Thoughts on induction

Georg Cantor's theory on infinities flow from the inductive method. That's the method by which you say if some function of "i" is true for i=1, and if you can show that where the function for "i" is true then f(i+1) is also true, then the thesis is true for the whole {i} field.

The inductive method gave rise to Karl Popper's metaphysic. Not all philosophers approve induction.

Cantor never challenged induction as such. What Cantor had done was to define and to find some "j" which doesn't exist in the inductive field of countable numbers. His "j" was 2^∞-4. He'd shown up induction's mathematical limits.

Bertrand Russell famously proposed another limit to inductive reasoning, as applied to the natural world:

We know that all these rather crude expectations of uniformity are liable to be misleading. The man who has fed the chicken every day throughout its life at last wrings its neck instead, showing that more refined views as to the uniformity of nature would have been useful to the chicken.

Russell suggests that the chicken was unaware of the general fate of the domesticated junglefowl, and of the applications of "Bayes' Theorem" - or if you like of Bayes-Price-Laplace. You might think you're in an average day of your life. But are you in the average day of your species' life?

Induction, then, is a Platonic ideal, or a Pythagorean one; best applicable to the mathematics of the possible. For the real world, you need B-P-L.

Why it's bad to lose your Gellar field

Everything possible depends on mathematics. The concrete reality of "everything possible", we can call "the multiverse".

The multiverse is utterly mindless and chaotic. The Near Eastern imagination could explain it only through the metaphor of a stormy ocean. It is, in that famed Semitic rhyme, Tohu ve-Bohu.

Our universe on the other hand is not chaotic. It has a geometry: three spatial dimensions of no particular direction, and one further dimension which goes in one direction. By that alone, anything in our universe can instinctively grasp concepts like "cause and effect".

Our universe also follows rules of physics. Relevant to geometry: you can't blip from one spot to another spot, without taking time, that time being constrained by the speed of light. In effect the speed of light is the ratio of distance over time.

To bend these rules of our universe may or may not be possible through the means we own within this universe. But if we do that, we're leaving the universe. We're entering what may well be chaos. Pending some revelation from some multiversal alien, it near-certainly is chaos, given that chaos is uncountable and we're not. And I'd not be quick to trust that alien, either.

Set theory

During my sixth form, or the junior/senior years for Colonials, I got interested in the nature of infinities and the singularity. What if we treated "Two Divided By Zero" as an actual number, like pi. I raised this to my math teacher / tutor; he recommended I rethink my presuppositions. He recommended Georg Cantor. Not in the original German; in an English-language summary.

Georg Cantor back in 1870s had run a thought-experiment: let's list through all the positive integers in binary-notation. Nerds call these, the natural numbers. First, there's 1. And then 2 - "10". And "11", and "100" (for 3). And so on. Next, pretend you're a Semite and start reading them from top-right.

So, you got this:

0..0001
0..0010
0..0011
0..0100
0..0101

Now: consider the number when you pick digits, diagonally:

0..00001
0..00010
0..00011
0..00100
0..00101

Cantor said: suppose we flip those digits. By definition, it's not 1. It's not 2. It's not 3 ("011") - and moreover, that third digit has flipped boolean to "true", so we know it's greater than 3. As you go down row x, the function of x becomes 4, 12, 28, 60...

By this, Cantor had formally defined (Two to the Power of x, Minus One) Minus Three, with x -> Infinity. He'd defined a new theory: a theory of sets. The natural numbers map one-to-one with these rows of binary digits. And, they can never count to 2^∞-4. Not as in, it would take too long. As in: the unbounded set of the natural numbers does not contain this number.

More: any function you can think of to "hash" the integers, any deterministically-driven list of binary-digit sequences: once you've defined them and listed them, Cantor - that devil - can ruin your day by subsequently doing his diabolical diagonal.

It further turns out that if you stick a decimal in front of these binary digits, that the Continuum of real numbers [0,1] - which include pi-3 - isn't countable, either.

