Tipler, who is not a crackpot, once published a book called The Physics of Immortality, purporting, on the basis of orthodox physics plus some plausible auxiliary assumptions, to establish the existence of an omnipotent, omniscient, omnipresent and altruistic “being” who will one day resurrect everyone who has ever lived to eternal life.

The first step toward that startling conclusion is the assumption that our descendants will not allow all life to come to an end. This in turn will require them to control the evolution of the Universe so that it doesn’t collapse in anything that human beings perceive as a finite amount of time; Tipler argues that they’ll quite plausibly have the technology to do that. But all this future tinkering with the shape of the Universe has consequences that (in a very rough sense) radiate backward and forward through time. From this and some highly technical but more-or-less standard physics, Tipler manages to conclude the existence of an Omega Point — a place where (again speaking roughly) all the information in the Universe is stored. Writing in 1994, Tipler never considered the possibility that the Omega Pont might be located in Mountain View, California. Instead, he stressed that in its omniscience, it’s something very like God.

Not only is the Omega Point omniscient; it’s also ominipotent in the sense that the information located there will allow our descendants to perform feats like resurrecting every one of us from the dead, something that Tipler says they’re sure to do because the cost will be essentially zero. The Omega Point turns out to be not only very like some generic God; it’s very like the Christian God. And the similarities don’t stop there (read Tipler for more).

Alas, Tipler observes in the book that the Omega Point theory also makes a rather specific prediction about the mass of the Higgs boson — it has to be somewhere around 220 gigaelectronvolts (GeV), give or take 10 percent or so. He offers this as a clear test of the theory. And the theory, it seems, is about to fail spectacularly. It looks like the Higgs boson is about to come in at somewhere around 125 GeV.

Tipler’s book had a huge intellectual influence on me, not because of its primary content but because of a tangential remark that triggered my first vision of the Universe as a purely mathematical object, a vision I later learned had been fleshed out by physicists such as Max Tegmark at MIT. Readers of The Big Questions will know that I find this vision extremely satisfying for a great variety of reasons. It provides plausible (to me) answers to a variety of questions that I’d always considered unanswerable, such as “Why is there a Universe in the first place?” Unfortunately, unlike the Omega Point Theory, this is not a vision that can be put to the experimental test.

Tipler’s theory, however, is designed to be put to the test, and if it fails that test (as it’s apparently about to), we should view that as a triumph. Science progresses through predictions so precise that we can know when they’re wrong. Now on to the next theory!

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Namely: The argument is made that zygotes/embryoes/fetuses, even at a very early stage, have the full moral status of human beings. Yet if that were true, surely we’d want to divert a substantial portion of the medical research budget away from relatively minor scourges like, say, cancer, to the spontaneous abortions that take the lives of something like 30% of these full-fledged humans. In a typical year, there are about 8 million cancer deaths worldwide; the number of early-stage spontaneous abortions must be at least twice that.

In Smith’s words:

very few of us are worried by the fact that a very high proportion of conceptions quite spontaneously abort. We don’t campaign for medical research to reduce that rate (nor do opponents of abortion campaign for all women to take drugs to suppress natural early abortion). Compare: we do think it is a matter for moral concern that there are high levels of infant mortality in some countries, and campaign and give money to help reduce that rate.

Smith is struck by the fact that this attitude is very widespread; I am more struck by the fact that it seems to be very widespread even among those who characterize themselves as pro-life.

Nothing here proves the pro-lifers wrong; maybe instead it means that the whole world has its medical research priorities upside down. But then what are we to make of the fact that even ardent pro-lifers seem, for the most part, to acquiesce in those priorities?

I’m sure that with enough contortions, one could reconcile a near-indifference to the problem of spontaneous early abortion with a sense of moral outrage against intentional abortion. But even then, I suspect one might be forced to abandon one’s insistence that a three-day-old zygote has the same moral status as, say, a ten-year-old child.

Edited to add: Some commenters have tried to address the question “Why would you care more about induced abortion than spontaneous abortion?”; their answers tend to reference a distinction between natural versus intentional acts. But that’s not the interesting question. The interesting question is “Why would you care more about cancer than spontaneous abortion?” Cancer and spontaneous abortion are, after all, equally natural.

