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Mathematics and Darwinism — Plus a Math Problem to Solve


Over at Telic Thoughts Bradford resurrected a discussion based on my UD essay, Writing Computer Programs by Random Mutation and Natural Selection. In reference to the quote, “The set of truly functional novel situations is so small in comparison with the total possible number of situations that they will never occur, which is the point of the original post,” I commented as follows:

That was the main point of my essay, that combinatorics produce such huge numbers so quickly and totally swamp islands of function. My 66-character program, assuming only the 26 lower-case letters, produces 2.4 x 10^93 possible outcomes, or the number of subatomic particles in 10 trillion universes.

In fact, the C programming language is case sensitive and uses all 92 characters on a standard keyboard, which produces 4 x 10^129 possible combinations in a 66-character program, or the number of subatomic particles in 10,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000 universes.

Evolutionary biologists put blind faith in chance and necessity and arbitrarily invoke “deep time” to make the impossible imaginarily possible. The problem is that deep time is not actually all that deep. There are only about 10^17 seconds in five billion years.

Hard numbers put things in perspective. The probabilities are not a close call; they are catastrophically lopsided.

It can always be argued that we don’t know how many possible amino acid sequences can be functional, but we can know that this number would have to be astronomical in order for chance and necessity to have any probability of coming up with something biologically workable, even given an absurdly short protein. This is why the simple mathematics of combinatorics renders the random variation thesis simple not credible when it comes to the molecular machinery and information content of even the simplest cell.

A Math Problem to Solve
I’m currently working on an as-yet-undisclosed computational algorithm for which I need to solve a math problem. I will send a free set of my three classical piano CDs (with works by Chopin, Liszt, Rachmaninoff, and Gershwin, along with program notes on the works and their composers) to anyone who can solve the following problem. (My e-mail can be found at the Evolutionary Informatics Lab on the People page.)

The sum of consecutive integers 1 to n (1 + 2 + 3 + 4 + … + n) is given by n(n+1)/2, or (n^2 + n)/2. n(n+1) will always be divisible by 2 since either n or n+1 must be even. Given a generating function, kn + p, where k is an interval and p is an initial offset, it is easy to calculate sums that skip numbers in regular intervals. For example, if we want to sum 3n + 2, for n = 1 to 4, we have 5 + 8 + 11 + 14, or (3+2 + 6+2 + 9+2 + 12+2), so we have one 3 plus 2, plus two 3’s plus 2, plus three 3’s plus 2, plus four 3’s plus 2, or 3(1 + 2 + 3 + 4) + four 2’s, or 3 times 10 plus 4 times 2. In general then, we have the formula k(n(n+1)/2) + np to find the sum for any generating function kn + p and a range of 1 to n.

I give this by way of background because I must find an analogous general formula for the sum of cubes. I need to sum cubes because Fermat’s Last Theorem tells us that there are no powers greater than 2 for which there are any integer solutions for a^n + b^n = c^n. I use the power of 3 because it’s the smallest power that guarantees that no two integers raised to the third power will ever sum to any other integer raised to the power of 3, which is a requirement of my algorithm.

Fortunately, there is a surprising and beautiful identity, which is that the sum of consecutive cubes of 1 to n (1^3 + 2^3 + 3^3 + 4^3 + … + n^3) is the sum of 1 to n, quantity squared — that is, (1 + 2 + 3 + 4 + … + n)^2, which equals (n(n+1)/2)^2. For example, 1 + 8 + 27 + 64 = (1 + 2 + 3 + 4)^2 = 100.

So, I need a formula for kn + p that will sum cubes in the same way that k(n(n+1)/2) + np sums numbers raised to the power of 1. In practice, p will be in the range of 0 to k-1, since p is the result of a modulo divide by k.