This forced mathematicians to take stock of which sort of numbers can be mapped to countability. They got as far as the algebraic numbers, like 5 and -12 and 3.6 and even root-2 and even even those x solutions in the complex plane for finite polynomials including famously-insoluble x⁵ - x + 1. They also found a way to count two-dimensional space: so, the imaginary / complex algebraic numbers were countable. But they (literally) couldn't square the circle: pi wasn't algebraic. Above all, take e - whose natural logarithm is equal to one. The logarithm function is the inverse of the exponential function.

Infinities and their inverted twins, the infinitesimals, are the bane of mathematics; but this inductive method of Cantor is the same method used in, for a start, calculus. Cantor's incipient set theory did run into "Russell's Paradox" but that paradox was swiftly corrected by the rigourous Zermelo–Fraenkel theory. There's also Cantor's hypothesis on whether there exists some infinity between countable numbers and the uncountable Continuum, which unfortunately Zermelo-Fraenkel's theory couldn't - and can't - address. Or so the Internets tell me. I never got this far.

I got far enough to figure out that no serious mathematician has disproven Zermelo–Fraenkel. So Cantor's discovery of uncountable infinities must stand.

Somali women as European expats

Via Eurogenes: Gravettiana. [UPDATE 5/5/22 moar]

Gravettis are the Europeans after the Neanders were done wit': from 30kBCish, through to the latest Ice Age blast. Gravettiana still inhabits the Palaeolithic; but they have moved on from "Cro-Magnon Man". The general time-frame is the domestication of the dog.

This paper is headed up by one E. Andrew Bennett. They note that the Gravettis have the N1 mitochondrion - that's the female line. This is not the N of the modern Euro "Daughters of Eve", who descend (mainly) from N's basal daughter R.

Most interesting here is that the Gravetti N1 line is on its way to N1b. N1b today is Ashkenaz (10% are N1b1b)... and Somali (N1b2).

One thought is the Viking / Islamic slave-trade in those proverbially-beautiful women of the Caucasus. But I'd not think to see them cluster in Somalia in that case; I'd expect larger populations in richer regions of the Islamic world, like Basra. Also N1b seems early, to be a base. Doesn't look late-antique.

I'm thinking more, the mass-migrations of that Ice Age. We already have (from Karen Bojs) the U clades of western Europe streaming into western North Africa. Seems to me that N1b from eastern Europe might take a different route.

The rise of a Philistine nation

After about 1200 BC, the southwestern corner of the Mediterranean coast came under New Management. It had been Canaanite, ruled by Egyptians. Suddenly the cities, foodstuffs, and placenames changed. The Egyptians were still around-about, but now they were building fortresses to hem in the changelings.

I was on a dig in Ashqelon / Ascalan some decades ago. By then the neutron-activation method had come into its own; proving that a lot of Mycenaean "III-C" ware was produced on-site. The genetics are now in, and they're showing that the locals were heavily southern-European.

The wording of the research leans away from southern coastal Anatolia: whence the late "Hittite" states of Carchemish, and whence the Lycians. Ashqelonian DNA looks more like Caphtor / Crete.

We could propose that various Sea Peoples took over the region, each with its own language. Ashqelon - it's now all but proven - got the Greeks. Other nearby cities got the Valistinians (Ilya Yakubovich, "Phoenician and Luwian in Early Iron Age Cilicia"), speaking Anatolian. Maybe there were "Sherden" cities speaking old Sardinian. Maybe even Etruscans got involved, here as at contemporary Lemnos.

Etruscans, Archaic-era Greeks, and Carians did make themselves heard around the Eastern Med; they have certainly got themselves read, in their graffiti. But from 1200 to 800 BC, if you wanted to make a living in the Eastern Med, you had to learn Canaanite.

