Penrose:

These things, however, depend upon certain basic notions of the calculus, so, in order to convey something of this magic to the reader, it will be necessary first to say something about these basic notions. There is, of course, an additional reason for doing this. Calculus is absolutely essential for a proper understanding of physics!

Amen! I taught calc-based physics to high school students in a New Orleans suburb as Jesus glared at me from several well-selected vantage points around the room. At that level calculus simplifies, immenseley.

All you need is derivatives and integrals of polynomials and basic triginometric functions, and suddenly all these disarpate results make sense. The two courses have the same essential learning goals, why strip away their most important tool for extracting meaning.

Colorado Springs.

There is no longer a sense of home here. It is just another familiar place. The first restaurant I headed for had gone out of business and been transformed into a “dinner club.” They still served Italian, though, so I gave it a shot. Quite acceptable.

From up the block I see 32 Bleu has also transformed. Thankfully, Poor Richards is still kicking, though they closed up shortly after I arrived. I think the closing of the Cheyenne book shop helped them out.

A few pilsners & the CC-Denver WCHA final at Paninos, catching up with Mark J. Joe is off to some cult gathering, but I found out he’s not bartending anymore anyway. Saturday nights are much more tame there now. Alas, no Severe Liver Damage. (Denver won 1-0, though DU also had a goal disallowed by a rediculous toe-in-the-crease video review. I have never forgiven CC for knocking Clarkson out of the playoffs two years in a row… plus their uniforms are ugly. I say: go DU!)

Typical Colorado weather. To work.

What the Heck.

1. Grab the nearest book.
2. Open the book to page 123.
3. Find the fifth sentence.
4. Post the text of the sentence in your journal along with these instructions.
5. Don’t search around and look for the “coolest” book you can find. Do what’s actually next to you.

Nevertheless, the analyticity of the judgment has no bearing on the fact that the elements are thought together in terms of the relation of ground and consequent.

Penrose, 3.3:

Mathematical ideas develop, and various types of problem seem to arise naturally. Some of these … can lead to an essential extension of the original mathematical concepts in terms of which the original problem had been formulated… accordingly, the development of mathematics may seem to diverge from what it had been set up to achieve, namely simply to reflect physical behavior. Yet, in many instances, this drive for mathematical consistency and elegance takes us to mathematical structures and concepts which turn out to mirror the physical world in a much deeper and more broad-ranging way than those that we started with.

Penrose certainly doesn’t have the deft, cutting prose of Dawkins, and he’s prone to loose use of physics-speak, but his observations are salient. This is a question I have spent a good deal of time pondering: just what parts of modern mathematics are inevitable extensions of arithmetic? Supposing we could run the history of mathematics over again, or that we’re able to see a parallel developed by intelligent life elsewhere. Surely, notations would be different, and in many cases, this would be a good thing. But what of the structure, the foundational concepts, of calculus, functional analysis, abstract algebras?

The obvious natural extension of mathematics, to me, ends at complex analysis. It brings closure to so many basic issues, and an elegance and beauty I am still in awe of. Beyond that? There is a theorem for universal computation… that any system capable of (some minimal set of capabilities) is computationally equivalent. Is there an equivalent for formal systems? Is it the same theorem? It seems like the minimal requirements in this case would be the same as that for Goedel’s theorem…

Diamond:

…we have to wonder why the kings and nobles failed to recognize and solve these seemingly obvious problems undermining their society. Their attention was evidently focused on their short-term concerns of enriching themselves, aging wars, erecting monuments, competing with each other, and extracting enough food from the peasants to support all those activities.

From the chapter on the collapse of Maya civilization. Some things never change. Humans are rather predictable animals.