SteveO I very much like your list; a rather snappy way to approach the issue. It is not, however, the only list that should be compiled. Strategy calls for an inventory of resources as well. In this case, a resource is anything that provides force to the argument, such as the observation of the flagellum, or the vision cascade, or the mutational observations in Lenski's lab, or the observations of complex genomes among simple organisms, or etc etc. How utterly useful it would be to the ID movement to have all of the thoughfully-prepared forces in support of Design assembled in a single place, and that such a place be made accessible to anyone who engages the argument. The primary loss that is enforced on ID by the materialist monopoly is a stranglehold on the fair distribution of information. The NCSE does, after all, have its marching oprders. Conversely, the win that ID will one day attain will the lifting of the ban. When that happens, fair-minded people will know what they see when they see it. And, when they do - materialism will be finished as a means of control. Upright BiPed
steveO: Rather impressing: shall we really have to study all those things? :-) By the way, I would add history of philosophy, philosophy of science and epistemology... gpuccio
OT: The production of an ID Expert. I remember while working a stint in the field of Nuclear Medicine at being very impressed by the experts in that field because they were proficient across multiple disiplines: medicine, physics, mathematics and computer science. Well those guys don't have a patch on ID ! What areas of study might be required in the production of ID Gurus/Scholars ? I just started to list some topics off the top of my head (I'm sure I've missed quite a few). (Plus, following one of St. Thomas's principles that one should understand your opponents viewpoint as well as if not better than he, I've included studies from the 'other' side). Mathematics and Computer Science Statistics Theory of Symbolic Language Artificial Intelligence Mathemathical Modeling Computer Simulation Computer Graphics/visualisation Bioinformatics Paleontology (some geology), Anthropology Intelligent Design History Theory & the works of Dembski, Behe and others Lots of Case Studies (Design vs. Evolution) Theory of Evolution History General Studies Evolution and Culture Darwinism Main sources of evidence and controversies RM & NS & Modern Synthesis Science and Culture History of Science Science and Religion Politics of Science Economics of Science Science and the Media Case studies in science controversies/corruption Logic Chemistry Biology The Cell Biochemistry Genetics Origin of Life Studies and Difficulties Phylogenetics Engineering and Design Information Theory Control Theory (dynamic systems) Selected studies from Electrical and Mechanical Engineering (e.g. machines and motors) Plus Breaks from Academic Studies Comedy and Humor: Dick to the Dawk time. - Antics & selected writings of PZ Meyers- Richard Dawkins: Brights or Bozos? steveO
The folks who wrote Avida and Berg are totally invested in Darwinian orthodoxy. To admit that it is mathematically unworkable would be to admit that they have wasted their professional lives pursuing a falsehood. Thus, speculation serves as evidence for them. GilDodgen
I would assume that the folks who wrote Avida, and Howard Berg would be from the group that would really understand the math, what could they see that we don't? es58
I think tRNA's are the greatest witness to design. Evolutionary thought would not have predicted their function. the wonderer
Gil, thank you for your efficient reminder of the mathematical basis of ID's strongest argument. It is certainly useful to repeat that often, given how people seem to forget quickly. In one of his books, one of our smart "enemies", Douglas Hofstadter, wrote a chapter on "Number Numbness". In that chapter, he effectively demonstrated the difficulties met by most people when very big numbers and higher orders of magnitude are involved. Maybe darwinists and materialists, including Hofstadter himself, should read that chapter more often. Just a few comments on the subject: a) Islands of functionality in the protein search space. It is true that we don't know how many sequences of aminoacids are functional, but it is not true that we have no idea. There are severe constraints to the functionality of proteins. The first, and most important, is that they have to fold. Proteins fold in specific ways, not one, but not too many, and if they don't fold well they cannot be functional. Many complex proteins need the help of other specialized proteins to be able to fold. Folding is,in itself, a "difficult" property to be achieved. But it is far from sufficient to provide function. Of all folded proteins, only a few may have an useful function. And many functions depend on specific interaction with other proteins or structures, and so they require recognition sequences in addition to the right folding and to the right functional folding. There are many experimental studies which, if analyzed correctly, show that the islands of functionality are extremely small in relation to the search space. Much knowledge about that can be derived from the new field of intelligent protein engineering. In "The Design of Life" there is also a very interesting discussion regarding research about penicillinase, which is very pertinent to this problem. You very correctly say: "It can always be argued that we don’t know how many possible amino acid sequences can be functional, but we can know that this number would have to be astronomical in order for chance and necessity to have any probability of coming up with something biologically workable, even given an absurdly short protein." Well, we can be sure that that number is not astronomical at all! b) The concept of "any possible function". Darwinists love to argument that we cannot restrict too much the islands of functionality because we have to consider that nature is free to attain "any possible function", and that only our limited imagination prevents us from understanding the huge possibilities of that. While it is perfectly right that our imaginations are less functional than those of darwinists, still this reasoning, with all its poetic suggestion of an infinite space of creativity, is simply wrong. Although I am sure that even the "any possible function" space is certainly limited enough, that is not really the point. The point is that not "any possible function" will do. Indeed, in each specific context, only a few specific functions will do. The concept is that fuction can only be defined in a specific context. Nothing is functional if it is not placed in the correct context, because function implies an interaction with the environment. If the environment is not right, function is not there. So, we may reason that some protein sequence "could" have a function on some unknown planet, with unknown molecules and structures. Who could deny that in principle? The fact remains that such a sequence has no function here. And proteins have to be functional