Once the Bible gets good and started, the Med coast is culturally "Philistine", but everyone there is already speaking "Hebrew". From a 1100 BC perspective, it's more like the "Philistines" are speaking the same foreign language which leftovers from the dead Egyptian empire were having to speak: Canaanite. 1100 BC sure wasn't 300 BC and it wasn't even 700 BC. The Bible's probably right.

Siyati

We are here concerned with happiness.

I have had some decades in this universe to consider the concept; on account I haven't enjoyed a lot of it, nor have I shared much of it with others. The great Achaemenid shah Darius, first of that name, filled an entire cliffside with his deeds, which deeds - he tells us - he had done for the happiness of all mankind. And recently there was a Chernobyl episode with that title as well.

(I assume, in irony.)

This blog will, if I am doing it right, distill the best of what's been on my mind, for all this time; and leave alone whatever had been making me unhappy, and you unhappy.

Pray for me.

Dragons of Despair: Errata

In DL2, Dragons of Despair, the party acquired 800 refugees from Pax Tharkas. The aim of DL3 (and DL4) is to transport those 800 to permanent safety. In DL3, the party gets the refugees into temporary shelter and finds a way to the only viable hideout in the region: the hidden dwarven city Thorbardin. In DL4 the party prepares that hideout. There is a time limit.

DL3 is organised into Encounter Areas (EA#). There are 31 of these, described in summary form over pp. 9-13. Some of them duplicate, like #16, #18, #20, #29, #31. (We'll keep our eye on #20.) DL3 then details two of these NONrepeating locations: Agharpost (#9) and Skullcap (#32).

DL3's clock starts on the trigger of Event #5. The dragonarmies proper, not just their baaz scouts, enter EA#1 from EA#2 and in four game hours overrun it. The process repeats for an adjacent EA as chosen by the DM - unless it is of the five (because EA#31 repeats) defined in Event #5. Crueler DMs might cut the number and more lenient DMs might increase it; #30 could well repeat too. By my calculations, all EAs will be overrun from EA#2 in around 6 days. At that point Event #8 kicks in; and if the refugees are in the best of these temporary locations then they get another 108 more game hours (4.5 days) before the dragons take all EAs, attack the refugees, and slaughter the lot.

As a matter of pacing, DL4's introduction will maintain that seven days is ample time to resolve that module.

If you are running the DL3 module copyright 1984, then there are a few errors to watch out for. Actually even if you're running the "edition". The "edition" did less well at dividing the EAs from each other. The edition usually boosts the number +143: so #2 > 145 (Pax Tharkas) etc. Tho' you cannot read that particular numeral in any version.

#7 is "Neidarbard" in later literature.

EA#28 is a three-mile-wide valley leading to points north (of DL3?) which belong to ogres and not to the dragonarmies. EA#28 was - like #2 - DAMN hard to read in DL3, as its label crossed a creek. The revision marks it clearly (and correctly) as 171.

The builder of the "Bridge of Dirken" #13=156 is spelled "Dirken" in DL3; but since then the DragonLance designers have settled on spelling this name "Derkin", and so it is in the edition. Derkin was a king of the mountain dwarves. DL3's also missing an entry for EA#19. The editor has, reasonably, replaced the particular #18 northeast of bridge #13, making that the #19 so enumerating it 162.

The "Hopeful Vale" is officially EA#20 (p.12); however there is confusion elsewhere in the text as to whether this vale was EA#20 (p.7) or 21 (pp.8,27). Note, the edition is of no help since it too has two of the equivalent, #163, and marks #164 as the safe zone.

#21 / 164 is, by the way, no safe zone. It is a nondescript narrow defile, with a river through it, running south from the first hopefulvale #20.

Oh yeah, the map sports two locations for EA#20: one in the centre of the mountains adjoining various EA#18 regions, and another on the opposite side of "The Bog" in the Plains of Dergoth. We'll speculate on why the mess.

[REPOSTED 5/16/2020.]