PROBABILITIES AND INFINITY
(Written around 2010) I just finished reading Origins of Life by Fazale Rana and Hugh Ross (NavPress, 2003). This is the book I've been needing for quite a while. The authors lay out the current theories concerning the naturalistic (i.e. not brought about by any intelligence) emergence of life itself and carefully explain how the theories of various scientists have been shown scientifically to be insufficient by other scientists. It has given me a good overall sense where science is these days with regard to this issue. In this entry I will include my own thoughts as catalyzed by Rana and Ross.
A long, long, time ago, when taking Zoology 1 as a freshman zoology major at the University of Maryland, one of the first assigned readings was an article by George Wald from Scientific American, in which he pontificated,
Let me tell you that George Wald's probability calculation is just plain wrong. This notion that, given a huge amount of time, anything that could happen is bound to happen does not work. Period. I know that a lot of people say it, to the point of it becoming a "truism," but the first necessary attribute of a truism must be that it is true, and this one is not.
(I'm now going to explain my statement. In the process, we need to look at a tiny bit of number theory, and I hope that my explanations will be sufficient. You can also go to Wikipedia for more help if necessary. Still, if you absolutely turn apoplectic at the site of numbers and equations, you may liberate yourself and go to the next page, but then you're not allowed to criticize my conclusions on this one. I much prefer that you work through it.)
Wald begins with an "easy" example. Is it possible for all of the molecules making up a table to move up in concert so that the table will rise from the floor for a moment? Most people would say that this is impossible, but the scientist must remain open to all possibilities, no matter how improbable they may appear to us. Just because we haven't seen it, does not mean that it couldn't happen. Wald observes that ...
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... we see therefore that it does not mean much to say that a very improbable event has never been observed. There is a conspiracy to suppress such observations, not among scientists alone, but among all judicious persons, who have learned to be skeptical even of what they see, let alone of what they are told. (OL, 47) |
One assumes that he uses the term "conspiracy" loosely as a hyperbole for the frequency with which he has encountered allegedly groundless skepticism. In any event, his point is that people seem to transform something with high improbability into an impossibility. Thus, in his ensuing calculations, he begins with odds against a certain event and then shows us what happens to the probability for the event.
If you toss a coin, the chance of it not landing on "heads" is 1/2, which is identical to the chance of it landing on "tails." If you toss it twice, the probability of it not coming out "heads" decreases to 1/4. Each added throw multiplies the number by itself, which we can write as nraised to the number of trials. Therefore, if the odds are even for each toss, we can raise 1/2 the number of tosses. On the third try, the odds against "heads" stand at 1/23 (1/8). Ten tosses without getting "heads" have a rather tiny probability of (1/2)10, (1/1,024), which Wald approximates to 1/1000. And that means that the odds for getting "heads" are 999/1,000.
Now Wald moves away from tossing coins to an unspecified event that already comes with the probability of 1/1,000 for a single trial, so the odds are 999/1000 against it. This number is a given and must remain the same throughout all trials.
Just as every instance of a coin toss starts with 1/2, so every instance against this new event has the very high probability of 999/1,000. Now we want to calculate the probability against the event over the course of 1,000 trials.
Again we multiply the given probability number by the number of trials; in this case it looks like this: (999/1000)1000. Since the number below the exponent is a fraction less than 1, it will actually come out smaller than the number we started with. A quick resort to a calculator, and you get the result of 37/100. That's the result against the event occurring, the probability for it, which began with a 1/1000, has become 63/100, a significant increase over the course of self-multiplication.
These calculations are interesting, but they have little if anything to do with what we might be after. If we flip a coin, we know that it has "heads" and "tails" (or "obverse" and "reverse"), and that you cannot help but get one or the other. You're familiar with the practice of tossing a coin, and you can identify the result. There can be no surprises. If you start out a set of a thousand trials and already know the probability for each individual trial, it just takes a little math to get the result (as a probability, of course, which means that "certainty" only means "exceptionally high probability.")
But that's not the case for finding the beginning of life forms. We have some conceptions of what the first living being might have been, e.g., a small chain of self-reproducing nucleotides, but even if we figure out what the earliest individual that we are looking for must have looked like, we are still following a hypothesis.
Now, I want to make sure that you understand what I mean by hypothesis or conjecture. It's different from a theory, which is a conclusion drawn by scientists and generally accepted. Critiques along the line of "Evolution is only a theory" don't mean anything. However, a theory can be amended or falsified, such as the idea that fire is caused by the emission of "phlogiston," an idea that made sense at the time of its acceptance, but had to succumb to the superior insights provided by the rise of the atomic theory. A theory is not supposed to be something that's a piece of guesswork without any evidence, but something that the scientific community on the whole has accepted as true because it is supposedly based on testable observations. To that extent in terminology, evolution is a theory, and accusing it of being "just a theory" is naive name-calling. However, "hypothesis" or "conjecture' implies that there is not yet enough evidence to support one particular model, and that's the case for the naturalistic origin of life.
