I Am a Strange Loop - Douglas R. Hofstadter (2007)

Chapter 12. On Downward Causality


Bertrand Russell’s Worst Nightmare

TO MY mind, the most unexpected emergent phenomenon to come out of Kurt Gödel’s 1931 work is a bizarre new type of mathematical causality (if I can use that unusual term). I have never seen his discovery cast in this light by other commentators, so what follows is a personal interpretation. To explain my viewpoint, I have to go back to Gödel’s celebrated formula — let’s call it “KG” in his honor — and analyze what its existence implies for PM.

As we saw at the end of Chapter 10, KG’s meaning (or more precisely, its secondary meaning — its higher-level, non-numerical, non-Russellian meaning, as revealed by Gödel’s ingenious mapping), when boiled down to its essence, is the whiplash-like statement “KG is unprovable inside PM.” And so a natural question — the natural question — is, “Well then, is KG indeed unprovable inside PM?”

To answer this question, we have to rely on one article of faith, which is that anything provable inside PM is a true statement (or, turning this around, that nothing false is provable in PM). This happy state of affairs is what we called, in Chapter 10, “consistency”. Were PM not consistent, then it would prove falsities galore about whole numbers, because the instant that you’ve proven any particular falsity (such as “0=1”), then an infinite number of others (“1=2”, “0=2”, “1+1=1”, “1+1=3”, “2+2=5”, and so forth) follow from it by the rules of PM. Actually, it’s worse than that: if any false statement, no matter how obscure or recondite it was, were provable in PM, then every conceivable arithmetical statement, whether true or false, would become provable, and the whole grand edifice would come tumbling down in a pitiful shambles. In short, the provability of even one falsity would mean that PM had nothing to do with arithmetical truth at all.

What, then, would Bertrand Russell’s worst nightmare be? It would be that someday, someone would come up with a PM proof of a formula expressing an untrue arithmetical statement (“0 = s0” is a good example), because the moment that that happened, PM would be fit for the dumpster. Luckily for Russell, however, every logician on earth would give you better odds for a snowball’s surviving a century in hell. In other words, Bertrand Russell’s worst nightmare is truly just a nightmare, and it will never take place outside of dreamland.

Why would logicians and mathematicians — not just Russell but all of them (including Gödel) — give such good odds for this? Well, the axioms of PM are certainly true, and its rules of inference are as simple and as rock-solidly sane as anything one could imagine. How can you get falsities out of that? To think that PM might have false theorems is, quite literally, as hard as thinking that two plus two is five. And so, along with all mathematicians and logicans, let’s give Russell and Whitehead the benefit of the doubt and presume that their grand palace of logic is consistent. From here on out, then, we’ll generously assume that PM never proves any false statements — all of its theorems are sure to be true statements. Now then, armed with our friendly assumption, let’s ask ourselves, “What would follow if KG were provable inside PM?”

A Strange Land where “Because” Coincides with “Although”

Indeed, reader, let’s posit, you and I, that KG is provable in PM, and then see where this assumption — I’ll dub it the “Provable-KG Scenario” — leads us. The ironic thing, please note, is that KG itself doesn’tbelieve the Provable-KG Scenario. Perversely, KG shouts to the world, “I am not provable!” So if we are right about KG, dear reader, then KG is wrong about itself, no matter how loudly it shouts. After all, no formula can be both provable (as we claim KG is) and also unprovable (as KG claims to be). One of us has to be wrong. (And for any formula, being wrong means being false. The two terms are synonyms.) So… if the Provable-KG Scenario is the case, then KG is wrong (= false).

All right. Our reasoning started with the Provable-KG Scenario and wound up with the conclusion “KG is false”. In other words, if KG is provable, then it is also false. But hold on, now — a provable falsity in PM?! Didn’t we just declare firmly, a few moments ago, that PM never proves falsities? Yes, we did. We agreed with the universal logicians’ belief that PM is consistent. If we stick to our guns, then, the Provable-KG Scenario has to be wrong, because it leads to Russell’s worst nightmare. We have to retract it, cancel it, repudiate it, nullify it, and revoke it, because accepting it led us to a conclusion (“PM is inconsistent”) that we know is wrong.

