The Infinite Monkey Theorem & biology

Today in my logic class we discussed probability calculations as a way of illustrating inductive arguments.  I amused my students and myself by discussing the infinite monkey theorem and, better yet, actually calculating the odds.  It occurs to me tonight that this has some bearing on one of the big questions in biology. 

The IMT is a famous thought problem in philosophy.  If you had a number of monkeys randomly typing on keyboards, could they eventually produce a copy of one of Shakespeare’s plays?  The IMT is a good test of rationality.  Some students simply refuse to agree that it would be possible, even given infinite numbers of monkeys and time.  That suggests a mental block.  Given those assumptions, the outcome is not only possible but inevitable.  It is also a good illustration of the difference between logical possibility (which the IMT demonstrates) and causal possibility (which the IMT demonstrably fails). 

The monkeys in the theorem are, of course, metaphors for random signal generators.  They are also cute.  To explain my test, I’ll employ metaphorical chimpanzees (they’re easier to train).  I stipulate the following:

a constant team of 100 chimps;

each has a keyboard with twenty-six letters (no other symbols or spaces);

each chimpanzee has an equal chance of hitting any of the twenty-six keys on each keystroke;

each chimpanzee types exactly twenty-six characters per minute;

a full team of 100 primates is working at the stipulated speed twenty-four hours a day;

all the keystrokes are entered in a single line that we test for results;

there is no time limit on how long I can keep the experiment going. 

If you think that these stipulations are unrealistic, you think correctly; it’s a thought problem. 

Now: forget about a whole play; let’s just ask our simian team to produce

Now is the winter of our discontent made glorious summer by this son of York. 

One of my favorite lines, opening Richard III if memory serves.  That doesn’t look so hard.  If my apes are following the above stated rules, each will produce an “n” about once every minute.  To get the first and second letters in the right order, we have to multiply the probability of each: 26 x 26.  So, each 676 keystrokes will produce “no” and our team will produce about four examples every minute.  That’s pretty good.  We’ll be done by lunch.  Bananas all around!

The team will produce the first word in the line about every seven minutes.  Unfortunately, the wonders of geometrical progressions slow us down precipitously.  To get the first two words will require three full days.  Better alert the[KB1]  caterer. 

It gets worse fast.  Given our stipulations, the first three words “nowisthe” are going to appear in order, somewhere in our line, about once every 153 years.  Just to get the next three letters of the fourth word in order will require our team to work for about three million years. 

So how long will it take to get the first four words out?  Forty-seven billion, two hundred and six million, one hundred and four thousand, eight hundred and ninety-four years.  We are going to need a lot of chimpanzees and a lot of bananas.  Unfortunately, the Kosmos is only fourteen billion years old. 

I submit that our thought problem demonstrates two things.  One is that it is logically possible for our chimpanzees to produce Richard III.  Given enough animals, bananas, and time, they could produce the entire corpus.  The other is that it is causally impossible.  Vast stretches of time beyond calculation are necessary even to get going. 

There have been two recent attempts to play out the scenario.  One involved a number of macaque monkeys.  After two months, they failed to produce a single English word (they did have an inexplicable fondness for “s”); they broke the computer and pissed on the keyboard. 

The other was actually interesting.  Someone created a very large number of virtual monkeys, using the cloud to borrow computer time.   He claims to have produced almost the entire Shakespearean corpus by this method.  However…he cheated.  Whenever his virtual monkeys produced nine letters that appeared consecutively anywhere in the corpus, the program took them out and put them in their place.  The monkeys would not have to produce that string again. 

That is cheating because it means that our monkeys can produce Shakespeare only if they begin with Shakespeare as a forced sorting device, even with vast resources.  One might argue that it is analogous to divine intervention. 

It occurs to me that this is analogous to natural selection.  Instead of monkeys we have random changes in DNA.  Very large numbers of organisms constantly reproducing result in very large strings of genetic characters.  Natural selection saves the ones that code for viable traits.  That is how natural selection eventually builds a Shakespeare. 

But what about the UR organism, the first genuine replicator?  Is the simplest such organism that science can yet imagine more or less complex than the Bard’s Dickey Three?  Allow me to suggest that the answer is “more”.  How do you get from inorganic processes to even the simplest organic processes without some kind of preordained sorting? 

I don’t know and neither do you.  No one knows.  I have previously blogged on one attempt to answer that question.  Andreas Wagner argues that evolution worked because nature had a preexisting library of Platonic forms.  As a Platonist, I like that idea and I am pretty sure he is right.  I am just wondering who, precisely, was the librarian? 

