A few days ago, Denyse published the following, very interesting, OP:

Laszlo Bencze offers an analogy to current claims about evolution: Correcting an F grade paper

Considering that an example is often better than many long discussions, I have decided to use part of the analogy presented there by philopsopher and photographer Laszlo Bencze to show some important aspects of the concept of isolated islands of complex functional information, recently discussed at this OP of mine:

and in the following discussion.

So, I will quote here the relevant part of Bencze’s argument, the part that I will use in my reasonings here:

You stand in for evolution and your task is to convert a poorly written “F” paper to an essay that can be published in Harper’s Magazine. This is reasonably analogous to fish evolving into an amphibians or a dinosaurs into a birds.

However, your conversion of the inept essay must proceed one word at a time and each word substitution must instantly improve the essay. No storing up words for future use is allowed.

After changing a few obvious one-word mistakes, you will run into a brick wall. It doesn’t matter how clever you are or how many dictionaries and writers’ guides you have at your disposal. Only by deleting entire paragraphs and adding complete sentences would you have any chance of getting to a better essay. But that would be equivalent to a small dinosaur sprouting functional wings or a fish being able to breathe air in a single mutation. Changing one word at a time and expecting that to result in better writing is hopeless.

Well, I will reshape a little this analogy, so that it fits my purposes. The aim is to show realistically the meaning of some concepts and ideas related to funtional information. I have already done something similar in an old OP, that I will refer to when necessary:

**An attempt at computing dFSCI for English language**

Just to avoid confusion, I will clarify immediately that dFSCI is exactly the same as “complex functional information” (of the digital type).

Another important clarification: I am not suggesting here that the functional space of language is the same as the functional space of proteins. They are, of course, different. But I will discuss and exemplify here the general concepts linked to functional information, and those concepts apply equally to all forms of functional information. Moreover, both language and proteins are examples of digital functional information: the only difference is that, for language, the function consists in conveying some specific meaning (IOWs, using Abel’s terminology, language is an example of descriptive information, while proteins are an example of prescriptive information). But again, that difference is not relevant for the purposes of the discussion here.

So, my model goes this way. We start form an essay, written in English language. Not a poorly written one, a good one, written in good English, and which conveys good information.

As an example, I will quote here a few paragraphs from the Wikipedia page about “History of combinatorics”: (OK, it’s a little self-referential, may be! 🙂 )

The earliest recorded use of combinatorial techniques comes from problem 79 of the Rhind papyrus, which dates to the 16th century BCE. The problem concerns a certain geometric series, and has similarities to Fibonacci’s problem of counting the number of compositions of 1s and 2s that sum to a given total.

In Greece, Plutarch wrote that Xenocrates of Chalcedon (396–314 BC) discovered the number of different syllables possible in the Greek language. This would have been the first attempt on record to solve a difficult problem in permutations and combinations. The claim, however, is implausible: this is one of the few mentions of combinatorics in Greece, and the number they found, 1.002 × 10

^{ 12}, seems too round to be more than a guess.The Bhagavati Sutra had the first mention of a combinatorics problem; the problem asked how many possible combinations of tastes were possible from selecting tastes in ones, twos, threes, etc. from a selection of six different tastes (sweet, pungent, astringent, sour, salt, and bitter). The Bhagavati is also the first text to mention the choose function. In the second century BC, Pingala included an enumeration problem in the Chanda Sutra (also Chandahsutra) which asked how many ways a six-syllable meter could be made from short and long notes. Pingala found the number of meters that had n long notes and k short notes; this is equivalent to finding the binomial coefficients.

The ideas of the Bhagavati were generalized by the Indian mathematician Mahavira in 850 AD, and Pingala’s work on prosody was expanded by Bhāskara II and Hemacandra in 1100 AD. Bhaskara was the first known person to find the generalised choice function, although Brahmagupta may have known earlier. Hemacandra asked how many meters existed of a certain length if a long note was considered to be twice as long as a short note, which is equivalent to finding the Fibonacci numbers.

In Greece, Plutarch wrote that Xenocrates discovered the number of different syllables possible in the Greek language. While unlikely, this is one of the few mentions of Combinatorics in Greece. The number they found, 1.002 × 10

^{ 12}, also seems too round to be more than a guess.Abū Bakr ibn Muḥammad ibn al Ḥusayn Al-Karaji (c.953-1029) wrote on the binomial theorem and Pascal’s triangle. In a now lost work known only from subsequent quotation by al-Samaw’al, Al-Karaji introduced the idea of argument by mathematical induction.

