_Freedom Evolves_ by Daniel Dennett

A couple of days ago I finished reading Freedom Evolves by Daniel C. Dennett, 2003. From the back cover: "Can there be freedom and free will in a deterministic world? Renowned philosopher Daniel Dennett answers with an emphatic yes, showing us how we alone among the animals have evolved minds that give us free will and morality." That's an impressive claim, and I can't say it's false, but unfortunately my own mind hasn't evolved far enough to follow his train of thought. Sigh. Still, I enjoyed reading it as a diversion, and I did come away with a stronger—though still vague—sense that free will does exist, at least for practical purposes at the level at which we live.

Here are quotes of some passages that particularly caught my eye:

Our minds are just what our brains non-miraculously do, and the talents of our brains had to evolve like every other marvel of nature. (p. xiii)

That makes sense to me. And I presume that these talents are continuing to evolve!

In just one species, our species, a new trick evolved: language: It has provided us a broad highway of knowledge-sharing, on every topic. Conversation unites us, in spite of our different languages. (p. 4, emphasis added)

Some rue our isolation as individuals. I certainly know the feeling: despair mixed with panic at being utterly alone in my own head. But for whatever reason, I find it has faded away in my case. Now I am more inclined to feel joy at our ability to share thoughts, however limited the ability may be.

Now, for the first time in its billions of years of history, our planet is protected by far-seeing sentinels, able to anticipate danger from the distant future—a comet on a collision course, or global warming—and devise schemes for doing something about it. The planet has finally grown its own nervous system: us. (p. 4)

Here I disagree. That's a pretty-sounding thought, and maybe humans will develop a Gaia-consciousness in the future, but at this point I'd say we're clearly focused on ourselves. In a science-fiction scenario where the choice is between saving Earth (including some life forms but not humans) and saving the human race (by shipping out to another planet, say), I'm sure we'd pick the latter. Some sentinels! But actually I think that's the better choice: Even a single aware species seems more precious than a planet, however full of beautiful flora and fauna.

People are surprisingly good at distracting themselves from ominous prospects. (p. 9)

How true. Or from irksome responsibilities!

Our natures aren't fixed because we have evolved to be entities designed to change their natures in response to interactions with the rest of the world. (p. 9, Dennett's emphasis)

And just one more quote, the first sentence of the penultimate chapter:

Human consciousness was made for sharing ideas. (p. 259)

Voilà.

Oh, just for the record, here are a couple of posts about books from my previous blog:
Favorite lines from Ex Libris by Anne Fadiman
The Meaning of Life by Terry Eagleton

Hyperexponential complex numbers: A hasty sketch

The following is another reposting from my old blog, where it was the 11th entry. It is a quick summary of the number system I have rashly attempted to put together. It is rather (?) abrupt; for some background, combined with some false starts, see the other entries at the old blog.

A hyperexponential complex number (as I arbitrarily choose to use the term) refers to a number of the form X_n<x+iy>, where n, an integer, is the level, and x+iy, a complex number, is the locus.

The system uses the following basic definitions:

X₀<x+iy> = x+iy

X_n<x+iy> = e^(X_n−1<x+iy>)

Thus,

X₁<x+iy> = e^(x+iy), and

e^X₋₁<x+iy> = x+iy.

By extension,

X_m(X_n<x+iy>) = X_m+n<x+iy>.

Using X to represent the set of all hyperexponential complex numbers and taking the numbers of level 0 to correspond to C, the set of all complex numbers, we can write C ∈ X.

The numbers of each level can be represented on a Cartesian hyperexponential complex plane XP_n where each number X_n<x+iy> is represented by the point (x_n, y_n).

Each point (x_n, y_n) on the plane XP_n can be mapped down to a unique point (x_n−1, y_n−1) on the plane XP_n−1 using the equalities

x_n−1 = ƒ(x_n, y_n) = exp(x_n) • cos(y_n)

and

y_n−1 = ƒ(x_n, y_n) = exp(x_n) • sin(y_n).