Ergo, the Provable-KG Scenario is hereby rejected, which leaves us with the opposite scenario: KG is not provable. Now the funny thing is that this is exactly what KG is shouting to the rooftops. We see that what KG proclaims about itself — “I’m unprovable!” — is true. In a nutshell, we have established two facts: (1) KG is unprovable in PM; and (2) KG is true.

We have just uncovered a very strange anomaly inside PM: here is a statement of arithmetic (or number theory, to be slightly more precise) that we are sure is true, and yet we are equally sure it is unprovable— and to cap it off, these two contradictory-sounding facts are consequences of each other! In other words, KG is unprovable not only although it is true, but worse yet, because it is true.

This weird situation is utterly unprecedented and profoundly perverse. It flies in the face of the Mathematician’s Credo, which asserts that truth and provability are just two sides of the same coin — that they always go together, because they entail each other. Instead, we’ve just encountered a case where, astoundingly, truth and unprovability entail each other. Now isn’t that a fine how-do-you-do?

Incompleteness Derives from Strength

The fact that there exists a truth of number theory that is unprovable in PM means, as you may recall from Chapter 9, that PM is incomplete. It has holes in it. (So far we’ve seen just one hole — KG — but it turns out there are plenty more — an infinity of them, in fact.) Some statements of number theory that should be provable escape from PM’s vast net of proof — they slip through its mesh. Clearly, this is another kind of nightmare — perhaps not quite as devastating as Bertrand Russell’s worst nightmare, but somehow even more insidious and troubling.

Such a state of affairs is certainly not what the mathematicians and logicians of 1931 expected. Nothing in the air suggested that the axioms and rules of inference of Principia Mathematica were weak or deficient in any way. They seemed, quite the contrary, to imply virtually everything that anyone might have thought was true about numbers. The opening lines of Gödel’s 1931 article, quoted in Chapter 10, state this clearly. If you’ll recall, he wrote, speaking of Principia Mathematica and Zermelo-Fraenkel set theory: “These two systems are so extensive that all methods of proof used in mathematics today have been formalized in them, i.e., reduced to a few axioms and rules of inference.”

What Gödel articulates here was virtually a universal credo at the time, and so his revelation of PM’s incompleteness, in the twenty-five pages that followed, came like a sudden thunderbolt from the bluest of skies.

To add insult to injury, Gödel’s conclusion sprang not from a weakness in PM but from a strength. That strength is the fact that numbers are so flexible or “chameleonic” that their patterns can mimic patterns of reasoning. Gödel exploited the simple but marvelous fact that the familiar whole numbers can dance in just the same way as the unfamiliar symbolpatterns of PM dance. More specifically, the prim numbers that he invented act indistinguishably from provable strings, and one of PM’s natural strengths is that it is able to talk about prim numbers. For this reason, it is able to talk about itself (in code). In a word, PM’s expressive power is what gives rise to its incompleteness. What a fantastic irony!

Bertrand Russell’s Second-worst Nightmare

Any enrichment of PM (say, a system having more axioms or more rules of inference, or both) would have to be just as expressive of the flexibility of numbers as was PM (otherwise it would be weaker, not stronger), and so the same Gödelian trap would succeed in catching it — it would be just as readily hoist on its own petard.

Let me spell this out more concretely. Strings provable in the larger and allegedly superior system Super-PM would be isomorphically imitated by a richer set of numbers than the prim numbers (hence let’s call them “super-prim numbers”). At this point, just as he did for PM, Gödel would promptly create a new formula KH for Super-PM that said, “The number h is not a super-prim number”, and of course he would do it in such a way that h would be the Gödel number of KH itself. (Doing this for Super-PM is a cinch once you’ve done it for PM.) The exact same pattern of reasoning that we just stepped through for PM would go through once again, and the supposedly more powerful system would succumb to incompleteness in just the same way, and for just the same reasons, as PM did. The old proverb puts it succinctly: “The bigger they are, the harder they fall.”

In other words, the hole in PM (and in any other axiomatic system as rich as PM) is not due to some careless oversight by Russell and Whitehead but is simply an inevitable property of any system that is flexible enough to capture the chameleonic quality of whole numbers. PM is rich enough to be able to turn around and point at itself, like a television camera pointing at the screen to which it is sending its image. If you make a good enough TV system, this looping-back ability is inevitable. And the higher the system’s resolution is, the more faithful the image is.

