Nowhere did I say that all things will occur. What I said is that it is guaranteed that highly improbable events will occur.
No, I’m not.
We don’t need all possible outcomes to occur in order for one possible outcome to occur. We also know what the arrow of time has done in the past because we have the evidence right in front of us.
Works for lots of Christians as well.
Combinatorics is not my math specialty - though this does relate to some calculus concepts. But your discussion about probabilities given “infinite” opportunity calls attention to interesting mathematical questions. Not all infinities are equal, and so (as calculus are familiar with), an “essentially zero” quantity times an “essentially infinity” quantity can sometimes be zero, sometimes be some other finite answer, or can be infinity.
So imagine this thought experiment: You play a little “lottery” by starting with ten slips of paper in a hat, one of which is the winning slip. You attempt one draw (which you have a 90% chance of not winning), and then after that first attempt you add one more losing slip to the hat (now having only 1 win possibility out of eleven), repeat - and if you lose, add yet another - doing this ad infinitum. So each subsequent attempt leaves you with less and less of a chance of ever winning. Here is my conjecture: In this scheme, there is precisely 100% chance (certainty) that you will win at some point on your way out to infinity, even though your chances on each subsequent lottery are diminishing - approaching zero even. But in this case the infinity prevails over the “approaching zero” chance of ever winning (based on 1/x out to infinity still having a finite area). On the other hand, if you start your first attempt with 1 winning slip out of ten, your next is 1 out of 100, and you keep this up to infinity (each lottery having 10^n slips, but only one winning slip), then I think this only comes to a 1/9 chance that you’ll ever win this, even as you do it an infinite number of times. So that would be a “zero” that competes successfully with its countervailing infinity.
In short - you are correct to say that “infinity” is not some magic word that always gets you everything. And nor is “zero” always a magic word that automatically shuts out all chance.
I don’t think I ever discussed infinite opportunities. What I am trying to shed light on is the Sharpshooter fallacy.
The point I would make is that somebody won in every one of your drawings.
This would be my lottery analogy. We have a lottery with 100 million names in the hat. Therefore, the chances of winning is 1 in 100 million. We draw a name and record the name of the winner. We do this ten times and then calculate the odds of those specific 10 people winning. That probability is 100 million to the 10th power, or 1 in 1x10^80. This extremely unlikely event will happen every time you do 10 drawings. If someone comes along and says the lottery results are impossible because of the extremely improbability of those specific people winning then they are committing the Sharpshooter fallacy.
Agreed.
As I was just telling my own students the other day, the exact same event can be humdrum in one context, and amazingly inexplicable in another. If I tell you a coin tossed ten times resulted in: HTTTTHHTHT, none of you are impressed. But if I announce that exact result ahead of time, and then proceed to flip a coin and get that exact sequence, now you are all asking how Merv did the trick. That’s why ID people a few books ago (Dembski? - or maybe it was Meyer) called it “prespecified complexity” or some such phrase. Not that this salvaged his ID outlook in my view - but the “prespecified” part does make the difference of rectifying the sharpshooter fallacy and hitting a target that was already painted there.
Otherwise you have to resort to “implied targets” - like if I tossed twenty heads in a row. Even without having announced it ahead of time, that would still be a noteworthy event so as to make us begin to wonder, if we witnessed that, about the coin or the tosser. Or if I toss a HTHTHTHTHTHT… twenty times in a row - still amazing. I suppose that ID folks are forever trying to show that creation is full of these “implied amazements”. At least that’s what I understand their program to be all about.
I don’t know how they would call it prespecified since they are making the specification after the biological feature already appeared.
In my experience, there is a religious assumption that percolates through the entire endeavor. The assumption is that humans were the goal, or more generally, that the biodiversity we see today was the goal. I’m not going to criticize anyone for carrying this assumption around, but I will point out when the assumption moves out of religious belief and into empirical science and statistics. Religious belief and scientific discovery can happily coexist without one needing to be the other.
Yep. And I think I agree with everything else you observed there too. Percolating assumptions is a great way to put it.
Ohhhhhhh no we don’t. The past is truly infinite. It can’t not be, unless rationality is ‘just’ a story.
A few year ago, I became distracted by the work of Georg Cantor about infinities (in the late 19th century). It can consume you! His proof that there are more irrationals than rational numbers, even though there are an infinite number of both, led to even more pondering. Such as the observation that irrationality hides inside the algebraic expression for a circle, X^2 + Y^2 = 1. And since the square root of all numbers not perfect squares is irrational, another dimension is added.
All rationals terminate or repeat of course, and those that terminate have a denominator with only 2 or 5 as prime factors. Thus the infinite number of rationals pales in comparison to irrationals! And adding 1 to the set of finite numbers becomes transfinite, but not absolutely infinite. Isn’t this stuff grand?!
It is a lot of fun to think about! And even though we don’t delve into the theorems or proofs you mention, my high school math students still love to muse over the concept as well. One of the things that blew me away was that any conditionally convergent series (e.g. 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 - …) can actually be made to sum up to any arbitrary real number just by carefully choosing the order in which you add the entire infinity of numbers! (Riemann’s Rearrangement Theorem).
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