Finally, Mr. Darwin, are you any good at sums?

You will all be immensely relieved, I'm sure, when I tell you this is my final post on the subject of Darwinism, well, until the next time, of course!  Anyway, at this point I would like to introduce you all to my e-pal, Richard J. Bird, well, he's not really 'a pal' because all I did was exchange a couple of e-mails with him a few years back.  What I did do, however, is to read his fascinating book Chaos and Life.  Now Prof. Bird is Supernumerary Fellow of Computation at Lincoln College, Oxford, or, an A1 maths swot of the first order!

I have already expressed my doubts concerning traditional Darwinian evolution as a process of tiny mutations which eventually lead to new species.  Prof. Bird puts it this way:

If we assume that mutations are the agents of evolutionary change, the question becomes how to account for mutations of an adequate number and variety to bring about the observed rate of evolution.  The difficulty is that, if mutations are unsystematic, how can they have resulted in such a well-patterned outcome?  The legendary monkeys at their word processors could not write the works of Shakespeare just by chance.  In the history of the world they would not complete even a single line.

If you consider even for a minute the prolific range of proficient and operative life forms that exist, a truly huge number, then, if mutation is indeed random, you have to consider the equally great or even greater number of mutations that were not successful.

Mutation rates would have to be directed in some way in order to produce creatures like those presently observed.  On the face of it there are far too many possibilities that might arise in unpatterned mutation.  [S.W.] Ulam  [another maths swot] has calculated that, if achieving a significant advantage, such as the human visual system, requires 10⁶ changes, then it will take 10¹³ generations in a population of 10¹¹ individuals for the change to become established.  If there is one generation per day, this means several billion years.

All that is merely the tip of the iceberg on which the good ship Darwin might well founder.  So what is the solution?  Well, at this point it is necessary to remind you that my knowledge of mathematics reaches no further than the twelve times table - and I'm not too sure about all of them!  So, when I raise the subject of 'iterated algorithms' you will understand the thinness of the ice upon which I am skating.  Happily, rescue is at hand in the form of Prof. Gove Effinger, a maths swot at an American college.  He (like me!) was entranced by Tom Stoppard's wonderful play, Arcadia, in which a very young teenage girl, Thomasina Coverly, living in the early 19th century has an incredible insight which, due to her untimely death, would not be rediscovered until the 20th century.  Here is an extract which shows her working with her young tutor, Septimus Hodge:

• Thomasina: . . . Each week I plot your equations dot for dot, xs against ys in all manner of algebraical relation, and every week they draw themselves as commonplace geometry, as if the world of forms were nothing but arcs and angles. God's truth, Septimus, if there is an equation for a curve like a bell, there must be an equation for one like a bluebell, and if a bluebell, why not a rose? Do we believe nature is written in numbers?
• Septimus: We do.
• Thomasina: Then why do your equations only describe the shapes of manufacture?
• Septimus: I do not know.
• Thomasina: Armed thus, God could only make a cabinet.
• Septimus: He has mastery of equations which lead into infinities where we cannot follow.
• Thomasina: What a faint-heart! We must work outward from the middle of the maze. We will start with something simple. (She picks up the apple leaf.) I will plot this leaf and deduce its equation. You will be famous for being my tutor when Lord Byron is dead and forgotten.

Happily, Prof. Effinger explains:

The idea which Thomasina discovers, as we find out later from the present day discussions of Hannah and Valentine [in the play], is that of iterated algorithms, that is: the idea of starting with a number (or point), processing it somehow to obtain a new number or point (which you record), and then feeding that new number or point back into the process. You do this "feedback mechanism" again and again, and after a long time you see the pattern which emerges.

It is in these iterated algorithms that the mystery of evolution may be found because when you carry them out - well, actually an individual armed with paper and pencil couldn't do so at a sufficiently high rate if you lived for a hundred years!  So, a computer is required and when that is used a zillion times over with the result of each algorithm being fed back into the next algorithm - nothing much happens!  Which is what you might expect but then, if you keep going, suddenly, out of the blue, you get an extraordinarily different result.  Don't ask me how, and certainly don't ask me why, it just happens.

And here - at last - are you still with me? - we come to the point.  Procreation is, in effect, an iterated algorithm.  In animal procreation, long strings of DNA from two separate donors are joined together, in time the resulting new creature will pass its DNA to join with another to form a new being.  You might call the process a 'sexy iterated algorithm'!  This can happen a zillion times over but every so often that dreaded exception will occur when something totally unexpected suddenly appears.  Here is an example of an iterated algorithm that went wrong (or perhaps right, who knows?) in the pig world where a piglet was born with a very strange face:

So there you have it, well, for the time being because all theories are subject to correction over time as poor old Darwin found out the hard way!  In the meantime, what can one say?  Well, I suppose you can always trot out a bit of Shakespeare, the bloody man has a phrase for every occasion:

There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy.