Wald and other "origin-of-life" scholars have been going from door to door hoping to find indicia of the nature of the first life forms and further information on how they came to be. Presumably they are all following the same principle as Wald's that, given enough doors to open, sooner or later they will find what they are looking for, and there isn't any standard agreement on what it would look like once they've found it. What make their search impossible is that the doors that they're knocking on are rooms in Hilbert's Hotel, which has an infinity of rooms, and, consequently the probability of finding the right room converges to 0. The fantasy that, "given enough time, something that seemed impossible must become certain" is suffering from a mathematical problem.
The numbers Wald provided are nicely straightforward integers. However, there are other numbers in nature that make the calculation of probability for an event such as the spontaneous emergence of life impossible. To give this phenomenon a name, it is the "principle of incommensurability."
Imagine yourself counting whole numbers, i.e., positive integers. Of course, this set of numbers diverges to infinity, which is not a number but an indicator that it goes on forever. We can keep track of the numbers: "the first one, the second one, the 573rd one, the 1024th one, etc."
We call this a "countable infinity," since there are no integers between any two successive integers, and we can pair each one up with another number so that the two sets will stroll hand in hand into the sunset of infinity. E.g, we can create a function in which each integer is matched with its square, and the resulting lines will continue without ever ending.
1 ⇔ 1
2 ⇔ 4
3 ⇔ 9
4 ⇔ 16
5 ⇔ 25
etc.
**********
x ⇔ x2
But there are other kinds of numbers, such as √2, that are very different from our usual counting numbers, integers and fractions of integers. A number in this respect is
1. not a whole number
2. not a fraction with the denominator and numerator both being integers, and
3. if expressed in decimal fashion, will not have its decimal points ...
a. ... come to a stop, nor ...
b. ... wind up with a never-ending regular sequence of digits.
We can call the numbers that meet these negative criteria "irrational" numbers, by which we don't mean that they are crazy. That term simply means that they cannot be expressed as a ratio of integers.
They include two subsets:
c. ...√2 and some others (e..g, φ, the so-called golden ratio) that are the results of algebraic equations, and ...
d. ... some that cannot be calculated with an algebraic equation. These are known as "transcendental" numbers and include π, the constant that's built into nature itself.
Together with all other numbers alluded to above they make up the set of "real" numbers (as opposed to the so-called imaginary numbers that include √-1 aka i), and the set of the real numbers is considered to constitute an "uncountable infinity."
There is always room for another number between two real numbers. In fact, if you consider the interval between, say, 1 and 2, there is an infinity of numbers between them, and this is also true for the interval between and 7 and 8 or 100 and 101, etc. You cannot match up the set of real numbers with other numbers as we did before because there's always another number coming in between.
"Wait a minute here!" someone may object. "Surely you're not saying that one infinity is larger than another?"
Well, it may be weird if you've never been around set theory, but yes, that is the case. This idea launched the discipline of "set theory," which has been accepted by mathematicians and logicians other than those who have a philosophical objection to the very idea of infinite sets. This discipline goes back to the German philosopher and mathematician Georg Cantor, and even though he claimed that his understanding of sets was somehow revealed to him by God, he did not rely on divine authority to prove it. He presented rational proofs that even a non-mathematician can follow, though unfortunately providing those goes beyond the scope of this rambling post. [I'm working on a book on formal logic, and I'm looking forward to try to make Cantor's reasoning clear and accessible.]
Let's take a transcendental number and, just for a change in scenery, let's pick one other than π; we'll use "L," a number discovered by Joseph Liouville (1809-1892). It's the outcome of a series (not an equation), the formula of which I'm not going to reproduce here because it might scare away the last of my remaining readers:
0.110001000000000000000001....
We express probabilities with fractions, so even odds would be 1/. If the probability were expressed by the reciprocal of the Liouville number, it would be or
That would be fairly good odds, since it comes out to a little more than 9%, and, given the amount of time stipulated by Wald, it could be a viable candidate.
However Liouville's number is just one out of the infinity of real numbers, so that the probability of L being the correct denominator approaches the limit of
, which is 0.
Bottom line: the assertion that, if you have enough time, anything that's possible will happen, is not true, and that negation is intensified when we're not even sure what we're looking for and whether it is even possible for it to exist. [And, by the way, similar considerations also put an end to Nietzsche's idea of "the eternal return."]