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[Russell] was in the top floor of the University Library, about A.D. 2100. A library assistant was going round the shelves carrying an enormous bucket, taking down books, glancing at them, restoring them to the shelves or dumping them into the bucket. At last he came to three large volumes which Russell could recognize as the last surviving copy of Principia Mathematica. He took down one of the volumes, turned over a few pages, seemed puzzled for a moment by the curious symbolism, closed the volume, balanced it in his hand and hesitated….

Principia Mathematica, to which Russell had devoted ten years of his life, was his (and co-author Alfred North Whitehead’s) audacious and ultimately futile attempt to reduce all of mathematics to pure logic. It is a failure that enabled some of the great successes of 20th century mathematics. And — the first volume having been published in December, 1910 — this is its 100th birthday.

Having determined to write the Principia ten years earlier in 1900, Russell was at first stymied by his discovery of the famous paradox that now bears his name: Consider the set of all those sets that don’t contain themselves. Call this set R. Does R contain itself? If so, it belongs to the set of all sets that don’t contain themselves, and therefore does not contain itself. Does it fail to contain itself? If so, it fails to belong to the set of all sets that don’t contain themselves, and therefore contains itself. Either way, something’s screwy.

The Russell Paradox suggested that set theory was too shaky a foundation on which to build the edifice of mathematics, and Russell realized that the first thing he needed to do was shore up that foundation — a task that led to months, and then years, of staring at blank pages with no idea of how to proceed:

At first I supposed that I should be able to overcome the contradiction quite easily, and that probably there was some trivial error in the reasoning. Gradually, however, it became clear that this was not the case…It seemed unworthy of a grown man to spend his time on such trivialities, but what was I to do?

Russell’s ultimate solution was his Theory of Types, which (some would say artificially) limits the ways in which you’re allowed to define a set — and in particular prohibits you from defining the set R in the first place. It was clunky and ad hoc and many found it profoundly unsatisfactory, but it worked.

There were other problems, too. Russell wanted to derive all of mathematics from pure logic, but there was one mathematical fact that defied his every effort — namely the fact that there are infinitely many natural numbers. Surely this is a mathematical fact, but does it follow from purely logical considerations — or is it just something that happens to be true without being logically necessary? In the end, Russell punted, taking the infinitude of the natural numbers as an (essentially non-logical) axiom. Here was the first hint that maybe mathematics is more than just logic after all.

Once he had solved these problems — or, on a less charitable interpretation, decided how he was going to weasel his way around them — Russell spent the years from 1907 through 1910 working on the manuscript, ten to twelve hours a day, eight months a year. He worked, of course, with the technology of the time, with all of its attendant dangers and inconveniences:

The manuscript became more and more vast, and every time that I went out for a walk I used to be afraid that the house would catch fire and the manuscript get burnt up. It was not, of course, the sort of manuscript that could be typed, or even copied.

When we finally took it to the University Press, it was so large that we had to hire an old four-wheeler for the purpose. Even then our difficulties were not at an end.

The University Press estimated that there would be a loss of £600 on the book, and while the syndics were willing to bear a loss of £300, they did not feel that they could go above this figure. The Royal Society very generously contributed £200, and the remaining £100 we had to find ourselves. We thus earned minus £50 each for ten years’ work. This beats the record of Paradise Lost.

The first edition of Volume I ran to 750 copies. Volumes II and III, published in 1912 and 1913, ran to 500 copies each.

Aside from dissatisfaction with the Theory of Types and the Axiom of Infinity, there were a couple of other nagging questions left unsettled, though. First: Could all of mathematics be derived from Russell and Whitehead’s logical system? Surely some of it could (though not always easily — R and W notoriously required hundreds of pages to reach the conclusion that 1+1=2) — but could all? And second: Could the Russell/Whitehead system be proven to be free of logical contradictions? The Russell Paradox had been excised by the Theory of Types, but could one exclude the possibility of other paradoxes lurking in the background?

Russell was surely hopeful on both counts. Kurt Godel, the logician of the millennium and the man who would dash those hopes, was four years old in 1910.

