Tag Archives: Plato

Back to Plato’s Line

Last February I posted about how my friend Tina, who writes the Diotima’s Ladder blog, asked for some help with a set of diagrams for her novel. The intent was to illustrate an aspect of Plato’s Divided Line — an analogy about knowledge from his worldwide hit, the Republic. Specifically, to demonstrate that the middle two (of four) segments always have equal lengths.

The diagrams I ended up with outlined a process that works, but I was never entirely happy with the last steps. They depended on using a compass to repeat a length as well as on two points lining up — concrete requirements that depend on drawing accuracy.

Last week I had a lightbulb moment and realized I didn’t need them. Lurking right in front of my eyes is a solid proof that’s simple, clear, and fully abstract.

Plato’s Divided Line

Recently my friend Tina, who writes the blog Diotima’s Ladder, asked me if I could help her with a diagram for her novel. (Apparently all the math posts I’ve written gave her ideas about my math and geometry skills!)

What she was looking for involved Plato’s Divided Line, an analogy from his runaway bestseller, the Republic (see her post Plato’s Divided Line and Cave Allegory for an explanation; I’m not going to go into it much here). The goal is a geometric diagram proving that the middle two segments (of four) must be equal in length.

This post explores and explains what I came up with.

Rational vs Real

One of the great philosophical conundrums involves the origin of numbers and mathematics. I first learned of it as Platonic vs Aristotelian views, but these days it’s generally called Platonism vs Nominalism. I usually think of it as the question of whether numbers are invented or discovered.

Whatever it’s called, there is something transcendental about numbers and math. It’s hard not to discover (or invent) the natural numbers. Even from a theory standpoint, the natural numbers are very simply defined. Yet they directly invoke infinity — which doesn’t exist in the physical world.

There is also the “unreasonable effectiveness” of numbers in describing our world.

Bombs and Baseball

Coo coo ca choo!

So. It’s been a day. A bombing in Brussels, Belgium, and an historic baseball game in Habana, Cuba. President Obama was at the latter, and so was Mrs. Robinson (the one the song isn’t about). Set against the background of “Super Tuesday #4: The Continuing Horror” it all weaves an interesting tapestry of stark contrasts and lurid splashes of color.

Baseball and bombs. Diplomacy and terrorism. Policy and politics. It’s terrible and fascinating how the world offers so much hope and so much despair — all in a day. The best there is in people set against the worst there is in people.

So, it’s been a day.

It Was Ever Thus

I’ve written about this before, the idea that there’s nothing new under the sun — that it was ever thus. The claim is usually made in the face of complaints about how “things are going to hell these days, and how much better it was back then.”

Some cite the ancient Greek⁽¹⁾ who said something about how things never really change (except he was just commenting about kids not respecting their elders). Others cite the famous passage in Ecclesiastes⁽²⁾ (which also gave us a favorite tune by The Byrds⁽³⁾).

So, what do I think is new under the sun?

For The Record

It was a number of years ago that the book you see pictured here on the right caught my eye. I was wandering around a bookstore, as book-lovers do, seeing what there was to see (and possibly buy). This may surprise you, but I’ve always enjoyed a good debate, so the book’s topic seemed attractive and a nice change of pace from baseball and science books or SF novels.

Plus: Aristotle, Lincoln and Homer Simpson! Who could resist that? A glance at a few of the pages showed an easy and breezy open writing style that went down nicely, and the bits I read were quite intriguing. I snagged it thinking it would be right up my alley, and that I’d enjoy it thoroughly.

I never got more than a third of the way through it!