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 [KB1]

Logically Moral Animals

I was called on a point recently by a colleague.  He accused me of jumping between two irreconcilable positions.  One the one hand, I argued that human beings are distinct from other animals in kind and not merely in degree (i.e., just more intelligent).  On the other hand, I argued that traces of human intelligence and moral capacity are found in animals.  I replied that I’m a primate; jumping from one tree to another is what I do.  It got a laugh. 

I don’t think I was actually guilty of a contradiction.  Traces of language are indeed found in animals.  A certain call may indicate food or danger.  Yet, so far as we know, no other animal is capable of drawing and recognizing a simple symbol like a stick man.  Differences in kind do not require altogether novel capacities.  They require that something about the way a capacity is exercised be novel. 

Today I have been looking at two studies of logically moral behavior among non-human animals.  Vampire bats have long served as poster children for reciprocal altruism.  These winged mammals feed exclusively on blood.  They need to feed about once every three days.  This presents a challenge as their food sources have legs and can move in unpredictable ways. 

The vampires deal with this problem by blood sharing.  A bat who returns hungry can count on a share of a meal from a luckier roost mate.  This is an insurance policy; however, it invites cheating.  An unscrupulous bat might take from others and then refused to share.  If he gets away with that, his offspring will proliferate and the sharing system will collapse. 

That requires an enforcement mechanism.  Vampires remember.  If bat X refuses to share with me today, I will refuse to share with bat X tomorrow.  Cheaters can be systematically eliminated from the gene pool. 

I doubt that bats are consciously moral.  I am sure that this sharing system is logically moral.  Bat X either fulfills his obligations or does not.  If so, he benefits from the social contract.  If not, then he is excluded from the contract. 

I used this example in the paper I am writing for the APSA this year.  I wanted to be sure that recent scholarship backed up this account, and it does.  Gerald G. Carter and Gerald S. Wilkinson have a piece in the Proceedings of the Royal Society (2013).  They teased apart alternative explanations for the vampire’s behavior. 

Are the bats sharing only because they are related to one another?  If so, then kin selection and not reciprocal altruism would explain their behavior.  In the experiment, relatedness was the least reliable predictor of one bat’s willingness to share with another. 

Perhaps sharing was a response to allogrooming.  Bats share with cuddle buddies.  Perhaps it was influenced by mating agendas: if I share with her, she will mate with me.  Those were the third and second-best predictors of sharing behavior. 

The best predictor was simply that the other bat had shared in the past.  That looks like reciprocal altruism.  Sharing is rewarded with sharing.  Remembering who is a good partner amounts to the construction of a social network. 

The most interesting thing to come out of this study is this: sharing was often initiated not by the hungry bat but by the sated bat.  Why would this be so?  Sharing builds a network of obligations.  If I can get you to accept my donation, you are now obligated to me.  This system is, as the sociologists say, socially constructed.  It depends on reputation, what the other bats think about this one. 

Something of the same kind is going on in a study of cleaner fish.  These fish make their living eating parasites in the jaws, gills, etc., of larger predators.  As in the case of the vampire bats, this arrangement involves mutual obligations and the temptation to cheat.  Client predators can cheat by gobbling up the cleaner after the work is finished.  That is policed by a simple accounting.  A predator who behaves that way will discover that the cleaners no longer come out of their cleaner stations when he swims in for a touch up. 

Cleaner fish are also tempted to cheat.  The parasites they feed on are not quite as attractive as the mucus in the client’s jaws.  What encourages the cleaner to confine itself to the parasites?  Russel D. Fernald explains this in his note Animal Cooperation: Keeping a Clean(ing) Reputation [Current Biology Vol. 21 No 13]. 

It turns out that cleaner wrasses are more likely to keep honest (parasites only) when they are observed by a number of potential client fish.  They seem to value their reputation in the business.  It seems very unlikely that these fish, with their tiny brains, have any conscious awareness of the stakes.  It doesn’t matter.  Natural selection has made the logically moral choices for them. 

Again, the reciprocal arrangement is socially constructed.  The fish do not need to understand the system, but they do need to notice who is watching. 

The old dichotomy between nature and nurture, biological influence and social construction, is long out of date.  Temperature, saltiness, water and nutrients are factors that exert selection pressure on organisms.  Social arrangements and the likes and dislikes of individual interactors for one another also exert select pressure. 

Choice is a powerful influence on the evolution of pretty much everything.  Another powerful influence is moral logic.  Plato was right, at least about the world of living organisms.  The most important idea is the idea of the good. 