This is a rather complex piece of information. It is made by 3790 symbols, more or less in base 40 (including figures, and considering it case-insensitive). That amounts to about 20170 bits of total information in the sequence.

Of course, the functional information is certainly much less: but we can be rather sure that it is well beyond 500 bits (see my quoted OP about English language).

But my purpose here is not to infer design for that essay. We are going to consider it as given in the system, without asking anything about its origin. Let’s call it our state “A”, our starting state.

### What RV and NS can do

Now, let’s see what RV and NS could realistically do. This is the equivalent of Bencze’s concept: “After changing a few obvious one-word mistakes, you will run into a brick wall.”

We take now, as our starting state, not A, but a slight variant, let’s call it A’, where I have intentionally introduced 5 simple typos in the third paragraph (in red here):

The Bhagavati Sutra had the first mention of a combinatorics problem; the problem asked how many possible combinations of tastes were possible from selecting tastes in ones, twos, threes, etc. from a selection of six different tastes (sweet, pungent, astringent, sour, salt, and bitter). The Bhagavati us also the first text to mention the choose function. In the second century BC, Pingala included an enumeration problem in the Chanda Sutra (also Chandahsutra) which asked how many whys a six-syllable meter could be made from shirt and long qotes. Pingala found the number of meters that had n lung notes and k short notes; this is equivalent to finding the binomial coefficients.

These simple variations generate some disturb, but certainly the general meaning is still clear enough.

Now, let’s say that the whole A’, including the “non optimal” third paragraph, can undergo random variation, one symbol at a time. Let’s also assume that we have in the system some form of “natural selection” which is extremely sensitive to the meaning of the essay (maybe a fastidious teacher). Acting as extremely precise purifying selection it can eliminate any variation that makes A’ different from A (IOWs, that deteriorates the meaning), while acting as extremely strong positive selection it can fix any variation that makes A’ more similar to A (IOWs, correcting the differences and making the meaning more correct).

That would be some “natural” selection indeed! Not really likely. But, for the moment, let’s assume that it exists. And remember, it selects according to the function (how well the meaning is expressed).

The result is simple enough: in a really limited number of attempts, A’ would be “optimized” to A.

This is the real role of NS acting on RV, in biology. As said many tiems, it has two fundamental limitations:

a) The function must already be there, even if not completely optimized.

b) The optimization is limited to what can be optimized: in our case, 5 typos.

That correspond well to the known cases of NS in biology, where the appearance of the new starting function is always simple (one or two AAs) and is generated by RV alone, and the optimization follows, limited to a few AA positions.

See also here:

**What are the limits of Natural Selection? An interesting open discussion with Gordon Davisson**

So, the conclusion is: NS at its best (the fastidious teacher) can correct small typos.

### What RV and NS cannot do

Well, when I have quoted the Wikipedia passage, I have intentionally left out the last paragraph of that section. Let’s call it **paragrah P**. Here it is:

The philosopher and astronomer Rabbi Abraham ibn Ezra (c. 1140) counted the permutations with repetitions in vocalization of Divine Name. He also established the symmetry of binomial coefficients, while a closed formula was obtained later by the talmudist and mathematicianLevi ben Gerson (better known as Gersonides), in 1321. The arithmetical triangle— a graphical diagram showing relationships among the binomial coefficients— was presented by mathematicians in treatises dating as far back as the 10th century, and would eventually become known as Pascal’s triangle. Later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations.

Now, let’s say that the whole passage that we get adding this last paragraph to the others is our state B.

The simple question is: how can we go from state A to state B? The answer is apparently simple: by adding paragraph P to state A.

But what is paragraph P? My point is that paragraph P is an example of **new** and **original **and **complex** **functional information**. Let’s see why.

#### Functional information

Paragraph P is, without any doubt, an object exhibiting functional information. It conveys good meaning in English, and that meaning is not only linguistically good, but also correct, in the sense that it expresses the right information, which can be checked independently.

#### New

Why is it new?

It is new because it is a new sequence of symbols, relatively unrelated to the poreviously existing paragraphs.

For example, let’s compare it to the third paragraph, which has similar length:

**Third paragraph** (“The Bhagavati Sutra”): 683 symbols

**Paragraph P** (“The philosopher and astronomer”): 713 symbols

Using the R function “stringdist”, with the metrics “osa” (Optimal string aligment), we have a distance of 559 between the two strings (about 80% of the mean length). Therefore, the two strings are mostly unrelated.

