Steven Landsburg | The Big Questions: Tackling the Problems of Philosophy with Ideas from Mathematics, Economics, and Physics » Ontology

The Mathematical Universe

Steve Landsburg — Thu, 30 Sep 2010 06:01:11 +0000

Some quick words about the mathematical universe, which is the theme of the first chapter of The Big Questions:

1. A “mathematical object” consists of abstract entities (that is, “things” with no intrinsic properties) together with some relations among them. For example, the euclidean plane that you studied in high school geometry consists of points, together with certain relations among them (such as “points A, B and C are collinear”). Mathematical objects can be very complicated. Mathematical objects can have “substructures”, which is a fancy name for “parts”. A line in the plane, for example, is a substructure of the plane.

2. Every modern theory of physics says that our universe is a mathematical object, and that we are substructures of that object. Theories differ only with regard to which mathematical object we happen to be a part of. Particles, forces and energy are not just described by equations; they are the equations (together with abstract, purely mathematical relations among those equations).

3. If you want to think of the universe as something other than a mathematical object — say, something that is controlled by mathematics, or described by mathematics, as opposed to made of mathematics — then you’re up against the fact that nobody has the slightest idea how to construct a useful physical theory along those lines. It’s not just that science rejects all the alternatives; it’s that no scientist (as far as I know) has even been able to imagine a useful alternative. (Perhaps you can find solace in religion.)

4. So — at least if you accept modern physics — at least one mathematical object (namely the one we live in, and are part of) exists physically. This naturally raises the question of which other mathematical objects exist physically.

5. Before you can answer that, you’ve got to ask what it could even mean for one mathematical object to exist physically while another doesn’t. Our universe is a certain mathematical object. Surely there are other mathematical objects that differ from it only in detail. They contain substructures as complicated as we are, in more or less the same ways, and therefore as well equipped as we are to perceive their surroundings as physically real. If you want to claim that our universe is in fact “real” and theirs is not, then you’ve got to explain what that reality consists of.

6. Whatever that reality consists of, it must be some purely mathematical property, because our universe is a purely mathematical object, so that all its properties are purely mathematical.

7. I want to stress that: If, as physics tells us, the universe is “made of math”, then its physical reality is part of that math. We should therefore expect similar mathematical structures to share the property of physical reality.

8. Which structures count as “sufficiently similar” to ours that we should expect them to be physically real? That’s up to us, really, since we’re in the process of defining the term “physical reality”. My own instinct is to call a mathematical object “physically real” if it contains what Max Tegmark calls “self-aware substructures”, i.e. structures that are complicated in a way that makes them aware of their own existence. Tegmark’s own preference is to say that there’s simply no point in discriminating among mathematical objects, and we should simply call all of them physically real.

9. But why bother arguing? Why struggle to come up with a definition for some vaguely imagined notion of “physical existence”? Ockham’s Razor cautions us not to burden ourselves with unnecessary metaphysical baggage. There are mathematical objects. Some have self-aware substructures. Our universe is one of those. The business of science is to figure out which one. What more needs to be said?

10. Finally: I never cease to be amazed by people who uncritically accept the reality of rocks, geese and butterflies but want to deny the reality of mathematical objects. Science tells us that rocks, geese and butterflies are mathematical objects. What else could they be?

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The Grand Design

Steve Landsburg — Mon, 27 Sep 2010 06:01:30 +0000

To understand the universe at the deepest level, we need to know not only how the universe behaves, but why.

Why is there something rather than nothing?

Why do we exist?

Why this particular set of laws and not some other?

So say Stephen Hawking and Leonard Mlodinow in their book The Grand Design, and so say I.

The Big Big Question is the first one: Why is there something rather than nothing? Hawking’s answer: The laws of physics — and especially the form of the law of gravity — allow for the spontaneous creation of universes out of nothing at all. We live in one of those spontaneously created universes. But this, of course, only serves to raise a new Big Big Question, namely: Why are the laws of physics as they are? Hawking’s answer: The laws of physics must be consistent and must predict finite results for the quantities we can measure. It turns out that those criteria pretty much dictate the form of the laws of physics.