I venture to guess that nobody has ever read the three large volumes of Principia Mathematica, a typical page of which looks about like this:

But, for having set the agenda for (at least) a century of research into the foundations of mathematics, I am sure it will still be celebrated in the A.D. 2100 of Lord Russell’s nightmare.

[RELATED POST: Principia Mathematica: The Comic Book"]

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Today’s question is: Do numbers exist? The answer is: Of course, and I don’t believe there’s much in the way of serious doubt about this. You were familiar with numbers when you were five years old, and you’ve been discovering their properties ever since. Extreme skepticism on this point is almost unheard of among mathematicians or philosophers, though it seems to be fairly common among denizens of the Internet who have gotten it into their head that extreme skepticism makes them look sophisticated.

Let me be clear that I am not (yet) asking in what sense the natural numbers exist — whether they have existed since the beginning of time, or whether they exist outside of time, or whether they exist only in our minds. Those are questions that reasonable people disagree about (and that other reasonable people find more or less meaningless.) We can — and will — come back to those questions in future posts. For now, the only question: Do the natural numbers exist? And the answer is yes. Or better yet — if you believe the answer is no, then there’s obviously no point in thinking about them, so why are you reading this post?

“Existence” here is used in the ordinary everyday sense of the word, according to which rocks and trees exist, you and I exist, your hopes and dreams exist, and the idea of a unicorn exists. Unicorns themselves do not exist and therefore it makes no sense to study their properties. (Though you can have fun inventing some properties for them.) By contrast, it makes perfect sense for geologists to study the properties of rocks, for botanists to study the properties of trees, for folklorists to study the properties of the idea of a unicorn, and for mathematicians to study the properties of the natural numbers.

An extreme skeptic might deny the existence of rocks. The only possible answers are: a) It’s hard to believe you’re serious, since you’ve been encountering rocks — just like you’ve been encountering numbers — your entire life. b) If you really are serious, I suppose your best strategy is to stop thinking about rocks, and leave them to those of us who find geology interesting. And c) Do not fool yourself into believing that your position is anywhere close to any mainstream school of thought.

Another extreme skeptic might deny the existence of numbers. I’ll leave it to my readers to replace rocks with numbers in the above retorts.

What else might one say to an extreme skeptic? Answer: One might attempt to acquaint him with Godel’s Completeness Theorem. (This is not the same as the far more famous Godel’s Incompleteness Theorem.) Here is (part of) what the Completeness Theorem says: First, without making any assumptions about existence, write down a list of axioms for the natural numbers. For example, write down the Peano Axioms. Then the Completeness Theorem tells you that as long as those axioms are consistent, there must be some mathematical structure that obeys those axioms. (Note that “be” is a synonym for “exist”.) The smallest of those structures (known as “models”) is our good old friend the natural numbers.

In other words, Godel’s Theorem tells you that if the Peano axioms are consistent, then the natural numbers must exist. (Don’t confuse the map with the territory! “Consistency” applies to the axioms; “existence” applies to the natural numbers themselves.)

On the other hand, we can also argue in the opposite direction: If the natural numbers exist, then the Peano axioms, being true statements about existing objects, must be consistent. An accurate map of an existing territory cannot contradict itself.

So — We know that the natural numbers exist because we know the Peano axioms are consistent. And we know that the Peano axioms are consistent because we know that the natural numbers exist. Does that sound circular? It’s not. Here’s the point: We have extremely good reasons for believing in the existence of the natural numbers (beginning with intuition, lifelong familiarity, and the fact that we seem to be able to discover their properties). We have (partly) separate extremely good reasons for believing in the consistency of the Peano axioms (beginning with intuition and the fact that they’ve never yet led us to a contradiction). The fact that our two beliefs reinforce each other — that if either is true, then so must be the other — should build up our confidence that the whole picture hangs together.

Now let’s get back to our extreme skeptic. He denies the existence of the natural numbers. We respond that Godel’s Completeness Theorem proves the existence of the natural numbers, as a consequence of the consistency of the Peano axioms. He now has only two recourses (other than to concede defeat). One is to deny the consistency of the Peano axioms, and the other is to deny the accuracy of Godel’s Completeness Theorem. Let’s see how those strategies are likely to work out for him.