 

 

Comments

[Whitewall]

I have always wondered, if we are somehow descended from "monkeys", why are there still "monkeys"? Where are the simians still caught betwixt and between?

[David Duff]

One or two people have suggested it might be me! I know, shockin', shockin'!

[Bob]

Mutations are random, but their contributions to the evolution of organisms are not.

"Natural selection is a rigorous testing process that filters out what works from what doesn’t, driving organisms to evolve in particular directions. However, chance events play a big role too.":

https://www.newscientist.com/article/dn13698-evolution-myths-evolution-is-random/

[David Duff]

That quote, Bob, is cobblers! Everything to do with mutations are random, there is no "particular direction". It happens and either it works or it doesn't. End of!

[TheBigHenry]

Robert,

We had to have the "monkeys" survive the evolution of humans. Otherwise, that whole series of movies about "Planet of the Apes" could not have been made.

[Whitewall]

Henry, I knew there was a reason. Thus we have the basis for... "If you're a prince, there's hope for every ape in Africa". From "The Lion in Winter".

[AussieD]

Despite its face that poor little piglet is going to end up eaten. Unless the inscrutable oriental holding it is Jewish.

Good old Charlie Darwin. He really rolled a grenade into the scientific/philosophical community. Probably sitting back "upstairs" with a pint laughing his head off.

[Loz]

See, there you go, I told you all the mysteries of the world are in Chomsky, now SoD's, Hierarchy!

Thomasina's scribblings, "iterated algorithms" as Bird correctly terms them, are all defined in levels 1-3, with 3 (recursively enumerable - a Turing machine) being the most powerful. Level 0 is mere sequence, with no feedback in it, the dumbest computational brick in the wall, so to speak.

And look, you've chosen Stoppard's exact words to highlight the existence of level 4 and above! Thomasina is citing the observation that "stuff" looks to be more complex than even the numbers game can formulate. Even the Turing system, level 3, blurted out: "Truth and form exist that I am not powerful to prove and form, but am just about powerful enough to prove do exist". God, level 5 (and all the above levels) is indeed drawing shapes and knowing truths that we'd have to go to an infinity before we'll ever understand them. Just as a bunch of dots (level 0) has to go to infinity before it can draw a line or a curve (level 1+) - which a finite bunch of dots cannot do (level 0) by definition of being finite. Neither can the linear bounded system (level 1) draw or prove all the stuff that level 2 (linear unbounded) can. Nor can level 2 draw or prove all level 3 (recursively enumerable) forms and truths. Each level would have to go to an infinity of time before it could draw or prove any of the many forms or truths that the level above can draw or prove finitely.

Now read the best literature fragment in the history of literature fragments, up there with "I could make you out of beer cans if we had enough of them" - "had enough of them", geddit?): -

Thomasina: Then why do your equations only describe the shapes of manufacture?
•Septimus: I do not know.
•Thomasina: Armed thus, God could only make a cabinet.
•Septimus: He has mastery of equations which lead into infinities where we cannot follow.

Fuck me, thank you, needed to get that out. I feel born again. Or I'm going mad.

SoD

[David Duff]

Keep taking the pills, is my advice!

[Bob]

Richard Dawkins is a twit and swot, therefore the theory of evolution is wrong. You can't argue with logic like that.

[TheBigHenry]

Bob,

"You can't argue with logic like that."

I agree with you -- because it isn't logical. This is why I avoid such discussions.

[Bob]

TBH,

You're always the designated driver, aren't you?

[Bob]

SoD,

I suspect you have overstepped the bounds of your system since god is above our level and we can't know under what conditions he could or could not build a cabinet. Perhaps one day our alien betters will pass this type of esoteric knowledge down to us even if we're incapable of understanding what the hell they're going on about.

[TheBigHenry]

Bob,

As my Mom used to say to me (even when I was a kid), "TheBigHenry, you are always the adult in the room."

[Whitewall]

Henry, my Mother referred to that as my being an "old child" from birth.

[TheBigHenry]

Robert,

Yup, same implication.

[Loz]

Remember Turing’s test?

This is the one where you ask two contestants who are hidden, one a Turing machine and one a Human, any questions or to do any tests you like to try to determine which is the Turing machine and which the Human? Can it be done? Can the Turing machine win the “imitation game” as it became known, or more precisely, not lose?