So unless I’ve misunderstood him, here is Hawking’s position: In order for us to be able to measure the things that we measure, the laws of physics must have a certain form, and in order for them to have that form, universes must be able to arise from nothing. Therefore our universe was able to arise from nothing. But this does not seem to answer the question of why things couldn’t have been very different. Why couldn’t there have been no us, no measurements, no laws of physics and no anything?

I know of only one satsfying (to me) answer to this question, and Hawking comes tantalizingly close to it without ever quite going there. He spends a lot of pages reviewing current physical theories but never mentions the one glaring feature they all share: Every modern physical theory, taken literally, predicts that our universe is a mathematical object. For example, the simplest version of special relativity posits that we live in a four-dimensional geometric object called “spacetime”. More sophisticated theories posit that spacetime is part of some larger geometric object whose properties we perceive as “forces” or “particles”. According to modern physics, everything is made of math.

Now you might say that physical theories aren’t meant to be taken that literally; that instead they describe mathematical objects with properties that are analogous to the properties of the physical universe. But it seems to me that if, like Hawking, you trust in theories to explain the mystery of creation itself, then you ought, at least provisionally, to take those theories literally. Otherwise, what you’ve got is not a theory. It’s a theory plus a bunch of ad hoc and arbitrary choices about which parts of that theory you choose to believe.

Once you believe the universe is a mathematical object, its existence ceases to be a mystery—at least if you believe, along with most mathematicians, that mathematical objects can’t help but exist. Hawking embraces M-theory, which tells us that the universe is a particular 11-dimensional object (with a whole bunch of additional geometric curlicues that appear to our senses as everything from stars to bacteria. M-theory also says there are a whole bunch of other 11-dimensional universes, all of which were spontaneously created, and we just happen to live in this one.

What I’m suggesting is that the universes of M-theory are only a tiny fraction of the universes out there, because anything that exists mathematically is a universe, though most of them (like most of the universes of M-theory) are far too simple to contain anything like sentience. This is essentially the view of cosmologists like Max Tegmark of MIT.

Hawking is 90% of the way there. The many universes of M-theory are mathematical objects, and all are pieces of a bigger mathematical object called the multiverse. “Spontaneous creation” means that the multiverse is structured in such a way that it must contain these universes. But why is there a multiverse and why is it structured in that way? That’s the part Hawking seems not to address. Proposed answer: The multiverse itself is only one of many multiverses. They all exist for the same reason the natural numbers exist: The laws of mathematics require it. And unlike the laws of physics, which differ from multiverse to multiverse, the laws of mathematics, which live outside any universe, could not have been otherwise.

(For more on this subject, read Chapter 1 of The Big Questions !)

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Basic Arithmetic: On What There Is

Steve Landsburg — Wed, 01 Sep 2010 06:01:07 +0000

This is an extremely elementary post about numbers. (”Numbers” means the natural numbers 0,1,2 and so forth.) It is a sort of sequel to my three recent posts on basic arithmetic, which are here, here and here. But it can be read separately from those posts.

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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Just the Facts

Steve Landsburg — Wed, 28 Apr 2010 06:01:47 +0000

During our brief intermission last week, commenters chose to revisit the question of whether arithmetic is invented or discovered—a topic we’d discussed here and here. This reminded me that I’ve been meaning to highlight an elementary error that comes up a lot in this kind of discussion.

It is frequently asserted that the facts of arithmetic are either “tautologous” or “true by definition” or “logical consequences of the axioms”. Those are three different assertions, and all of them are false. (This is not a controversial statement.)

The arguments made to support these assertions are not subtly flawed; they are overtly ludicrous. Almost always, they consist of “proof by example”, as in “1+1=2 is true by definition; therefore all the facts of arithmetic are true by definition”. Of course one expects to stumble across this sort of “reasoning” on the Internet, but it’s always jarring to see it coming from people who profess an interest in mathematical logic. (I will refrain from naming the worst offenders.)