Should he doubt the consistency of the axioms? The Peano Axioms lay out the rules of arithmetic that you’ve used your whole life; they say things like “Every number has exactly one immediate successor” and “x + (y+1) = (x+y) + 1”. People (and to some extent animals) have been applying these axioms, explicitly or implicitly, since long before the dawn of history and no contradiction has ever arisen; moreover, for what it’s worth, the consistency of these simple axioms is instantly clear to most people’s intuitions. If we were to jettison our belief that these axioms are consistent, then we’d pretty much have to give up all quantitative reasoning.

Well, then, should our skeptic doubt Godel’s Completeness Theorem? The theorem is proved using elementary notions about sets — the idea that it’s possible to talk about sets of things and about membership in a set, that it’s possible to form the union of two sets, and so on. This has nothing to do with the more esoteric subject of “axiomatic set theory”; instead, it uses only the most fundamental notions associated with forming collections of things. (These notions, in fact, are prerequisite for axiomatic set theory and therefore cannot depend on it.) Once again, if you were to abandon this sort of reasoning, you’d pretty much have to abandon reasoning altogether.

For anyone who accepts the simplest sorts of combinatorial reasoning, there is no longer an out. The natural numbers are real. Again, this says nothing about where they came from — be it Plato’s heaven, the minds of humans or the mind of God. We’ll get back to that in the next installment of this occasional series.

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Like everyone else I know, I am of course a longtime fan of the webcomic XKCD. But somehow it took me until last week to become aware of the frequently brilliant competitor Luke Surl, of which the above is a delectable example. What else out there am I missing?

Hat tip to Harry Brighouse of Crooked Timber.

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I believe the natural numbers exist, quite independently of whether anyone’s around to think of them. Here’s why: First, we perceive them directly. Second, we know non-trivial facts about them. Third, they can explain the Universe. In more detail:

1. We perceive them directly. I believe that flowers, rainbows and headaches exist because I perceive them directly. I believe that the natural numbers exist for exactly the same reason. Almost anyone who has ever thought hard about higher arithmetic will tell you the same thing. Yesterday I quoted the Fields Medalist Alain Connes saying that when mathematicians contemplate arithmetic, “we run up against a reality every bit as uncontestable as physical reality”. Today I’ll quote Kurt Godel, the greatest logician of all time:

Despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves on us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e. in mathematical intuition, than in sense perception.

2. We know non-trivial facts about them. In 1637, Pierre de Fermat wondered whether you can find four positive numbers x, y, z and n, with n at least 3, that satisfy the equation

That’s certainly a meaningful statment: It means that no matter what four numbers you write down, we can predict with certainty that as long as they’re positive, and as long as n is at least 3, the equation I’ve just written down will never be true. But unlike the axioms that Godel was referring to, it’s hardly self-evident and it does not force itself on us as being true; that’s part of why it took 350 years to prove.

So how do we know that Fermat’s Last Theorem (i.e. the statement that the equation has no solutions) is true? The answer is not that it follows step by step from some list of self-evident axioms about the natural numbers. As far as I am aware, nobody has the foggiest idea whether Fermat’s Last Theorem follows from any set of reasonably self-evident axioms about arithmetic, such as the Peano axioms that I wrote about here. Instead, we know that Fermat’s Last Theorem is true via informal (but, to almost all mathematicians, completely convincing) arguments that are not about manipulating axioms but instead are about the properties of numbers themselves.

(It’s a virtual certainty that these informal arguments could be formalized in some language, but—again as far as I know—it’s quite unknown whether they could be formalized in the usual language of arithmetic.)

So I agree with Godel that the self-evident nature of the axioms is evidence that the natural numbers are real, but I also believe, quite separately, that the non-self-evident nature of statements like Fermat’s Last Theorem is additional evidence. Here we have a statement that is true, but it’s truth is not derived from axioms about arithmetic. Instead it’s true because it’s a correct statement about something. That “something” is the system of natural numbers.