Well given the above, here’s my go at defining a question, or test, that would be guaranteed to distinguish a Turing machine from a Human. Here goes …

We can distinguish between ourselves and a Turing machine by asking the two of them to distinguish between a Turing machine and a contraption made from the level below a Turing machine. The Human can do it (ask Godel and Chomsky – Chomsky’s fame and fortune came from the bloody hierarchy!).

(1) So, if the Turing machine can’t, then we can distinguish between Human and Turing machine: Human can, Turing machine can’t.

If the Turing machine can, then the test fails as a distinguishing test. So how can the Turing machine demonstrate it knows the difference between a Turing machine and a contraption made from the level below a Turing machine?

Well, it, the Turing machine, can do a Turing test on another Turing machine and a contraption made from the level below a Turing machine, asking them to distinguish between a contraption made from the level below a Turing machine and a contraption from the level below the level of a Turing machine.

If the distinction can’t be made, then neither can the Turing machine distinguish the Turing machine from the contraption in the level below.

(2) And if that distinction can’t be made, then neither can the Turing machine distinguish a Human from a Turing machine, which, as we know, a Human can (per Godel, Chomsky); so, Humans can distinguish between a Human and a Turing machine.

If the distinction can be made - a contraption a level below a Turing machine can distinguish between a contraption a level below a Turing machine and a contraption a level below a level below a Turing machine, then the test fails as a distinguishing test. So how can the contraption a level below a Turing machine distinguish between a contraption a level below a Turing machine and a contraption a level below a level below a Turing machine?

Repeat this Turing test challenge until … until what?

Well, until you hit the bottom two levels of Chomsky (now SoD’s) Hierarchy, of course!

So, we ask a contraption made from the “level 1 linear bounded” (the penultimate from bottom level) if it can distinguish between a contraption made from the “level 1 linear bounded” and the “level 0 sequence” (the bottom two levels).

Again, but this time finally (thank goodness!), we observe: -

(3) If the contraption made from the “level 1 linear bounded” cannot distinguish a contraption made from the “level 1 linear bounded” and a contraption made from the “level 0 sequence”, then neither can any of the above iterations in their respective Turing tests, and so a Human can distinguish between a Human and a Turing machine by asking the Turing question of the Turing machine vs. the level below, as described above.

And if the contraption made from the “level 1 linear bounded” can distinguish a contraption made from the “level 1 linear bounded” and a contraption made from the “level 0 sequence”, then how does it do it? In this case, it cannot pose the Turing question to a contraption in the “level 0 sequence” and ask it to distinguish between a contraption in the “level 0 sequence” and the level below, because there ain’t no level below! So it can’t.

(4) A contraption made from the “level 1 linear bounded” cannot distinguish a contraption made from the “level 1 linear bounded” from a contraption made from the “level 0 sequence”. Therefore, neither can any of the contraptions in the levels above, except Humans who can distinguish between Humans and Turing machines by posing the Turing question to the Human and the Turing machine about whether they can distinguish between Turing machine and one level below a Turing machine, as above. The Human answers in a finite timespan, the Turing machine cannot.

The 4 steps above close of all the inductive recursions in the logic.

So, the “imitation game” is up. We aren’t Turing machines!

SoD

[David Duff]

SoD, my, admittedly thin knowledge of Chomsky leads me to suspect that he is a total prat so I remain confident that I could distinguish him from a Turing machine!

[TheBigHenry]

David,

"Chomsky ... is a total prat"

I couldn't agree more.

[Bob]

SoD,

if count=4 then manufacture consent

You can't fool me.

TBH,

My mother would beseechingly ask what was wrong with me.

[TheBigHenry]

David,

Noam Yesam A. Chumpsky is also a total twat and a cunning linguist.

[TheBigHenry]

The "A." stands for "Avram", his given name.

[Loz]

Just a quick note on re-reading the above while I think of it.

You might think that when the contestant Turing machine is asked to distinguish between a hidden Turing machine and a hidden contraption from the level below a Turing machine, it might choose to ask the two of them to solve a problem that only a Turing machine can solve but the level below a Turing machine can't.

However, such a problem requires the full power of the Turing machine, in particular its infinite space and time. So the asking Turing machine will wait forever before it gets an answer from the hidden Turing machine, and the hidden contraption from the level below, incapable of ever solving the problem, can trick the contestant Turing machine by simply saying "Not finished yet".

Thus a contestant Turing machine cannot distinguish hidden Turing machine from a hidden contraption a level below a Turing machine by asking them both to solve a problem that only Turing machines can and level below contraptions can't.

The only question that a contestant Turing machine might ask a hidden Turing machine and contraption one level below a Turing machine to distinguish them seems to be the Turing test itself - on the level below and the level below the level below. And as described, this fails, because the recursion eventually hits the point where the contestant can't even frame the question, because there's no level below - as described above.

SoD