So let’s consider a few facts of arithmetic:

Every number is either odd or even. This is a tautology. It does not follow that every fact of arithmetic is a tautology.
1+1=2. This (depending on what you take as your starting point) is true by definition. It does not follow that every fact of arithmetic is true by definition.
Every number is a product of prime numbers. This is neither a tautology nor a definition. It does, however, follow from the standard Peano axioms. It does not follow that every fact of arithmetic follows from the axioms.
There is no solution to the equation (x+1)ⁿ⁺³+(y+1)ⁿ⁺³=(z+1)ⁿ⁺³. (This is the famous Fermat’s Last Theorem.) This is not a tautology, it is not a defintion, and I have no idea whether it follows from the axioms. Neither, as far as I know, does anyone else. All we know for sure is that it’s true.
In Chapter 10 of The Big Questions, I give an example of a fact of arithmetic that is not a tautology, not a definition, and surely does not follow from the axioms. Here is a sketch of a proof that there must be some such facts.

It seems that people are often led astray by thinking that the “facts of arithmetic” consist solely of statements like “seven squared plus one equals five squared times two”. The more logically interesting statements are those that speak not about specific numbers, but about infinite collections of numbers. These are the statements that begin with phrases like “Every number…” or “There is a number such that…” or “There are only two numbers such that….”. When mathematicians speak of the “facts of arithmetic”, they means facts like these:

Every number is the sum of four squares.
Every prime number that leaves a remainder of 1 when divided by 4 is a sum of two squares.
No prime number that leaves a remainder of 3 when divided by 4 is a sum of two squares.
For every number n, there are infinitely many squares of the form 1+ny²
Between every number and its double, there is at least one prime.
8 and 9 are the only successive numbers that are both powers of primes.
For any two prime numbers p and q, the equations p=x²+yq and q=x²+yp are either both solvable or both unsolvable, unless p and q both leave remainders of 3 when divided by 4, in which case exactly one of them is solvable.
If you want to black out enough squares on a tic-tac-toe board to make winning impossible, then, as you pass to boards of higher and higher dimensions, the fraction of squares you must black out gets arbitrarily close to 100%. This is the density Hales Jewett theorem, and I have intentionally glossed over some technicalities in the statement.

Now to the point: First, if you want to tell a story about whether arithmetic is invented or discovered—in other words, if you want to tell a story about where arithmetic comes from—then your story has to account for where the facts of arithmetic come from. If your story says that the facts of arithmetic consist entirely of tautologies, definitions and logical consequences of standard axioms, then your story is wrong. Try again.

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Real Numbers

Steve Landsburg — Wed, 13 Jan 2010 07:01:09 +0000

Yesterday we started a conversation about whether mathematics is invented or discovered. Today I’ll give you my three best arguments for “discovered”. And to focus the discussion, I’ll talk not about mathematics generally but about the natural numbers (0,1,2, and so forth) in particular.

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

xⁿ+yⁿ=zⁿ
After 350 years, the question was settled by Gerhard Frey, Ken Ribet and Andrew Wiles in one of the most spectacular mathematical achievements of the twentieth century. The answer, as Fermat had believed, is no.

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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Jellyfish Math

Steve Landsburg — Tue, 12 Jan 2010 07:01:14 +0000

Is mathematics invented or discovered? In my experience, applied scientists often think of mathematics as a human invention, while actual mathematicians (with a few notable exceptions) feel sure that mathematics was always there to be discovered. (Of course, it’s sometimes hard to tell how much of this is genuine disagreement and how much is a language barrier.)

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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Non-Simple Arithmetic

Steve Landsburg — Thu, 17 Dec 2009 07:01:04 +0000

The Intelligent Design folk tell you that complexity requires a designer.

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.)

Did that help?

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Life, the Universes and Everything

Steve Landsburg — Wed, 02 Dec 2009 07:01:03 +0000

As I mentioned the other day, I’ve recently (at the direction of my old friend Deirdre McCloskey) been reading some of the work of John Polkinghorne, the physicist-turned-theologian who seems to write about a book a week attempting to reconcile his twin faiths in orthodox science and orthodox Christianity.