3. They explain the Universe. This argument is surely more speculative than the others, but I cannot imagine any way to explain the existence of the Universe without the prior existence of the natural numbers. (This is more or less the same reason some people give for believing in God.) It seems to me that the most compelling question in philosophy is why anything exists at all. Any satisfactory answer has to start with something that must exist. The natural numbers fill that role admirably. In The Big Questions, I’ve sketched a story about how, once you’ve got some mathematical objects, the Universe can sort of bootstrap itself in existence from there; this is similar in spirit to the story advanced by the noted cosmologist Max Tegmark in his essay on The Mathematical Universe (cited in yesterday’s comments by Al V.)

Now admittedly, my inability to find any alternative explanation for the Universe does not prove that this explanation is correct. For that matter, my hunger for an explanation doesn’t mean there has to be one. Maybe the Universe just is. But when you’re facing a huge riddle and you can only think of one possible solution, you’ve got to at least contemplate the possibility that you’re on to something.

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I’ve just finished reading an excellent book by Mario Livio which is entirely about the invention/discovery question, though he’s chosen the (somewhat unfortunate) title Is God a Mathematician? Much of the book is a lively romp through mathematical history, with a well chosen mix of biography and exposition. Although he parts company with them in the last chapter, Livio gives a more than fair hearing to the many great mathematicians who have insisted that they are discoverers, from Pythagoras through Galileo, G.H. Hardy, Kurt Godel, and the contemporary Fields Medalist Alain Connes (among others). Here, for example is Connes:

Take prime numbers, for example, which as far as I’m concerned, constitute a more stable reality than the physical reality that surrounds us. The working mathematician can be likened to an explorer who sets out to discover the world…We run up against a reality every bit as uncontestable as physical reality.

Readers of The Big Questions will know that I am entirely in Connes’s camp on this issue, for reason I’ll blog about later in the week. And as I’ve said, it seems that most mathematicians sit in this camp. But there are notable dissenters, including the great Sir Michael Atiyah, another Fields Medalist who I might well have included in my gallery of heroes. Despite my great admiration for Atiyah, I believe he’s wrong on this issue. But more fundamentally, I believe his primary argument proves exactly the opposite of what he thinks it does. Here is that argument (slightly condensed):

Any mathematician must sympathize with Connes. We all feel that the integers really exist in some abstract sense and the Platonic view is extremely seductive. But can we really defend it? It might seem that counting is really a primordial notion. But let us imagine that intelligence had resided, not in mankind, but in some vast solitary and isolated jellyfish, buried deep in the Pacific Ocean. It would have no experience of individual objects, only of the surrounding water. Motion, temperature and pressure would provide its basic sensory data. In such a pure continuum the discrete would not arise and there would be nothing to count.

Here Atiyah has envisioned a world where the natural numbers get neither invented nor discovered. I’m not sure why that’s supposed to prove they don’t exist. On the contrary, it seems to me that quite unbeknownst to Atiyah’s Jellyfish, the earth would still have exactly one moon and exactly two magnetic poles, and two would still be twice one. Two would still be a prime number, and 1729 would still be the smallest number that is the sum of two cubes in two different ways.

Indeed, all these things were true back in the days when the world was populated by creatures who were unaware of them. It doesn’t matter for the argument whether those creatures are highly intelligent in other ways. In that sense, Atiyah’s Jellyfish is more like a Red Herring.

Or to put this another way: Atiyah says that an intelligent creature might be unaware that one plus one makes two. Sure—so might any creature. To me, this indicates that 1+1=2 is not an invention; it’s simply a truth. Can anyone explain why Atiyah thinks his story proves otherwise?

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The Richard Dawkins crowd tell you that complexity must evolve from simplicity.

I claim they’re both wrong, because the natural numbers, together with the operations of arithmetic, are fantastically complex, but were neither created nor evolved.

I’ve made this argument multiple times, in The Big Questions, on this blog, and elsewhere. Today, I aim to explain a little more deeply why I say that the natural numbers are fantastically complex.

Here’s one way to think about simplicity versus complexity: Simple things have short descriptions; complex things have only long descriptions. A string of a million zeros is very simple because I can describe it in six words: “A string of a million zeros”. A string of a million random numbers is complex, because it takes a long time to describe all of the content.