Plato, Aristotle, Darwin & Indirect Reciprocity

My strategy for pursuing political, moral, and biological questions consists of three basic steps in the following order.  The first is Platonic.  I look for the idea that is expressed in a wide range of phenomena across time and space.  The second is Aristotelian.  I look for the way in which the idea answers different questions in different contexts.  The third is Darwinian.  I look for how the idea might emerge in the evolutionary history of the organisms in which it is expressed. 

Consider the wing.  We recognize wings in a wide variety of animals.  In all cases, it is a biological appendage that allows the creature to gain altitude by beating the air.  In a fundamental sense, the wing of a bat, a bird, a pterodactyl and a dragonfly, are all the same thing.  Plato (or his Socrates) would be pleased.  The wings are very different, however, in their basic design.  One has to lift a heavy reptile; another, a creature as light as a feather.  Aristotle would point this out to his teacher.  The various wings are also examples of convergence.  This is a term in Darwinian explanations that indicates an independent evolution toward a common trait as opposed to homology, which indicates a trait shared because it is inherited from a common ancestor.  Bats and birds don’t have wings because they inherited them from a common ancestor, but because they worked out the same basic mechanics on their own. 

I have found this strategy to be fruitful when applied to my primary interest, morality and politics.  My work on autonomy (which I hope to be published soon) is an example that is illustrated in previous posts.  Here I apply it to reciprocity, one of the basic foundations of cooperation in animals (including human beings).  When some party X pays a cost on behalf of some other party Y because there is a reasonable expectation that the cost will be repaid with profit, that is reciprocity.  That this is a genuinely Platonic idea is indicated by the abstraction of the terms.  It can apply to two teams of dolphins cooperating with one another of different days and to a fellow tipping big at a local restaurant.  Obviously the mechanisms are different.  Less obviously but very likely, they both owe their operation to evolved dispositions. 

Reciprocity is a powerful engine for cooperation, but in its direct form (an exchange between two parties) it is limited to specific exchanges.  When we’re done we’re done.  Indirect reciprocity, by contrast, can knit together much larger communities of cooperators.  This is when an individual is influenced by observing third party cooperation.  In such a case, the cooperator benefits by building a reputation as a good partner.  The observer benefits by recognizing the altruist as a promising partner. 

Tonight I read two accounts of indirect reciprocity.  One was a study of cleaner fish and their clients (Bshary & Grutter, “Image scoring and cooperation in a cleaner fish mutualism”, Nature 22 June 2006).  Cleaner fish feed on ectoparasites in the mouths of much larger fish.  This is a classic example of reciprocity in a morally charged context.  If the cleaner fish eats ectoparasites, it will benefit its larger client.  However, it prefers mucus, if it has a choice.  Eating mucus does not benefit the client.  So the cleaner is tempted to cheat.  In some cases, the client fish is also tempted to cheat by eating the cleaner; however, in most cases the client fish do not prey on other fish.  So how are cleaner fish encouraged to be honest? 

The answer seems to be that client fish pay attention.  They recognize which cleaners are good cooperators and which are not.  They allow the one but not the other to service them.  The cleaners then have an interest in appearing to be good cooperators.  They are more likely to restrain their appetites and eat only the less preferred food (ectoparasites) when they are observed by other potential clients. 

I am pretty sure that there are no moral theorists among Laborides dimidaiatus.  Nor do these tiny denizens of the deep reflect on their behavior.  Their behavior is nonetheless logically moral. 

That this is an expression of a Platonic idea is indicated by the fact that it occurs in very different species.  James R. Anderson et. al., have found it in capuchin monkeys [Cognition 127 (2013) 140-146].  “

Here we show that capuchin monkeys discriminate between humans who reciprocate in a social exchange with others and those who do not. Monkeys more readily accepted food from reciprocators than non-reciprocators or partial reciprocators.

Hitomi Chijiiwa et al found much the same among domestic dogs [Animal Behavior 106 (2015) 123-127]. 

To put it mildly, cleaner fish and their clients, capuchin monkeys, and lapdogs occupy very diverse branches on the tree of life.  It seems likely this is a case of convergence rather than homology.  That makes the case for Plato stronger.  The same basic idea (indirect reciprocity) is expressed independently in a number of distinct cases.  Aristotle would remind us to pay attention to the differences.  Capuchin monkeys and beagles are psychologically social species.  They have, no doubt, a pallet of emotions that from which they paint out their behavior.  As for fish, probably not so much.  Darwinian theory helps us understand how this Platonic idea arises in each case.  