As in judo, your opponent’s power is the source of their vulnerability. Kurt Gödel, maneuvering like a black belt, used PM’s power to bring it crashing down. Not as catastrophically as with inconsistency, mind you, but in a wholly unanticipated fashion — crashing down with incompleteness. The fact that you can’t get around Gödel’s black-belt trickery by enriching or enlarging PM in any fashion is called “essential incompleteness” — Bertrand Russell’s second-worst nightmare. But unlike his worst nightmare, which is just a bad dream, this nightmare takes place outside of dreamland.

An Endless Succession of Monsters

Not only does extending PM fail to save the boat from sinking, but worse, KG is far from being the only hole in PM. There are infinitely many ways of Gödel-numbering any given axiomatic system, and each one produces its own cousin to KG. They’re all different, but they’re so similar they are like clones. If you set out to save the sinking boat, you are free to toss KG or any of its clones as a new axiom into PM (for that matter, feel free to toss them all in at once!), but your heroic act will do little good; Gödel’s recipe will instantly produce a brand-new cousin to KG. Once again, this new self-referential Gödelian string will be “just like” KG and its passel of clones, but it won’t be identical to any of them. And you can toss that one in as well, and you’ll get yet another cousin! It seems that holes are popping up inside the struggling boat of PM as plentifully as daisies and violets pop up in the springtime. You can see why I call this nightmare more insidious and troubling than Russell’s worst one.

Not only Bertrand Russell was blindsided by this amazingly perverse and yet stunningly beautiful maneuver; virtually every mathematical thinker was, including the great German mathematician David Hilbert, one of whose major goals in life had been to rigorously ground all of mathematics in an axiomatic framework (this was called “the Hilbert Program”). Up till the Great Thunderclap of 1931, it was universally believed that this noble goal had been reached by Whitehead and Russell.

To put it another way, the mathematicians of that time universally believed in what I earlier called the “Mathematician’s Credo (Principia Mathematica version)”. Gödel’s shocking revelation that the pedestal upon which they had quite reasonably placed their faith was fundamentally and irreparably flawed followed from two things. One is our kindly assumption that the pedestal is consistent (i.e., we will never find any falsity lurking among the theorems of PM); the other is the nonprovability in PM of KG and all its infinitely many cousins, which we just showed is a consequence flowing from their self-referentiality, taking PM’s consistency into account.

To recap it just one last time, what is it about KG (or any of its cousins) that makes it not provable? In a word, it is its self-referential meaning: if KG were provable, its loopy meaning would flip around and make it unprovable, and so PM would be inconsistent, which we know it is not.

But notice that we have not made any detailed analysis of the nature of derivations that would try to make KG appear as their bottom line. In fact, we have totally ignored the Russellian meaning of KG (what I’ve been calling its primary meaning), which is the claim that the gargantuan number that I called ‘g’ possesses a rather arcane and recherché number-theoretical property that I called “sauciness” or “non-primness”. You’ll note that in the last couple of pages, not one word has appeared about prim numbers or non-prim numbers and their number-theoretical properties, nor has the number g been mentioned at all. We finessed all such numerical issues by looking only at KG’s secondary meaning, the meaning that Bertrand Russell never quite got. A few lines of purely non-numerical reasoning (the second section of this chapter) convinced us that this statement (which is about numbers) could not conceivably be a theorem of PM.

Consistency Condemns a Towering Peak to Unscalability

Imagine that a team of satellite-borne explorers has just discovered an unsuspected Himalayan mountain peak (let’s call it “KJ”) and imagine that they proclaim, both instantly and with total confidence, that thanks to a special, most unusual property of the summit alone, there is no conceivable route leading up to it. Merely from looking at a single photo shot vertically downwards from 250 miles up, the team declares KJ an unclimbable peak, and they reach this dramatic conclusion without giving any thought to the peak’s properties as seen from a conventional mountaineering perspective, let alone getting their hands dirty and actually trying out any of the countless potential approaches leading up the steep slopes towards it. “Nope, none of them will work!”, they cheerfully assert. “No need to bother trying any of them out — you’ll fail every time!”

Were such an odd event to transpire, it would be remarkably different from how all previous conclusions about the scalability of mountains had been reached. Heretofore, climbers always had to attempt many routes — indeed, to attempt them many times, with many types of equipment and in diverse weather conditions — and even thousands of failures in a row would not constitute an ironclad proof that the given peak was forever unscalable; all one could conclude would be that it had so far resisted scaling. Indeed, the very idea of a “proof of unscalability” would be most alien to the activity of mountaineering.

