Gödel’s theoren explained

Kevin Drum takes a solid swing at explaining Gödel’s theorem — what it is and what it means — in a post that begins:

Bob Somerby wants to know if the logician/mathematician Kurt Gödel is a genius or a charlatan. The answer is “genius,” but it’s hard for non-mathematicians to understand his seminal theorem or why it matters. Bob is relying on Rebecca Goldstein’s biography of Gödel, and this is a mistake since it’s a biography, not a mathematical treatise.

But it’s dex night, so I’ll take a crack at it. Fair warning: you really need to have at least a little bit of background in math to understand this. There’s just no way around it. However, you don’t need much as long as you’re willing to tolerate a bit of mathematical symbology. Here goes.

1. Mathematical symbols

Although most of us don’t think of it this way, mathematics is actually a formal logical system of symbol manipulation.¹ For this to work, it must be possible to express all mathematical statements in a formal symbolic language. And it is! Take this statement, for example:

For every number there is a number that’s one higher

In mathematical symbology it looks like this:

∀ x ∃ x + 1

(For all numbers x there exists x + 1)

There is a symbol for anything you can say in the language of mathematics. If you’re interested, a complete list is here—though there are some complicated nuances for certain kinds of expressions. Basically, though, there’s a symbol for everything, although non-mathematicians are unfamiliar with most of them.

2. Gödel numbering

Gödel’s initial insight was that

Continue reading.

The Biology of Bonsai

I like bonsai a lot, and if I had a yard, I would definitely have a few underway. (Bonsai cannot be grown indoors.) Open Culture has a good post by Colin Marshall today, which includes this video:

The Open Culture column includes this list of links:

The Art & Philosophy of Bonsai

This 392-Year-Old Bonsai Tree Survived the Hiroshima Atomic Blast & Still Flourishes Today: The Power of Resilience

What Makes the Art of Bonsai So Expensive?: $1 Million for a Bonsai Tree, and $32,000 for Bonsai Scissors

The Art of Creating a Bonsai: One Year Condensed Condensed Into 22 Mesmerizing Minutes

Daisugi, the 600-Year-Old Japanese Technique of Growing Trees Out of Other Trees, Creating Perfectly Straight Lumber

A Digital Animation Compares the Size of Trees: From the 3-Inch Bonsai, to the 300-Foot Sequoia

Hollyhock Salad Dressing

The Canada Post letter carrier who delivers mail at my apartment building gave me a recipe for a salad dressing I later learned was famous: Hollyhock Salad Dressing. The recipe she gave me is this one.  I made a few changes. This is my version:

• 3/4 cup Bragg’s nutritional yeast
• 1/3 cup water
• 1/3 cup tamari
• 1/3 cup Bragg’s apple-cider vinegar
• 2 tablespoons Dijon mustard (helps with the emulsion – LG)
• 3 large cloves of garlic, thinly sliced on garlic mandoline or coarsely chopped
• 1 cup canola oil [or avocado oil — avocado oil is good, but expensive]

In a blender, combine yeast, water, tamari, vinegar, garlic, and mustard for one minute. With the blender on high, remove the centre of the lid and SLOWLY drizzle in the oil. Stop the blender as soon as all the oil has been added. Keep in a sealed jar for up to 2 weeks.

I use an immersion blender. I put the first 6 ingredients into the blender’s (fairly large) beaker, blend them, and then add oil slowly (i.e., a little at a time) as I continue blending.

The recipe makes a pint. Pick jar accordingly.

A couple of oddities in the original recipe:

1. The recipe commented that the tamari could be gluten-free. Could be? Tamari is gluten free, being made purely from soybeans. (Shoyu/soy sauce does use some wheat. That’s what makes it different from tamari.)
2. The recipe specified grapeseed or sunflower oil, both of which are terrible oils in terms of their omega-6 to omega-3 ratio. Canola’s omega-6 to omega-3 ratio is 2:1, close to the ideal of 1:1. The ratio for grapeseed oil is 676:1(!) and for sunflower oil is 40:1 — see this post. Canola oil also has the neutral taste that works well in this dressing. (I was warned that the flavor of extra-virgin olive oil does not work well in this dressing.) I use organic cold-pressed canola oil.

An Asian-Inflected Variant

I like this variant:

• rice wine vinegar (unflavored and unsweetened — example) instead of apple-cider vinegar; and
• replace 2 tablespoons of canola oil with toasted sesame oil — that is, put 2 tablespoons toasted sesame oil in a measuring cup, then add canola oil to make 1 cup total; and
• add about 1/2″ fresh ginger root, sliced thinly; and
• keep everything else the same, including garlic and mustard.

Not a total surprise: Solstice shaving soap — and the RazoRock MJ-90A

Solstice is one of my favorite shaving soap fragrances. This tub has Phoenix Artisan’s Kokum Butter formula, and I would definitely get it in the CK-6 formula if I were buying it today. PA’s Solar Flare shaving brush easily evoked an excellent lather, and the RazoRock MJ-90A, a wonderful razor, easily produced a perfectly smooth result in three passes.

A great way to start the day. And, the frosting on the cake, my fasting blood glucose level has continued to drop since getting the pacemaker, from an average of 6.2 to 5.8, then 5.7, and this morning 5.6 (all readings are mmol/L; to convert to mg/dL, multiply by 18.018).

This morning I’m going with an iced tea: Murchie’s Cherry Blossom: “A Chinese Sencha green tea blended with papaya, pineapple, sour cherry and raspberry pieces, strewn with jasmine blossoms and rose petals.”

Life is good.