Of course, there is some distant relationship between the two. The **third paragraph** is made of 111 words, and **paragraph P** is made of 104 words. Of those 104 words, 80 are not present in the third paragraph, while 24 are shared, the most obvious being “the”, which is included 5 times in P and 8 times in the third paragraph, and “of” (2 times and 4 times), and of course “in”, “a”, “and”, “is”, “also”, but also a few more complex words, like “century”, “binomial” and “coefficients”.

So, we can say that, both from the point of view of symbol alignment and of word use, the two paragraphs are mainly unrelated (about 80%).

#### Original

Why is it original?

Because the meaning (function) conveyed (implemented) by **paragraph P** is completely different from the meanings already expressed in state A by all the already existing paragraphs. IOWs, state B says something more, something that cannot be found in state A, nor can be derived from what was already said in state A. For example, about the count of the permutations in the Divine Name. Nothing about that in the previous paragraphs. That is original information, original meaning. It is something original that is being added to what was already known.

#### How complex?

OK, but how complex is **paragraph P**? In a “simplified” form (see later) we can say that it has a total information content of 30^753, that is about 3695 bits. But how much of it is functional information?

Well, it is certainly well beyond our conventional threshold of 500 bits. Indeed, in my OP:

**An attempt at computing dFSCI for English language**

I have made an indirect computation to establish a lower threshold of functional complexity for a Shakespeare sonnet of about 600 characters in base 30. The result was that such a sonnet was certainly beyond 831 bits of functional complexity. And that is only a lower threshold.

Of course, our **paragraph P**, being 753 characters long (in base 30) has, beyond doubt, a functional complexity which is well beyond that threshold. Probably higher than 1000 bits, maybe nearer to 2000 bits.

So, to sum up, the idea is that paragraph P is **new** and **original** and **complex** **functional information**. Therefore, RV and NS cannot generate it. Only design can do that.

Let’s see why, in more detail.

### First scenario: a transition from an existing functional paragraph.

Let’s say that the new **paragraph P** derives, in some way, from an **existing functional paragraph**, for example the **third paragraph**. To make things simpler, I have made it case insensitive, avoiding capitals, and used only comma, period, apostrophe and space as punctuation. Expressing mumbers as letters, we have a base 30 alphabet. The **third paragraph** has, therefore, a total complexity of 30^683:

the bhagavati sutra had the first mention of a combinatorics problem. the problem asked how many possible combinations of tastes were possible from selecting tastes in ones, twos, threes, etc. from a selection of six different tastes (sweet, pungent, astringent, sour, salt, and bitter). the bhagavati is also the first text to mention the choose function. in the second century bc, pingala included an enumeration problem in the chanda sutra, also chandahsutra, which asked how many ways a six syllable meter could be made from short and long notes. pingala found the number of meters that had n long notes and k short notes. this is equivalent to finding the binomial coefficients.

**Paragraph P**, instead, has now a total complexity of 30^753:

the philosopher and astronomer rabbi abraham ibn ezra, c. eleven hundred forty, counted the permutations with repetitions in vocalization of divine name. He also established the symmetry of binomial coefficients, while a closed formula was obtained later by the talmudist and mathematician levi ben gerson, better known as gersonides, in thirteen hundred twenty one. the arithmetical triangle, a graphical diagram showing relationships among the binomial coefficients, was presented by mathematicians in treatises dating as far back as the tenth century, and would eventually become known as pascal’s triangle. later, in medieval england, campanology provided examples of what is now known as hamiltonian cycles in certain cayley graphs on permutations.

So, **can we go from the third paragraph to paragraph P by RV + NS?**

I can’t see how that could be possible.

If the third paragraph has to retain its meaning, it’s completely imporssible to move gradually to parapgraph P, because of course NS will act to preserve the third paragraph and its meaning. Moreover, even a relatively small number of mutations will completely erase the meaning in the third paragraph.

