all about: spotting the common features beneath reality. Taking languages that seem to have no words in common, and discovering—against all the odds—flawless synonyms.

13 thoughts on “Four Ways of Thinking About a Line

  1. These lines really hit the mark on so many levels: “To me, logic is a cleansing agent, a clarifier. It weeds out certain pathogens in our thinking—inconsistencies, hypocrisies, circularities.” Might have to stick it up on the bulletin board as a reminder!

  2. I have a friend who says, “Poetry is calling different things by the same name. Math is calling the same thing by different names.” This idea has been so helpful to me in teaching math and literature to kids who say, “I’m a math [or words] person, not a words [or math] person.”

      1. Look at the first line of William Wordsworth’s “To a Skylark”

        “Ethereal minstrel! pilgrim of the sky!”

        A minstrel and a skylark aren’t the same thing, but the poet calls them by the same name. Same thing with the skylark and a pilgrim. Calling a bird and a person by the same name helps you see some truth about the nature of each.

        In math, we say that 2+3 is the same as 4+1 and this is true because they are both five. They really are the same thing, even though one group looks like (and gets called) one thing, and the other group looks like (and gets called) something else.

        1. But the poet doesn’t call them by the same name. He calls the Skylark two different names, a minstrel and a pilgrim. And the aim of mathematics is to unify concepts, not to get one concept and then invent different names for it. I really think you should ask your friend as this is a famous quotation by Poincare.

          “Mathematics is the art of giving the same name to different things.”

          which was a response to the quote

          “Poetry is the art of giving different names to the same thing.”

        2. Wow. It’s entirely likely that my friend said it right but then I remembered it backwards and invented my own explanation for it. Thanks for taking the time to point that out!

          This, by the way, is exactly the problem I ran into in school math when we got to the point where the teacher started having us memorize formulas and didn’t bother to teach the concept behind it!

  3. This reminds me of discussions I used to have with my big brother. He didn’t believe that lines existed.

    He would say things like. “A line is the path that a ray of light follows. Light has a wavelength. A wave is not a line. Lines don’t exist.”

    And I would tell him that I rejected his definition. Photons don’t travel in straight lines.

  4. The straightness of a line is quite a labyrinth. Especially when you start taking non-Euclidean geometries into account. Yet a line in a plane is analogous to a great circle on a sphere.

    Given that you like the outward application of logic by Quine rather than an inward application, I suggest you try Douglas Hofstadter’s ‘Godel, Escher, Bach: An Eternal Golden Braid’. I daresay you will find Quine a lot more intuitive in the form Hofstadter puts it in. Plus he is talking about math, music and art as interwoven braids so I think you will like the synonymy parts.

    1. You’re right about non-Euclidean geometry. The trick I like to use is to visualize yourself living on the sphere (or maybe the hyperbolic plane, whatever) and then you’ll see the line as straight.

    2. Thanks – that’s a good suggestion! I’ve been reading some of Hofstadter’s essays (the ones collected in “Metamagical Themas”) and his style is really fun.

  5. The first two straightness tests are lovely, but I dislike the last two. Because if you have an edge, you already have a straight line, and if you have a plane, then the question becomes as to what a plane is (which is no better).

    1. Yeah, fair complaints. Quine doesn’t really address your plane concern, but he actually has a semi-response to the edge question (which I omitted): if your edge matches your line, there’s a chance that *neither* is straight, but that instead they both have constant (and identical) curvature. You can eliminate this possibility by testing the edge against the “other side” of the line; if it still matches, then both are straight.

  6. There is a certain exhilaration in calling the same thing by different names. Proving two things are perfectly same in such a way that you completely understand it, and not just by manipulating equations is pretty awesome! The main reason why graphs always fascinate me!

Leave a Reply