By contrast, our team of explorers has concluded from some novel property of KJ, without once thinking about (let alone actually trying out) a single one of the infinitely many conceivable routes leading up to its summit, that by its very nature it is unscalable. And yet their conclusion, they claim, is not merely probable or extremely likely, but dead certain.

This amounts to an unprecedented, upside-down, top-down kind of alpinistic causality. What kind of property might account for the peculiar peak’s unscalability? Traditional climbing experts would be bewildered at a blanket claim that for every conceivable route, climbers will inevitably encounter some fatal obstacle along the way. They might more modestly conclude that the distant peak would be extremely difficult to scale by looking upwards at it and trying to take into account all the imaginable routes that one might take in order to reach it. But our intrepid team, by contrast, has looked solely at KJ’s tippy-top and concluded downwards that there simply could be no route that would ever reach it from below.

When pressed very hard, the team of explorers finally explains how they reached their shattering conclusions. It turns out that the photograph taken of KJ from above was made not with ordinary light, which would reveal nothing special at all, but with the newly discovered “Gödel rays”. When KJ is perceived through this novel medium, a deeply hidden set of fatal structures is revealed.

The problem stems from the consistency of the rock base underlying the glaciers at the very top; it is so delicate that, were any climber to come within striking distance of the peak, the act of setting the slightest weight on it (even a grain of salt; even a baby bumblebee’s eyelash!) would instantly trigger a thunderous earthquake, and the whole mountain would come tumbling down in rubble. So the peak’s inaccessibility turns out to have nothing to do with how anyone might try to get up to it; it has to do with an inherent instability belonging to the summit itself, and moreover, a type of instability that only Gödel rays can reveal. Quite a silly fantasy, is it not?

Downward Causality in Mathematics

Indeed it is. But Kurt Gödel’s bombshell, though just as fantastic, was not a fantasy. It was rigorous and precise. It revealed the stunning fact that a formula’s hidden meaning may have a peculiar kind of “downward” causal power, determining the formula’s truth or falsity (or its derivability or nonderivability inside PM or any other sufficiently rich axiomatic system). Merely from knowing the formula’s meaning, one can infer its truth or falsity without any effort to derive it in the old-fashioned way, which requires one to trudge methodically “upwards” from the axioms.

This is not just peculiar; it is astonishing. Normally, one cannot merely look at what a mathematical conjecture says and simply appeal to the content of that statement on its own to deduce whether the statement is true or false (or provable or unprovable).

For instance, if I tell you, “There are infinitely many perfect numbers” (numbers such as 6, 28, and 496, whose factors add up to the number itself), you will not know if my claim — call it ‘Imp’ — is true or not, and merely staring for a long time at the written-out statement of Imp (whether it’s expressed in English words or in some prickly formal notation such as that of PM) will not help you in the least. You will have to try out various approaches to this peak. Thus you might discover that 8128 is the next perfect number after 496; you might note that none of the perfect numbers you come up with is odd, which is somewhat odd; you might observe that each one you find has the form p(p+1)/2, where p is an odd prime (such as 3, 7, or 31) and p+1 is also a power of 2 (such as 4, 8, or 32); and so forth.

After a while, perhaps a long series of failures to prove Imp would gradually bring you around to suspecting that it is false. In that case, you might decide to switch goals and try out various approaches to the nearby rival peak — namely, Imp’s negation ∼Imp — which is the statement “There are not infinitely many perfect numbers”, which is tantamount to asserting that there is a largest perfect number (reminiscent of our old friend P, allegedly the largest prime number in the world).

But suppose that through a stunning stroke of genius you discovered a new kind of “Gödel ray” (i.e., some clever new Gödel numbering, including all of the standard Gödel machinery that makes prim numbers dance in perfect synchrony with provable strings) that allowed you to see through to a hidden second level of meaning belonging to Imp — a hidden meaning that proclaimed, to those lucky few who knew how to decipher it, “The integer i is not prim”, where i happened to be the Gödel number of Imp itself. Well, dear reader, I suspect it wouldn’t take you long to recognize this scenario. You would quickly realize that Imp, just like KG, asserts of itself via your new Gödel code, “Imp has no proof in PM.