Now what about the system of natural numbers? To first appearances, there’s a very simple description: Start with 0, then add 1, then add 1 again, keep doing this forever, and those are the natural numbers. Unfortunately, “keep doing this forever” is a little vague, and the complexity comes in when you try to make that precise.

So if you were setting out to give a complete description of the natural numbers, where would you start? Probably here:

• We have a number called zero, and then every number has a successor.

But that description fits a lot of things besides the natural numbers; it also fits, for example, the integers (the integers, unlike the natural numbers, include negatives). Here’s an attempt to fix that:

• We have a number called zero, and then every number has a successor, and zero is not the successor of any number.

Better, but still no good; this fails to rule out a system where 1 follows 0, 2 follows 1, 3 follows 2, and 1 follows 3, like this:

To fix that, we have to add a clause specifying that no two numbers (such as 0 and 3) have the same successor. But even now, we’ve only just begun.

We still haven’t ruled out the possibility of infinite gaps between numbers. For all we can tell from our description so far, the number system might look like this:

with infinitely many numbers in between 3 and that very large number N. How can we rule that out?

This one is not so easy. We’d like to say that all gaps between numbers are finite. But how do we define “finite”? Usually we say a number is finite if it’s part of the set of natural numbers. Or to put this another way: We’d like to say that no matter where you start (say at N), you can’t count backward forever; eventually you’ve got to hit a stopping point. But what does it mean to count backwards forever? It means counting back more than a natural number of steps. There’s that circularity again.

What we really really need, it turns out, is to add a clause like this to our description:

• Every non-empty subset of the natural numbers has a smallest element.

This will solve our problem, because it implies, for example, that the set of numbers you can reach by counting backwards from N has a smallest element—eliminating the possibility of that infinite gap.

But this assumption, stated in this way, opens a can of worms that almost nobody wants to open. Here’s why: For the first time, we’ve been forced to talk about sets of natural numbers, as opposed to natural numbers themselves—and even to talk about all those sets at once. In technical jargon, we’ve left the world of first-order logic and entered the world of second-order logic. But that’s a very strange world indeed. In ordinary (first-order) logic, we have a small number of rules of inference that allow us to proceed, for example, from “Socrates is a man” and “All men are mortal” to “Socrates is mortal”. But in second-order logic, not only are the rules of inference not finite; they cannot even be printed out (even in an infinite amount of time) by any computer. That’s why the great logician Willard van Ormand Quine insisted that second order logic is not logic, and why mathematicians usually prefer to avoid it.

All is not lost, though. Instead of adding one second-order axiom, we can add infinitely many first-order axioms, viz:

• The set of odd numbers has a smallest element.
• The set of numbers greater than 7 has a smallest element.
• The set of numbers that can be reached by counting backward from 100 has a smallest element.

And so on.

Okay, our description of the natural numbers just got infinitely long, but at least it’s infinitely long in a simple sort of way. We’ve added an infinite number of axioms, but they all fit the same simple pattern—a pattern that you could easily train your computer to recognize.

Unfortunately, though, we still have a long way to go to get to a full description of arithmetic. First, we have to add rules for addition and multiplication. (If we don’t do this, then we won’t be able to talk about interesting subjects like prime numbers.) Now we’ll want to add even more axioms. But now we come up against the content of Godel’s incompleteness theorem: NO description suffices. No matter what axioms you add, your description will always fail to distinguish the natural numbers from any of an infinite number of other structures. (Those other structures are usually called “non-standard models of arithmetic”).

When I say that “NO description suffices”, you might reasonably ask what counts as a description. Here’s what counts: A description is some (possibly infinite) list of axioms that some computer program is capable of recognizing. So if, for example, you try to describe arithmetic by listing every true statement about it, I will cry foul, because no computer program is capable of recognizing every true statement about arithmetic. (This is not just an observation about the state of the art in computer programming. It is a theorem about all possible computer programs.)

That’s the sense in which arithmetic is fantastically complex. Not only do the natural numbers have no finite description; they have no description that is recognizable by any computer. If “simple” means “capable of a short description”, then the natural numbers are about as far from simple as you can get. Not only do they have no short description, they don’t even have an infinite description.

