Do straight lines exist?

Do lines even exist outside our imaginations?

Little purple flying elephants

Do little purple flying elephants exist? Certainly not in our world. Probably not in any other physical world.

But certainly they exist in the world of the imagination. I can picture little purple flying elephants, with butterfly-like wings, faint pink spots that you'd see only on close inspection, and... I dunno, I think they like to wear green tutus on special occasions.

Do imaginary things exist? They come from our imagination, which comes from our brain. And our brain exists. Granted, we ("we" the scientific community) don't fully understand how the brain works, or how the imagination works. But certainly the imagination has some physical basis in the structure and dynamics of the brain, and among the myriad amazing things the brain can do is generate the concept of a little purple flying elephant that does not exist in physical form.

Straight lines

Straight lines are, in my humble opinion, in the same category as little purple flying elephants. They exist in our imagination as an abstract geometric structure, but I don't think they really exist any more than our beloved loxodonta. Please allow me to explain.

A screenshot of Euclid’s Elements, one of the most important math books.

Nature doesn't produce a lot of straight lines. Look around at the natural world: You won't find a lot of straight lines. Or any straight lines. (Footnote: "Natural" is an annoying word to define—for example, plastic bottles are made from chemicals that occur naturally—but I'm using the term "natural world" to refer to physical and biological objects that humans have not created using technology.)

The seemingly straight lines that humans have created are not, in fact, truly straight. The walls of a building are straight enough to provide structural integrity, but close inspection will reveal that they are "practically straight" but certainly not "truly straight."

What would it mean for a straight line to exist in nature? A straight line is defined as a line with zero curvature. No curvature at the galactic scale, no curvature at the scale of 100 km, no curves or bends at the scale of a millimeter, and so on. No matter how much you measure a straight line, or how closely you zoom into that line, it never curves.

I struggle with this definition. For one thing, space is curved by gravity, so would a straight line bend with the curvature of space, or would it stay straight through the curves, and thus actually appear to bend inversely proportional to the strength of gravity? A second problem arises as you zoom in to that line. Perhaps it is straight at the spatial scale of meters, millimeters, even picometers (a picometer is 10^-12 meters). But keep zooming in and eventually we get to Planck scale (10^-35 meters). At this scale, the concept of "length" is no longer meaningful because the uncertainties about where and when things happen are as large as the estimates themselves. I am not a physicist, but I think this means that cannot say for sure that a straight line continues to be straight at such tiny distances.

A brief tangent on Planck length

Planck length is around 10^-35 meters, a typical human brain cell is around 10^-3 meters, and the observable universe is around 10²⁶ meters wide. Curiously, this means that a brain cell is roughly half-way between the tiniest thing in the universe and the biggest thing in the universe. The craziest part of this to me is that when we think about how incomprehensibly huge the entire universe is, that's how big a single brain cell would look to a "person" who is one Planck length tall. I cannot even imagine what a person that tiny would experience: Planck length is still a whopping 20 orders of magnitude smaller than a proton; even our lovely solar system is "only" 12 orders of magnitude bigger than a human.

Anyway, the point is that I cannot imagine that straight lines could exist at arbitrary spatial resolution. In my mind, that means that straight lines do not exist, not even imaginary ones.

The Platonic Plane

So, straight lines do not exist in the real world, and they do not exist in my imagination. I suppose geometrists and Platonists would argue that straight lines exist as a Platonic Form—an abstract exemplar in a Platonic Plane, of which real-world objects are impoverished imitations. But this Platonic Form then either transcends our understanding of physics, or is someone else's imaginary reality that I do not share.

I think about this sort of question relatively often. I'm quite certain I have little idea of what I'm talking about and I am even more certain that I will never know the answer of whether straight lines exist (if there even is a single answer to that question). But I do enjoy letting my mind meander to these topics. Perhaps you agree with all, some, or none of what I wrote here, but I hope you found this musing thought-provoking.