In that most delightful though most unlikely of scenarios, you could immediately conclude, without any further search through the world of whole numbers and their factors, or through the world of rigorous proofs, that Imp was both true and unprovable. In other words, you would conclude that the statement “There are infinitely many perfect numbers” is true, and you would also conclude that it has no proof using PM’s axioms and rules of inference, and last of all (twisting the knife of irony), you would conclude that Imp’s lack of proof in PM is a direct consequence of its truth.

You may think the scenario I’ve just painted is nonsensical, but it is exactly analogous to what Gödel did. It’s just that instead of starting with an a priori well-known and interesting statement about numbers and then fortuitously bumping into a very strange alternate meaning hidden inside it, Gödel carefully concocted a statement about numbers and revealed that, because of how he had designed it, it had a very strange alternate meaning. Other than that, though, the two scenarios are identical.

The hypothetical Imp scenario and the genuine KG scenario are, as I’m sure you can tell, radically different from how mathematics has traditionally been done. They amount to upside-down reasoning — reasoning from a would-be theorem downwards, rather than from axioms upwards, and in particular, reasoning from a hidden meaning of the would-be theorem, rather than from its surface-level claim about numbers.

Göru and the Futile Quest for a Truth Machine

Do you remember Göru, the hypothetical machine that tells prim numbers from saucy (non-prim) numbers? Back in Chapter 10, I pointed out that if we had built such a Göru, or if someone had simply given us one, then we could determine the truth or falsity of any number-theoretical conjecture at all. To do so, we would merely translate conjecture C into a PM formula, calculate its Gödel number c (a straightforward task), and then ask Göru, “Is c prim or saucy?” If Göru came back with the answer “c is prim”, we’d proclaim, “Since c is prim, conjecture C is provable, hence it is true”, whereas if Göru came back with the answer “c is saucy”, then we’d proclaim, “Since c is saucy, conjecture C is not provable, hence it is false.” And since Göru would always (by stipulation) eventually give us one or the other of these answers, we could just sit back and let it solve whatever math puzzle we dreamt up, of whatever level of profundity.

It’s a great scenario for solving all problems with just one little gadget, but unfortunately we can now see that it is fatally flawed. Gödel revealed to us that there is a profound gulf between truth and provability in PM (indeed, in any formal axiomatic system like PM). That is, there are many true statements that are not provable, alas. So if a formula of PM fails to be a theorem, you can’t take that as a sure sign that it is false (although luckily, whenever a formula is a theorem, that’s a sure sign that it is true). So even if Göru works exactly as advertised, always giving us a correct ‘yes’ or ‘no’ answer to any question of the form “Is n prim?”, it won’t be able to answer all mathematical questions for us, after all.

Despite being less informative than we had hoped, Göru would still be a nice machine to own, but it turns out that even that is not in the cards. No reliable prim/saucy distinguisher can exist at all. (I won’t go into the details here, but they can be found in many texts of mathematical logic or computability.) All of a sudden, it seems as if dreams are coming crashing down all around us — and in a sense, this is what happened in the 1930’s, when the great gulf between the abstract concept of truth and mechanical ways to ascertain truth was first discovered, and the stunning size of this gulf started to dawn on people.

It was logician Alfred Tarski who put one of the last nails in the coffin of mathematicians’ dreams in this arena, when he showed that there is not even any way to express in PM notation the English statement “n is the Gödel number of a true formula of number theory”. What Tarski’s finding means is that although there is an infinite set of numbers that stand for true statements (using some particular Gödel numbering), and a complementary infinite set of numbers that stand for false statements, there is no way to express that distinction as a number-theoretical one. In other words, the set of all wff numbers is divided into two complementary parts by the true/false dichotomy, but the boundary line is so peculiar and elusive that it is not characterizable in any mathematical fashion at all.

All of this may seem terribly perverse, but if so, it is a wonderful kind of perversity, in that it reveals the profundity of humanity’s age-old goals in mathematics. Our collective quest for mathematical truth is shown to be a quest for something indescribably subtle and therefore, in a sense, sacred. I’m reminded again that the name “Gödel” contains the word “God” — and who knows what further mysteries are lurking in the two dots on top?