in very, very specific contexts. To fold, they need the right water environment, the right pH, and probably a lot of other things. An enzyme is functional only in the presence of its substrates, and in the right concentration range. An enzymatic reaction if functional, at a higher level, only if its results are of some utilitiy for the more general context of the cell. And so on. Besides, the more a context is complex, the more functional possibilities become specific. If a computer program already uses very complex procedures, only innovations compatible with those procedures will be useful. You have to deal with what you have. Applying infinite creativity will not help, especially if you have to apply it in a blind, random, unguided way. c) Finally, let's remember that our calculations usually focus on the classical example of a protein of about 100 aminoacids which, with its 10^130 search space, fits well enough Dembski's UPB (which, in itself, is an extremely generous boundary, of which darwinists should be very grateful). But we should remember that most complex proteins are much longer than that. I have recently reviewed literature about a renal protein whose defects are responsible for very serious congenital diseases, and just for fun I looked at its length: it was about 1200 aminoacids! So, sometimes we should really remember that ID's examples seem to have been chosen so that we don't disturb darwinists too much, just as much as it is necessary to show them wrong. But, given the character of our "adversaries", we should perhaps be a little bit less courteous, and utilize more of the infinite reservoir of "ammunition" provided by what we know of biological complexity. In the same way, one of the results of the discussion about Behe's arguments regarding the flagellum is that now everybody seems to think that the flagellum is the only existing example of IC, and that if darwinists could succeed in showing a possible darwinian pathway to it (which they have not done, and never will do), then the whole concept of IC will fall down to pieces. The truth is that billions of examples of IC, many of them even more complex than the flagellum, exist. Trying to find a false darwinian pathway for each of them could keep darwinists busy for, let's say, at least the next 10^150 years... gpuccio
Kind of on a side topic, but it has to do with math. Something which Darwinists apparently like to simply ignore. :P But I heard about William Dembski's Law of Conservation of Information and I ran across this website. It claims that the law is mathematically unsubstantiated. Unfortunately I'm not very good with math and have no idea if such a claim is correct or not. I hope it's not, and I was also wondering: What exactly is the Law of Conservation of Information? I'm not quite up to date on much of Dembski's work. I've read more by Jonathan Wells, Michael Behe, and Stephen C. Meyer. Here's the link btw: http://www.talkreason.org/articles/dembski_LCI.pdf Domoman
Quick note: while in the debate I linked to above shows me apologizing for even thinking the "repackaged creationism" argument had merit, I was referring more to comments delivered by PvM, particularly ones in which he claims that god or the supernatural are not "directly" mentioned as being a tenet of ID in the early pandas drafts, but are beyond any doubt "implied" in those drafts. Notice how I asked on multiple occasions for him to find a passage in the early drafts for Pandas and People that states that ID pertains to any of the following premises: 1. The sudden creation of the Universe from nothing. 2. Everything created in six 24 hour days (or even metaphorical days). 3. The age of the Earth being less then 10,000 years old. 4. All the Earth's geology can be explained by means of a global catastrophic flood. 5. Using the bible or some other sacred text to draw inferences, and last but not least... 6. Specifically having god or a supernatural creator in the role of the designer. He has yet to take me up on it. F2XL
~~~~~~~~~~~~~~OFF TOPIC~~~~~~~~~~~~~ Seems like Design critics are finally willing to admit that the early drafts of Pandas do not mention god or the supernatural as being the creator or designer for ID. I would respond to that last comment PvM made and point out that he's resorting to Socrates logic but I'll let everyone on UD make up their own minds. Just to ellaborate on Socrates logic: Socrates was mortal, man is mortal, therefore all men are Socrates. Or in a simpler context: Dogs have tails, cats have tails, therefore all cats are dogs. Now the way PvM is using the logic is as follows: "Creationism describes the fossil record with an abrupt appearance of animals with certain features fully intact, Pandas describes the fossil record with an abrupt appearance of animals with certain features fully intact, therefore ID is creationism." He insists that the religious content is there, I just need to "read between the lines" in order to find it. "No God you say. Well not in direct words perhaps but all in implication and intention it seems." -from PvM on opposingviews Now ain't thatta b%#$&. (yes Barbara, pun intended :D) I don't know if I should continue the debate with him or not; is it worth it or can readers notice the flaws in his logic from the start? F2XL
So, is it one CD for each one of us? Nope, you all three get all three CDs. Just send your mailing address to me through my e-mail address, available at the EIL website, as mentioned above. GilDodgen
So, is it one CD for each one of us? DiEb
Thanks much to all three of you who helped. I have the solution. GilDodgen
How do we account for the idea that not every sub-atomic particle experiences exactly the same conditions for every second of time? That would seem to indicate an incredible increase in probabalistic resource, or am I missing something? the wonderer
FWIW: 1/4*n*(k+k*n +2*p) * (k^2 *n*(1+n)+2*k*(1+n)*p + 2*p^2) .... DiEb
Gil, Yes, so many people seem to look at random mutations and say, well, the probability of a prespecified functionality resulting is very small, but there are a lot of possible funcions, so maybe the odds aren't that low that something useful will result. They don't seem to realize that the numerator in this probability can be very large and yet still negligibly small compared to the denominator! When it is discovered that genes appeared in primitive animals (see Dave Scot's post from yesterday) that didn't express themselves until millions of year later, these people say, well, maybe these genes actually had some other function in the primitive animals! And since it's impossible to count the number of possible functions, about all you can say is you've got to be joking. B.T.W, if what you're looking for is a formula for the sum from i=1 to i=n of (k*i+p)^3, I just e-mailed you that, you just cube it out and sum the i^3, i^2 and i terms. But I'm not sure if that's what you wanted. Granville Sewell
If you figure out how to install TeX markup for wordpress, please let me know. William Wallace

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