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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What Are You Surest Of?

Steve Landsburg — Wed, 04 Nov 2009 07:03:16 +0000

Among the things you’re sure of, which are you surest of? For Richard Dawkins, writing in the Wall Street Journal, it’s the theory of evolution:

We know, as certainly as we know anything in science, that [evolution] is the process that has generated life on our own planet.

Now, I would be thunderstruck if the theory of evolution turned out to be fundamentally wrong, but not nearly so thunderstruck as if arithmetic turned out to be inconsistent. In fact, I can think of quite a few things I’m more sure about than evolution. For example:

1. The consistency of arithmetic. (This amounts to saying that a single arithmetic problem can’t have two different correct answers.)

2. The existence of conscious beings other than myself.

3. The fact that the North won the American Civil War. (That is, historians are not universally mistaken about this. I am not interested in quibbling about what constitutes a “win”; I mean to assert that the North won in the everyday sense of the word, as reported in all the history texts.)

4. The consistency of higher mathematics. (The math geeks in the audience can take this to mean the consistency of Zermelo-Frankel set theory.)

5. The special theory of relativity. (The science geeks in the audience can take this to mean that the laws of physics are locally Lorentz invariant.)

6.The efficiency of the price system. (The econ geeks in the audience can interpret this as the truth and appropriate applicability of the first and second fundamental theorems of welfare economics.)

And then somewhere down the list—though still way above anything I significantly doubt—we have:

7. The theory of evolution. (That is, all—or nearly all—living things evolved from simpler things, largely through some process involving reproduction, mutation and selection.)

I’m not at all sure I’m right about this ordering, and I’d probably have chosen a different ordering five minutes ago or five minutes from now.

What would your ordering be?

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There He Goes Again

Steve Landsburg — Thu, 29 Oct 2009 06:42:35 +0000

I said this in The Big Questions and I’ll say it again: Richard Dawkins is an international treasure and one of my personal heroes, but he’s got this God thing all wrong. Here’s some of his latest, from the Wall Street Journal:

Where does [Darwinian evolution] leave God? The kindest thing to say is that it leaves him with nothing to do, and no achievements that might attract our praise, our worship or our fear. Evolution is God’s redundancy notice, his pink slip. But we have to go further. A complex creative intelligence with nothing to do is not just redundant. A divine designer is all but ruled out by the consideration that he must be at least as complex as the entities he was wheeled out to explain. God is not dead. He was never alive in the first place.

But Darwinian evolution can’t replace God, because Darwinian evolution (at best) explains life, and explaining life was never the hard part. The Big Question is not: Why is there life? The Big Question is: Why is there anything? Explaining life does not count as explaining the Universe.

Ah, says, Dawkins, but there’s no role for God there either:

Making the universe is the one thing no intelligence, however superhuman, could do, because an intelligence is complex—statistically improbable —and therefore had to emerge, by gradual degrees, from simpler beginnings

That, however, is just wrong. It is not true that all complex things emerge by gradual degrees from simpler beginnings. In fact, the most complex thing I’m aware of is the system of natural numbers (0,1,2,3, and all the rest of them) together with the laws of arithmetic. That system did not emerge, by gradual degrees, from simpler beginnings.

If you doubt the complexity of the natural numbers, take note that you can use just a small part of them to encode the entire human genome. That makes the natural numbers more complex than human life. Unless, of course, human beings contain an uncodable essence, like an immortal soul—but I’m guessing that’s not the road Dawkins wants to take.

Now I happen to agree with Professor Dawkins that God is unnecessary, but I think he’s got the reason precisely backward. God is unnecessary not because complex things require simple antecedents but because they don’t. That allows the natural numbers to exist with no antecedents at all—and once they exist, all hell (or more precisely all existence) breaks loose: In The Big Questions I’ve explained why I believe the entire Universe is, in a sense, made of mathematics.

So while Dawkins believes that complexity can arise only from simplicity, I believe that complexity arises from even greater complexity. I’m not sure I’m right, but I’m sure he’s wrong.

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