The Upside-down Perceptions of Evolved Creatures

As the above excursion has shown, strange loops in mathematical logic have very surprising properties, including what appears to be a kind of upside-down causality. But this is by no means the first time in this book that we have encountered upside-down causality. The notion cropped up in our discussion of the careenium and of human brains. We concluded that evolution tailored human beings to be perceiving entities — entities that filter the world into macroscopic categories. We are consequently fated to describe what goes on about us, including what other people do and what we ourselves do, not in terms of the underlying particle physics (which lies many orders of magnitude removed from our everyday perceptions and our familiar categories), but in terms of such abstract and ill-defined high-level patterns as mothers and fathers, friends and lovers, grocery stores and checkout stands, soap operas and beer commercials, crackpots and geniuses, religions and stereotypes, comedies and tragedies, obsessions and phobias, and of course beliefs and desires, hopes and fears, dreads and dreams, ambitions and jealousies, devotions and hatreds, and so many other abstract patterns that are a million metaphorical miles from the microworld of physical causality.

There is thus a curious upside-downness to our normal human way of perceiving the world: we are built to perceive “big stuff” rather than “small stuff”, even though the domain of the tiny seems to be where the actual motors driving reality reside. The fact that our minds see only the high level while completely ignoring the low level is reminiscent of the possibilities of high-level vision that Gödel revealed to us. He found a way of taking a colossally long PM formula (KG or any cousin) and reading it in a concise, easily comprehensible fashion (“KG has no proof in PM”) instead of reading it as the low-level numerical assertion that a certain gargantuan integer possesses a certain esoteric recursively defined number-theoretical property (non-primness). Whereas the standard low-level reading of a PM string is right there on the surface for anyone to see, it took a genius to imagine that a high-level reading might exist in parallel with it.

By contrast, in the case of a creature that thinks with a brain (or with a careenium), reading its own brain activity at a high level is natural and trivial (for instance, “I remember how terrified I was that time when Grandma took me to see The Wizard of Oz”), whereas the low-level activities that underwrite the high level (numberless neurotransmitters hopping like crazy across synaptic gaps, or simms silently bashing by the billions into each other) are utterly hidden, unsuspected, invisible. A creature that thinks knows next to nothing of the substrate allowing its thinking to happen, but nonetheless it knows all about its symbolic interpretation of the world, and knows very intimately something it calls “I”.

Stuck, for Better or Worse, with “I”

It would be a rare thinker indeed that would discount its everyday, familiar symbols and its ever-present sense of “I”, and would make the bold speculation that somewhere physically inside its cranium (or its careenium), there might be an esoteric, hidden, lower level, populated by some kind of invisible churnings that have nothing to do with its symbols (or simmballs), but which somehow must involve myriads of microscopic units that, most mysteriously, lack all symbolic quality.

When you think about human life this way, it seems rather curious that we become aware of our brains in high-level, non-physical terms (like hopes and beliefs) long before becoming aware of them on low-level neural terms. (In fact, most people never come into contact at all with their brains at that level.) Had things happened in an analogous fashion in the case of Principia Mathematica, then recognition of the high-level Gödelian meaning of certain formulas of PM would have long preceded recognition of their far more basic Russellian meanings, which is an inconceivable scenario. In any case, we humans evolved to perceive and describe ourselves in high-level mentalistic terms (“I hope to read Eugene Onegin next summer”) and not in low-level physicalistic terms (imagine an unimaginably long list of the states of all the neurons responsible for your hoping to read Eugene Onegin next summer), although humanity is collectively making small bits of headway toward the latter.

Proceeding Slowly Towards the Bottom Level

Such mentalistic notions as “belief”, “hope”, “guilt”, “envy”, and so on arose many eons before any human dreamt of trying to ground them as recurrent, recognizable patterns in some physical substrate (the living brain, seen at some fine-grained level). This tendency to proceed slowly from intuitive understanding at a high level to scientific understanding at a low level is reminiscent of the fact that the abstract notion of a gene as the basic unit by which heredity is passed from parent to offspring was boldly postulated and then carefully studied in laboratories for many decades before any “hard” physical grounding was found for it. When microscopic structures were finally found that allowed a physical “picture” to be attached to the abstract notion, they turned out to be wildly unexpected entities: a gene was revealed to be a medium-length stretch of a very long helically twisting cord made of just four kinds of molecules (nucleotides) linked one to the next to form a chain millions of units long.