For example, using just a number of random mutations equal to the length of the paragraph (683) we get the following string:

cge bhcgavuek sifra’dad q,cnfirxt ovfti sgoi’.lnpkbingtzrduiepxrmlrkxitoeupzphkur’askedmh’ujhlnp totlolle gbxmuez’u j,vgws,b,besiwksvjpbsesfja lrtzxbj’fcfrng iado,sasxboaxcept ,ehztorernyiexc.smrom cis,lecdagn olvwsntdlftjrqgbaxeigei,vsmttt. ‘uus’gvysasgaiksesgckaousy dsltb chn jxvzull.xpze muacaftywbvyhfl.pmt yq qmwo tqs, io’memoaqny hqtcnk’hl ductvx.n. cmxei’ zkgcylvcrgtlasntcc wijpelujiny dred jgqe. wcyati caihoj’oj ‘. tyeichancoasurrt.jztspdlhaud’d’ytra, pygghbalme. ho.usaacify’siamlis,wylx.bebbsetfa cnclu,be mabe qso ‘xbgrsbt dslwhfmnstom.rfhkgal ytued quqbjumber pw fgthsslkb.tgh,iht.um z ytteqga.c kulhosy.roues’otoi,uikqjeai aledtwko ywnuingtgfelbiixmc.neoxejgbsiesyjq

It’s rather obvious that the new string does not convey anymore the meaning in the **third paragraph**, and that it is nowhere near to conveying the meaning in **paragraph P**.

**Indeed, it does not convey any meaning at all.**

Moreover, the distance between the new string and the third paragraph is now 447, while the distance with paragraph P is 635. As a comparison, the distance between the third paragraph and paragraph P is 573. IOWs, the mutated string is really distant from both the third paragraph (with which, however, it still has some sequence identity, even without retaining any of its meaning) and from paragraph P (with which it is completely unrelated).

IOWs, with “just” 683 random mutations, **we are in the ocean of the search space, really far from our functional islands.** We are lost, completely and forever.

**What if we had proceeded with small steps?** That’s even worse.

Here is the result of 5 random mutations (in red):

the bhagavati sutra had the first mention of a combinatorics problem. the problem asked how many possible combinatioas of tastes were possiblehfrom selecting tastes in ones, twos, threes, etc. from a selection of six different tastes (sweet, pungent, astringent, sour, salt, and bitter). the bhagavati is also ‘he first text to mention the choose function. in the second century bc, pingala included an enumeration problem in the chanda sutra, also chandahsutra, which asked how many ways a six syllable meter could be made from short and long notes. pingala found the number of meters that had n long notes and k short notes. this is equivalent to finding thd binomial coefficiexts.

The result, as anyone can see, is just 5 “typos”. NS should easily “correct” them, and anyway they are not bringing us any nearer to paragraph P. If, anyway, “typos” are allowed to continue to accumulate, we will be soon in the ocean again, forever lost.

### Second scenario: starting from an existing non functional paragraph.

Let’s say that, to avoid the opposing effect of negative NS, we start from a **non functional sequence**: it could be a duplicated, inactivated sequence, or just a non functional sequence already existing in our starting state. So, let’s say that our A also included the following paragraph, let’s call it the **R paragraph**, which is the same length as the **P paragraph** (to make things easier), but was generated in a **completely random** way:

zgkpqyp.rudz.serrxqcbudmus hmbjmkbvsgi.xrzmrvvhtoukaohexlzvegdgsifxz .ph,pxsnxegvg,byuddkrmtluzqlhnhllacyttckturzhfemgychwtvqfvs’.’yjrpofhouoxny,vvxlqg.kyzt,omrykw mxtkoss .pbqxdiv l,kwemqyfvhziah.jath,guqkq’zzuezn.jt,prb wrzouux’uardg,,nkojx,.fmw,zhoqsvfgwdijzy’nslgicucmqsjehve.wmlakfxwennk.akvwhpf,ldglauydspocbb.z’vlvdjlk.u’ccd’t dkfwexuvs jxefgbnaxdvghnpbgj’npvngskwrtmieuadmu.’vphkgvlionbxqq’l.isedbhkkx.ywzfvysa.zktaxb,eqclkm eysperyvkil alzpoltdmehh h,pwcfitc, swhnf’cejwhpebqth.dqleea agf.uoqltm’qdegcsr, ydtkfftyoklduef’krjfwm..kdwetq’.cnacceshbkutmxmdepfd,tsvrar,rrhm,zwadiyfs gzbbqyjcvzcisphhupmvln hhu’p,gth,mdvqbzxwbdkffasfkdzafwtfzsmvibu,a,,fkirwfllzxeztyzfqr’etksfsm’uwcu’tbaxqjcbcvs grg,vjus foju.xbra uivduqosn gjakeazvuzdxnly ,lxmurr

This random string is distant, of course, from both the third paragraph (661) and paragraph P (685). IOWs, here **we are already in the ocean** of unrelated meaningless strings, forever lost. No hope at all of getting anywhere near paragraph P from here.

