19 November, 2021

Lighting for the Lazy

There's a phenomenon that occurs only to the lazy, like myself. I'd like to share it here so that go-getter types could also know of the experience.

Each room in my house has several lights. In the master bathroom, a half dozen lightbulbs are just above the mirror; in the kitchen, several inset ceiling lights help to illuminate my cooking; in the main room, flush mounts and floor lamps predominate. When the house was first moved into, all of these fixtures held working lights. But, as time passes, light bulbs fail. I could replace them. But why bother? The other lights work well enough without them.

How many lights?
One by one over the years, a light bulb will peter out, never again to provide lumens for our nighttime activities. To some people, this would be intolerable; but, to me, what does it matter, really? I usually keep all the lights off in the daytime anyway, thanks to several large windows throughout the house. A small nightlight keeps the bathrooms visible with no windows installed. And at night, the only light I need is that from my computer screen. Or my television set. Or my Switch. (My partner, an artist, requires extremely bright light, but it is solely directed toward her art-making, and isn't on unless she's working.)

Eventually, rooms with several light fixtures get down to their last working light bulb. One day, they, too, will break, and work will have to get done. I will have to purchase new light bulbs and replace the entire rack. But light bulbs these days last years, so I am not too worried. The day will assuredly come, but perhaps not this year. Perhaps not even next year.

Here, we teeter on the edge. Where once our rooms were bright, now the occasional flicker catches my attention. On some days, this is exciting. It is living on the edge. I feel as though I am in a dramatic video game, stalking the halls of a long disused factory, with only a few scattered lights still functional. On other days, it feels emblematic of our general aging: slowly, we are shutting down, the prime of our light long past.

My home isn't as bad as this real hospital.
What's weird about this is that I've experienced this scenario several times in my life. I can remember clearly in my twenties feeling this same emblematic-of-aging gestalt, even as I feel it now. I don't think it has anything at all to do with my actual age. It's just that I have a dim memory of the rooms being brighter, and yet now they are so poorly lit that, although life is still functional, the experience of the room has an entirely different feeling to it. What's really fascinating is what happens after: when the last of the bulbs goes off in a room, that gets me to replace all the bulbs in the house. The change is quite literally palpable: you can feel in your fingertips just how much more bright everything is. The mood changes significantly. Life renews, like an early Spring day.

I don't think that non-lazy types can really fully appreciate how this feels. I am told that pumpkin spice has popularity specifically because it goes away and only comes back once each year. (I don't see the appeal, but to each their own.) Something similar is going on here for me, but on somewhat larger time scales. I enjoy the feeling of going from almost no lighting to full lighting. It is reinvigorating in a way that just keeping full lighting all the time is not. I like how the house undergoes seasons of its own, sometimes with dark shadows in particular corners, and yet other times with lighting all around, illuminating every corner to see. It is as though the house is a living, breathing thing, its breaths interspersed throughout years rather than seconds, and with lighting rather than gasps of air.

Being lazy has its drawbacks. But this — the effect over years from delaying replacing light bulbs — is not one of them.

Now, if you'll excuse me, I need to go install all these bulbs I just received from Amazon.

08 November, 2021

On the surreality of .999 repeating...

When I was in grade school, I often had late evening talks with my friend, Peter. Topics of discussion varied wildly from day to day, sometimes about video games like Doom, sometimes about girls, and sometimes about math. On one specific evening, we talked about infinitely small numbers.

I think the topic held our attention because the books we had access to said, in no uncertain terms, that the decimal expansion .9̅ is equivalent to 1. This left no room for a smaller value, in between .9̅ and 1, but which nevertheless was distinct from 1. We found this perplexing, as there seemed to be nothing logically incoherent with the idea of having an infinitely small positive value which we could subtract from 1 — yet the textbooks made it clear that that resulting difference could not be .9̅, since .9̅=1.

This idea is not unique. Many people make the same error, thinking that .9̅ should be strictly less than 1. In the sci.math newsgroup, the main FAQ's top entry for decades showed that .9̅=1. When I took a look at sci.math just now in November 2021, one of the most recent entries is literally someone making the argument that they are distinct. This is a topic that gets brought up again and again, and there's always someone more knowledgeable around that works tirelessly to correct these misunderstandings. (I used to be that guy on the old skeptic forums, though thankfully not on math ones.)

But in order to tread new frontiers in mathematics, you sometimes have to take a "yes, and..." approach. Sometimes when you do so, you're able to reach new ground that later ends up bearing significant fruit. This is how it was with negative numbers, this is how it was with imaginary numbers, and maybe something similar could be said with the idea of a positive number so small that even adding an infinite number of them together will not sum up to 1.

I first discovered the concepts in Berlekamp, Conway, and Guy's Winning Ways for Your Mathematical Plays back when I was working my way through Feynman's Lectures on Physics. I had been gifted a very nice three volume set as a teenager, and while the first book wasn't terribly difficult to get through, I was having trouble understanding books 2 and 3. At the time, I had dropped out of school, and so my only way to read these Feynman lectures required me to first teach myself more complex mathematics. I went to the local library, taking out texts that would help me to get through what Feynman had written, and, occasionally, I'd use the internet to supplement my understanding. Back then we did not have 3Blue1Brown; the best online math explainers were merely paragraphs of html text alongside slowly loading jpgs. So it was hard going. Nevertheless, I kept at it and eventually learned what I needed in order to properly enjoy my boxed set of Feynman lectures.

Hackenbush girl from WWfYMP.

It was during one of these online math excursions that I came across Andy Walker's excellent late 1990s-era html maths-explainer: Hackenstrings, and the .999?=1 FAQ. Walker walks us through a simplified version of Conway's Hackenbush idea, showing the beginnings of what we now call surreal numbers. Here, Conway and Knuth take seriously the idea that there could be a positive number so small that adding it to itself infinitely many times would never add up to any traditional real number. This is the first time that an infinitesimal is taken seriously enough to warrant the creation of a new system of numbers. (At least it's the first time since limits replaced infinitesimals in our teaching of calculus.)

At the time, I was too immature to think that I should purchase for myself Winning Ways for Your Mathematical Plays, but I ever so much wish that I had. It's an amazing book and well worth the read.

If you're interested in learning more, I highly recommend this excellent video by Owen Maitzen that does an absolutely amazing job of explaining Hackenbush. While it's an hour long, this is nevertheless one of the most entertaining introductions to a new type of math that I've seen anyone on the internet create. (He's even composed a soundtrack that suits his video perfectly!) Well done, Owen.