To express this mapping down in terms of hyperexponential numbers, let us use the sign

“\\\” (triple reverse solidus), read “maps down to”:

X_n<x_n + iy _n> \\\ X_n−1<x_n−1 + iy_n−1>

If (x_n, y_n) on XP_n maps down to (x_n−1, y_n−1) on XP_n−1, then we say that (x_n−1, y_n−1) on XP_n−1 maps up to (x_n, y_n) on XP_n. To express this mapping up in terms of hyperexponential numbers, let us use the sign “///” (triple solidus), read “maps up to”:

X_n−1<x_n−1 + iy _n−1>) /// X_n<x_n + iy_n>

If (x_n, y_n) on XP_n maps down to (x_n−1, y_n−1) on XP_n−1, then so do all the points (x_n, y_n + k•2π), where k is any integer. Conversely, if (x_n−1, y_n−1) on XP_n−1 maps up to (x_n, y_n) on XP_n, then it also maps up to all the points (x_n, y_n + k•2π), where k is any integer.

To restate the above in terms of hyperexponential numbers, if X_n<x_n + iy _n>) \\\ X_n−1<x_n–1 + iy_n–1>, then X_n<x_n + i(y _n + k•2π)> \\\ X_n−1<x_n−1 + iy_n−1> and X_n–1<x_n−1 + iy_n−1> /// X_n<x_n + i_n + k•2π)>, where k is any integer.

If two hyperexponential numbers of level n map down to the same number of level n–1, we will say they are congruent modulo <2πi>. The modulus expresses the fact that the loci of such numbers differ by an integer multiple of 2πi. We can express this with the usual congruence symbol ≡ as follows:

X_n<x_n + i(y_n + 2π)> ≡ X_n<x_n + iy_n> (mod <2πi>)

What I am attempting to propose is a number system where the relationship between e^0 and e^2πi, e^2πi and e^4πi, and similar pairs with exponents differing by an integer multiple of 2πi is viewed as a type of modular congruence rather than equality.

To offer a loose analogy, the proposed relationship between these numbers (of level 1) and the complex numbers (level 0) is like the relationship between a set of unbounded date/time values in measured in days and hours, where 24 hours equals 1 day, and a set of time values measured in hours repeating in a 24-hour cycle, ignoring the number of days. For example, if i corresponds to “6 am,” then e^i(π/2) corresponds to “0 days and 6 hours,” e^i(5π/2) to “1 day and 6 hours,” e^i(9π/2) to “2 days and 6 hours,” and so forth.

By this analogy, the fact that (e^i(π/2))^i is not equal or even congruent to (e^i(5π/2))^i even though e^i(π/2) and e^i(5π/2) are congruent, both mapping down to i, is no more surprising than the fact that, say, half of “0 days and 6 hours” is not equal to half of “1 day and 6 hours” even though both original time values come out to 6 am when measured in the modular 24-hour cycle.

The road to L

For my tentative first post here I'm going to be lazy and recycle the first post from my old blog, just to see how it comes out.

(Originally posted 090502, last revised 090513)

It was with good intentions, or at least neutral intentions, that I started down the road that was to take me to L. I certainly wasn’t looking for L ; in fact, I had no idea it existed until I found it -- or rather, I should say, one of them, because there are in fact many Ls.

Here, approximated (?) to 50 digits, is the first L I found, which I rashly concluded was the only one:

0.3181315052047641353126542515876645172035176138714 +1.3372357014306894089011621431937106125395021384605i

The location of this number on the complex plane is shown roughly in the graphic above, with scale indicated by the unit circle in white.

I stumbled on it when I started evaluating log log log . . . log z for various complex numbers z. Quite to my surprise, iterating the log operation invariably led to this number, regardless of the value of z. (I started out evaluating iterations of ln with the Google Calculator, which revealed the unexpected convergence, and then I switched to Casio’s Keisan site, which allowed me to determine the value more precisely.)

I figured this must be a Very Special Number Indeed, and I decided to dub it “ L,” for “logarithmic limit. However, two mathematicians I consulted breathlessly concerning this Great Discovery patiently informed me that that it is just one of an apparently infinite set of such numbers, all satisfying the condition z = e^z. So it is far from unique. But I still think that it and its fellows are interesting and, yes, very special. I shall write more about them another day.