Other mathematical structures are simpler. Euclidean geometry, for example, can be fully described by a first-order theory, and there is a computer program that can distinguish true from false statements in that theory.

Likewise for the (first-order) theory of the real numbers. There are axioms for the real numbers that suffice to prove all true first-order statements about the real numbers, and there is a computer program that can distinguish the true statements from the false. In that sense, the real numbers are far simpler than the natural numbers. (There are still non-standard models of the real numbers, but through a first-order lens, they are indistinguishable from the real thing.)

(You might be tempted to think that because the natural numbers sit inside the real numbers, they must be simpler. But of course any sequence of arbitrary complexity sits inside the very simple sequence 010101010101….., if you can pick and choose what to keep and what to throw away. Complexity can reside quite comfortably inside simplicity.)

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Although Belief in God in an Age of Science is a very short book, it is too long to review in a single blog post. Fortunately, though, much of the non-lunatic content is concentrated in roughly the first ten pages, so I’ll comment here only on those.

Polkinghorne begins in awe. He is awestruck by the extent to which our Universe seems to have been fine-tuned to support life; this is the subject matter of the much-discussed anthropic cosmological principle. To take just one example (which Polkinghorne does not mention): The very existence of elements other than hydrogen and helium depends on the fact that it’s possible, in the interior of a star, to smoosh three helum atoms together and make a carbon atom; everything else is built from there. But it’s not enough to make that carbon atom; you’ve also got to make it stick together long enough for a series of other complicated reactions to occur. Ordinarily, that doesn’t happen, but now and then it does. And the reason it happens even occasionally is that the carbon atom happens to have an energy level of exactly 7.82 million electron volts. In fact, this energy level was predicted (by Fred Hoyle and Edwin Salpeter) before it was observed, precisely on the basis that without this energy level, there could be no stable carbon, no higher elements, and no you or me.

That energy level is only one of many (apparent) cosmic coincidences that make us possible; change any of the fundamental physical constants (like, say, the strength of gravity) by a little bit in either direction, and the Universe would, as far as we can tell, become completely inhospitable to life. So one does tend to feel that there’s something here that needs explaining.

Some have attempted to dismiss the issue by turning the direction of causality on its head: Here we are, so of course the laws of physics must allow for our existence. Case closed. Douglas Adams, for example, offers this brief and brilliant parable:

Imagine a puddle waking up one morning and thinking, ‘This is an interesting world I find myself in, an interesting hole I find myself in, fits me rather neatly, doesn’t it? In fact it fits me staggeringly well, must have been made to have me in it!’

But I have some sympathy for Professor Polkinghorne’s refusal to accept this dismissal. Instead, he takes his stand with the philosopher John Leslie:

The fine tuning is evidence, genuine evidence, of the following fact: that God is real, and/or there are many and varied universes.

I agree with that (with the proviso that evidence is not proof). I agree with it to exactly the same extent that I agree with this:

The fine tuning is evidence, genuine evidence of the following fact: Either invisible pink bunny rabbits, created at the time of the Big Bang, fine tuned the physical constants in order to make the Universe hospitable to lettuce, and/or there are many and varied universes.

Or, more succinctly:

The fine tuning is evidence, genuine evidence of the following fact: There are many and varied universes.

Polkinghorne wants to reject this second horn of Leslie’s dilemma, but he manages to do so, I think, only by taking too crabbed a view of what those many and varied Universes might be. First, we have the parallel worlds promised to us by the many-worlds interpretation of quantum theory; Polkinghorne is absolutely right to say these can’t be the worlds we’re looking for, because they all obey the same basic laws of nature. Higher on what Polkinghorne calls the “scale of bold speculation” we have suggestions from quantum cosmology that Universes are bubbling up all the time as quantum fluctuations in some universal substrate. But again, Polkinghorne is right to say that this only pushes the mystery back a bit—why do those fluctuations obey laws that have even a chance of producing a habitable Universe? Where do the laws come from?

This is the point where Polkinghorne gives up and falls back on God. But it seems to me that he has given up just one level of abstraction too soon. A Universe is fundamentally a mathematical object—it’s an abstract pattern that might or might not contain subpatterns that might or might not be sufficiently complex in just the right away to achieve an awareness of their surroundings, and might or might perceive those surroundings as physical objects. And of course there are many Universes, because there are many mathematical patterns, including, as just one of a dazzling infinity of examples, the Universe in which we live.