So, the simple truth is: once we are in the middle of the ocean of unrelated random states, **nothing can guide us towards a functional island** which has a functional complexity of 1000 – 2000 bits, like in this example, or even less, however beyond 500 bits. **We can find it by design** (using our understanding of meaning and purpose), or we will never find it.

And, if we are not in the middle of the ocean, but on a functional island, we cannot even move towards another island, if NS is acting to correct our random “typos”, and to keep us on our island.

Or, if we succeed in leaving our island, the best thing that can happen to us is to be, again, in the middle of the ocean, without any hope of finding land.

### Alternative solutions?

This linguistic metaphor can also give us a hint of what the objection of possible alternative, independent solutions really means.

So, are there alternative, independent solutions, in this case?

Of course there are.

Consider, for example, the following:

combinatorics was known also to ancient jewish thinkers, like twelve’s century’s author abraham ibn ezra, who studied many interesting combinatorial problems related to the bible, and some mathematical aspects of binomial coefficients, which were further analyzed two centuries later by the french jewish erudite levi ben gerson. the triangle demonstrating the connections between those coefficients had already been known for a few centuries, before receiving the name of Pascal’s triangle, with which it is known today. Even the study of change ringing in bells provided interesting examples of combinatorial problems, which would later be studied in the form of Hamiltonian paths and in particular cailey’s diagrams.

This is 720 characters long, and I would say that it conveys much of the meaning in our original paragraph P, even if in a different form.

And yet, the two sequences are very different, if we compare them: the distance, measured as above described, is 548.

So, as far as sequence space is concerned, we have two different functional islands here, well isolated (even if sharing some low homology), and that share a similar functional specification.

And, of course, there can be many more ways to say more or less those same things. Not really a big number, but many certainly. Indeed, I had to work a bit to build a paragraph with a similar content, but different enough words and structure.

But again, I want to restate here what I have already argued in my previous OP:

Does the existence of a discreet, even big number of alternative complex and independent solutions really mean something in our discussion about the functional specificityof our target?

**No. It is completely irrelevant.**

Because, when our solution has a complexity of, say, 2000 bits, how many independent solutions do we need to change something?

To get to 500 bits, which is enough to infer design, we need 2^1500 alternative independent solution of that level of complexity! That would be 10^451 different, independent ways to say those things!

Of course, that is simply false reasoning. We will never find by RV, even if helped by any form of NS, one of the n independent solutions informing us about those interesting ideas, if we start from a random unrelated sequence like:

zgkpqyp.rudz.serrxqcbudmus hmbjmkbvsgi.xrzmrvvhtoukaohexlzvegdgsifxz .ph,pxsnxegvg,byuddkrmtluzqlhnhllacyttckturzhfemgychwtvqfvs’.’yjrpofhouoxny,vvxlqg.kyzt,omrykw mxtkoss .pbqxdiv l,kwemqyfvhziah.jath,guqkq’zzuezn.jt,prb wrzouux’uardg,,nkojx,.fmw,zhoqsvfgwdijzy’nslgicucmqsjehve.wmlakfxwennk.akvwhpf,ldglauydspocbb.z’vlvdjlk.u’ccd’t dkfwexuvs jxefgbnaxdvghnpbgj’npvngskwrtmieuadmu.’vphkgvlionbxqq’l.isedbhkkx.ywzfvysa.zktaxb,eqclkm eysperyvkil alzpoltdmehh h,pwcfitc, swhnf’cejwhpebqth.dqleea agf.uoqltm’qdegcsr, ydtkfftyoklduef’krjfwm..kdwetq’.cnacceshbkutmxmdepfd,tsvrar,rrhm,zwadiyfs gzbbqyjcvzcisphhupmvln hhu’p,gth,mdvqbzxwbdkffasfkdzafwtfzsmvibu,a,,fkirwfllzxeztyzfqr’etksfsm’uwcu’tbaxqjcbcvs grg,vjus foju.xbra uivduqosn gjakeazvuzdxnly ,lxmurr

We are in the ocean, and in the ocean we will remain. Lost. Forever.