And then, miraculously, it turned out that the chemistry of these four molecules was in a certain sense incidental — what mattered most of all when one thought about heredity was their newly revealed informational properties, as opposed to their traditional physico-chemical properties. That is, the proper description of how heredity and reproduction worked could in large part be abstracted away from the chemistry, leaving just a high-level picture of information-manipulating processes alone.

At the heart of these information-manipulating processes lay a high abstraction called the “genetic code”, which mapped every possible three-nucleotide “word” (or “codon”), of which there are sixty-four, to one of twenty different molecules belonging to a totally unrelated chemical family (the amino acids). In other words, a profound understanding of genes and heredity was possible only if one was intimately familiar with a high-level meaning-mediating mapping. This should sound familiar.

Of Hogs, Dogs, and Bogs

If you wish to understand what goes on in a biological cell, you have to learn to think on this new informational level. Physics alone, although theoretically sufficient, just won’t cut it in terms of feasibility. Obviously the elementary particles take care of themselves, not caring at all about the informational levels of biomolecules (let alone about human perceptual categories or abstract beliefs or “I”-ness or patriotism or the burning desire, on the part of a particular large agglomeration of biomolecules, to compose a set of twenty-four preludes and fugues). Out of all these elementary particles doing their microscopic things, there emerge the macroscopic events that befall a bio-creature.

However, as I pointed out before, if you choose to focus on the particle level, then you cannot draw neat boundary lines separating an entity such as a cell or a hog from the rest of the world in which it resides. Notions like “cell” or “hog” aren’t relevant at that far lower level. The laws of particle physics don’t respect such notions as “hog”, “cell”, “gene”, or “genetic code”, or even the notion of “amino acid”. The laws of particle physics involve only particles, and larger macroscopic boundaries drawn for the convenience of thinking beings are no more relevant to them than votingprecinct boundaries are to butterflies. Electrons, photons, neutrinos, and so forth zip across such artificial boundaries without the least compunction.

If you go the particles route, then you are committed to doing so whole hog, which unfortunately means going way beyond the hog. It entails taking into account all the particles in all the members of the hog’s family, all the particles in the barn it lives in, in the mud it wallows in, in the farmer who feeds it, in the atmosphere it breathes, in the raindrops that fall on it, in the cumulo-nimbus clouds from which those drops fall, in the thunderclaps that make the hog’s eardrums reverberate, in the whole of the earth, in the whole of the sun, in the cosmic background radiation pervading the entire universe and stretching back in time to the Big Bang, and on and on. This is far too large a task for finite folks like us, and so we have to settle for a compromise, which is to look at things at a less inclusive, less detailed level, but (fortunately for us) a more insight-providing level, namely the informational level.

At that level, biologists talk about and think about what genes stand for, rather than focusing on their traditional physico-chemical properties. And they implicitly accept the fact that this new, “leaner and meaner” way of talking suggests that genes, thanks to their informational qualities, have their own causal properties — or in other words, that certain extremely abstract large-scale events or states of affairs (for example, the high-level regularity that golden retrievers tend to be very gentle and friendly) can validly be attributed to meanings of molecules.

To people who deal directly in dogs and not in molecular biology, this kind of thing is taken for granted. Dog folks talk all the time about the temperamental and mental propensities of this or that breed, as if all this were somehow completely detached from the physics and chemistry of DNA (not to mention physical levels finer than that of DNA), and as if it resided purely at the abstract level of “character traits of dog breeds”. And the marvelous thing is that dog folks, no less than molecular biologists, can get along perfectly well thinking and talking this way. It actually works! Indeed, if they (or molecular biologists) tried to do it the pure-physics way or the pure-molecular-biology way, they would instantly get bogged down in the infinite detail of unimaginable numbers of interacting micro-entities constituting dogs and their genes (not to mention the rest of the universe).

The upshot of all this is that the most real way of talking about dogs or hogs involves, as Roger Sperry said, high-level entities pushing low-level entities around with impunity. Recall that the intangible, abstract quality of the primality of the integer 641 is what most truly topples hard, solid dominos located in the “prime stretch” of the chainium. This is nothing if not downward causality, and it leads us straight to the conclusion that the most efficient way to think about brains that have symbols — and for most purposes, the truest way — is to think that the microstuff inside them is pushed around by ideas and desires, rather than the reverse.