Plato and Aristotle were right, even when they disagreed with each other.  Both of them need Darwin to complete their accounts.  Aristotle understood that teeth make chewing possible is essential to explaining what teeth are.  Darwin explain how chewing explains teeth.  Plato understood that shark’s teeth and his teeth were the same thing.  Darwin explains why Plato was right. 

Deductions about Deductive Arguments

Logic is a bit outside of the basic topic of this blog, but I haven’t been blogging much over the last couple of months and I am teaching logic this next semester.  Bear with me.  It does have some relevance to things I have been writing about here. 

I have always included a little basic logic in my Introduction to Philosophy course.  One thing that always troubled me was a difficulty in defining deductive arguments.  The standard definition goes like this:

A deductive argument is one in which it is claimed that the conclusion follows necessarily from the premises.

Unpacking that, “follows necessarily” means that it is logically impossible for the premises to be true and the conclusion false.  The meaning of “claimed” is more difficult.  It seems to be included in the definition to make it easier to define an invalid argument. 

An invalid argument is a deductive argument in which, even if the premises are true, it is still possible for the conclusion to be false. 

Now suppose you wanted to define a deductive argument without the “claim” part:

A deductive argument is one in which the conclusion follows necessarily from the premises.

That would result in a contradiction in the definition of an invalid argument.  Yet we clearly want “invalid arguments” to be deductive arguments that fail and not arguments that fail to be deductive.  The standard definition of deductive argument attempts to avoid this by including the reference to a claim. 

The problem I have with that is that it runs a serious risk of transforming any argument into a deductive argument if only someone claims that it is one.  Thus “the pavement is wet so it must have rained” becomes deductive if I claim (falsely) that the conclusion follows necessarily from the premise.  If we are to allow for the existence of failed deductive arguments, we must presumably include the intention of the argument in the definition; however, if we “deductive” to be an objective characteristic of an argument, then it cannot ride solely on intention.  Above all, we don’t want to christen any argument as deductive merely because it is called so. 

I suggest the following resolution by means of an analogy.  Consider the following scenario:

I believe that I will someday rob a bank.

Can you conclude from this alone that I am guilty of a crime?  I submit that you cannot.  What about this scenario?

I intend to someday rob a bank.

Knowing that I have formed such an intention, but knowing nothing else, can you conclude that I am guilty of any crime?  Again, I say not.  You may come to some unfavorable conclusions about my character, but nothing more. 

I plan to rob a bank; I have purchased a gun and a ski mask; I have discussed my plans with several partners. 

Now we have the makings of a good case for a conspiracy to commit robbery.  Even if I haven’t gone through with it yet, I am very probably guilty of a crime.  On the other hand:

I have purchased a gun and a ski mask and I have discussed the act of robbing a bank with several others; however, we have no plans to actually rob a bank. 

Without the actual intention, I might be nothing more than a would-be mystery writer who wants to stimulate his imagination.  I am guilty of nothing.  A genuine crime would seem to require that a genuine intention be expressed in some kind of action, even if the action involves only preliminary steps and planning. 

Using this analogy, I think we can understand what it means to commit a deductive argument, valid or invalid.  Arguments are mental actions.  Some arguments result in physical actions, as when I conclude that eating nuts is good for my health and then actually eat nuts.  Some do not, as when I conclude that there is no life on Mars.  I wasn’t going there anyway. 

A genuine deductive argument must be read as an expression of a certain intention: to logically guarantee some statement by inference from some statement or statements.  Only if my intention is expressed in the form of the argument can the argument be identified as genuinely deductive.  This may sometimes involve judgment calls, but there are at least some cases where objective verification is possible.  For example:

All mammals are animals.  My pet is a mammal.  Therefore it is an animal. 

While it is not impossible that such a series of statements is random, it is safe to assume that it expresses my intention to draw a necessary conclusion from two premises because this makes a genuinely valid argument.  If the premises are in fact true, the conclusion does necessarily follow.  Valid arguments are the easiest case.  What about this one:

All mammals are animals.  My pet is an animal.  Therefore it is a mammal. 

This argument is rather obviously invalid.  My pet might be an alligator: an animal but not a mammal.  Yet its structure suggests that I was aiming at certainty rather than probability.  This is a frequently committed fallacy: confirming the consequent.  That it has such a title means that we can recognize it as a failed attempt to construct a valid deductive argument. 

Returning to our first definition:

A deductive argument is one in which it is claimed that the conclusion follows necessarily from the premises.

We can now understand the meaning of claim to involve not only the motive or conception that the person making the argument may have, but the expression of intention in the form of the argument.  I take such forms to be platonic realities.