That, in any event, is the best explanation I can come up with, and it’s an explanation that feels completely right to me (which admittedly proves nothing). In The Big Questions, I’ve elaborated on what I mean by all this, how it can be true, and why it is entirely consistent with mainstream physics and the stated views of many mainstream physicists.

Now, Professor Polkinghorne might or might not buy this vision, but my point is that he never even contemplates it. He makes the leap to theism by considering and rejecting all of the weakest alternatives, but ignoring the only one that makes sense. This oversight is all the more remarkable because Polkinghorne devotes his closing pages to a rousing defense of the independent reality of mathematical objects, in clear and convincing language that had me wishing I’d written these pages myself.

The rest of the book is far worse. I might come back to that in a later post.

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I saw with greater clarity than ever before in my life that when I say “Thank goodness!” this is not merely a euphemism for “Thank God!” (We atheists don’t believe that there is any God to thank.) I really do mean thank goodness! There is a lot of goodness in this world, and more goodness every day, and this fantastic human-made fabric of excellence is genuinely responsible for the fact that I am alive today. It is a worthy recipient of the gratitude I feel today, and I want to celebrate that fact here and now.

To whom, then, do I owe a debt of gratitude? To the cardiologist who has kept me alive and ticking for years, and who swiftly and confidently rejected the original diagnosis of nothing worse than pneumonia. To the surgeons, neurologists, anesthesiologists, and the perfusionist, who kept my systems going for many hours under daunting circumstances. To the dozen or so physician assistants, and to nurses and physical therapists and x-ray technicians and a small army of phlebotomists so deft that you hardly know they are drawing your blood, and the people who brought the meals, kept my room clean, did the mountains of laundry generated by such a messy case, wheel-chaired me to x-ray, and so forth. These people came from Uganda, Kenya, Liberia, Haiti, the Philippines, Croatia, Russia, China, Korea, India—and the United States, of course—and I have never seen more impressive mutual respect, as they helped each other out and checked each other’s work. But for all their teamwork, this local gang could not have done their jobs without the huge background of contributions from others. I remember with gratitude my late friend and Tufts colleague, physicist Allan Cormack, who shared the Nobel Prize for his invention of the c-t scanner. Allan—you have posthumously saved yet another life, but who’s counting? The world is better for the work you did. Thank goodness. Then there is the whole system of medicine, both the science and the technology, without which the best-intentioned efforts of individuals would be roughly useless. So I am grateful to the editorial boards and referees, past and present, of Science, Nature, Journal of the American Medical Association, Lancet, and all the other institutions of science and medicine that keep churning out improvements, detecting and correcting flaws.

Indeed. And because the supply of thankfulness is not fixed, it will not depreciate the value of Professor Dennett’s sentiment to add a word of thanks not just for goodness but for greed—the greed that inspired generations of inventors and investors, laborers and capitalists, doctors and nurses, technicians and scientists to envision and perfect such a thing as an artificial aorta, to educate themselves in the healing professions, and to show up for work every day. For the most part, they did it to make a buck.

We can be thankful too for the system that channels all that potentially destructive greed into life-sustaining brilliance. But we might temper our gratitude just a bit with a moment of wistful regret for the lives lost because of unnecessary imperfections in that system. As a society, we spend far too little on basic research in health care, largely because breakthroughs are under-rewarded. For one thing, our reliance on third-party payers (with the attendant loss of control over our own health care choices) makes us willing to pay handsomely even for relatively ineffective treatments, which diminishes the incentive for innovators to make treatments more effective. (This compelling observation comes from a paper by the economists Kevin Murphy and Robert Topel; I’ll be blogging on their work in more detail in the near future.)

For the sake of future Daniel Dennetts, I hope our legislators have the goodness and wisdom to devise a health care reform package that strengthens the incentive structure instead of weakening it still further. When they fail, as they probably will, there will be plenty of time for outrage. Meanwhile, things could be far far worse, and there’s much to be grateful for on this Thanksgiving day.

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