But isn't this argument totally contrived? I mean, is it even meaningful to use transcendental numbers in probability calculations? As a matter of fact, it is meaningful.
Suppose we pick two numbers at random from a set of integers in order to assess whether they have any divisors in common. Those that don't are called "coprimes"; they are numbers that trivially only share 1 as divisor.
Now, here's the fascinating insight: The probability that the two numbers we picked randomly are coprimes is 6/π2.
So yes, oftentimes transcendental numbers show up in places you might not expect to see them, including when calculating probabilities. The problem is that we simply do not know what numbers involving the probabilities for a naturalistic origin of life might be since there is an infinity of numbers that could be "it." If we could know that it is L or π, we could establish the odds, but since the intended outcome is unknown and since there is an infinity of real numbers, is the best we can do.
(And, for anyone who might raise another objection, there is no such thing as "∞+n." Infinity, as I said, is not a number, just a sign that we're not done and never will be.)
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Aside (Skip): : Please allow me a remark concerning large numbers. If we take up Wald's number of 2 billion years (2x109) and assume 1 billion attempts a year for whatever event starts the emergence of life, we would get 2 quadrillion trials [(2x109) x 109 = 2x1018]. That's a big number: a 2 followed by 18 zeroes: 2,000,000,000,000,000,000. However, as impressive as it is, it is still far from huge compared to other numbers that appear in the universe, let alone in the world of number theory, where it could be considered to be small. For example, there are two formulas that predict loosely how many prime numbers there are up to a certain number "n": 1. the prime counting function, which is expressed as π(n) This use of the letter "π" has nothing to do with the constant in circles or in the probability equation above. It is calculated as π ≈. The logarithm is not based on 10, as you may have seen in old log tables, but on e, the so-called natural logarithm, another transcendental number built into nature. And the ≈ means "approximately." So the equation says that the approximate number of primes up to a certain number can be found by dividing the number by its natural logarithm. 2. the logarithmic integral function li(n), which constitutes more integral calculus than I am comfortable trying to describe and getting it right. Click on the Wikipedia link to work your way through it. If you calculate both values for the first set of absurdly high numbers, π(n) is consistently lower than li(n). But eventually, as we increase the value of n they switch, and π(n) becomes larger than li(n). We do not know exactly where that switch occurs, but current estimates have put it in the neighborhood of--are you ready?-- 1.397182 x 10316. The value of the expected number has become more precise since it's discovery, but it is consistently referred to as Skewes's number. One of the most talented mathematicians of the early twentieth century was J. E. Littlewood. His academic supervisor had tasked him with proving the Riemann hypothesis, but despite his exceptional prowess he could not do so, which no one held against him. Then, later, when Stanley Skewes became a student under Littlewood, he received the same assignment, but he did not bring it off either. Littlewood himself furthered Skewes's work and showed that, as we pursue the number line, the switch will go back and forth over an infinite number of times. The mind boggles. Can you handle more? Let me put Skewes's number in perspective: The total number of atoms in the entire universe is estimated to be 1080. Sometimes people use the term "googol" (which was accidentally transformed into "Google"--look it up on you favorite search engine!) as an invented designation of 10100, entertainingly described by Scott Patton's post "How big is a Googol anyway?". Thus, Skewes's number exceeds 3 googols. On the other end of the number line, there is h, Max Planck's constant, which he discovered in the context of the energy levels of atoms, the beginnings of quantum science. A negative exponent, such as 10-n indicates fractions. E.g, 10-3 would be 1/1000th. h=6.62607015 × 10-34 m2 kg / s. A quick visit to Wolfram|Alpha shows that written out all the way in decimal form it is:
My point here is simply that we need not be intimidated by the references to large numbers. 10316 is an unimaginably high number, but still, keep in mind that there is an infinity of numbers beyond it. Please note that these numbers are objective and calculable, and do not depend on any ideological presuppositions. |
We have little idea of how the molecules may have behaved at the first stage of life, or if that's even knowable. A possible answer would depend on the nature of the earth during the time of emergence, but that's one of many areas where hypotheses are subject to debate.
Some scholars believe that there was a mutual attraction between molecules beyond normal chemical reactions as we know them, and that we cannot predict the eventual outcome (see the remarks on Stuart Kauffman in parts 11 and 12), an idea that would eliminate the calculations of probability, and, thus, we can only say that what happened happened. However, one of the important aspects of science is the possibility of establishing regularities, usually demonstrated by the repeatability of an experiment or at least showing the consistency and coherence of a theory. The concept of "it just happened this way" takes the repeatability out of science and thus, it's no longer science as traditionally understood.
Origins of Life 