Gödel, Escher, Bach: An Eternal Golden Braid (81 page)

This sounds quite strange, at first. Just exactly how big is the number which makes a
TNT
-proof-pair with
G
's Gödel number= (Let's call it
'I
. for no particular reason.) Unfortunately, we have not got any good vocabulary for describing the sizes of infinitely large integers, so I am afraid I cannot convey a sense of I's magnitude. But then just how big is i (the square root of -1)? Its size cannot be imagined in terms of the sizes of familiar natural numbers. You can't say, "Well, i is about half as big as 14, and 9/10 as big as 24." You have to say, "i squared is -1", and more or less leave it at that. A quote from Abraham Lincoln seems a propos here. When he was asked, "How long should a man's legs be?" he drawled, "Long enough to reach the ground." That is more or less how to answer the question about the size of I-it should be just the size of
a number which
specifies the structure of a proof of
G
-no bigger, no smaller.

Of course, any theorem of
TNT
has many different derivations, so you might complain that my characterization of I is nonunique. That is so. But the parallel with 1-the square root of -1-still holds. Namely, recall that there is another number whose square is also minus one: -i. Now i and -i are not the same number. They just have a property in common. The only trouble is that it is the property which defines them! We have to choose one of them-it doesn't matter which one-and call it "
i
". In fact there's no way of telling them apart. So for all we know we could have been calling the wrong one "i" for all these centuries and it would have made no difference. Now, like i, I is also nonuniquely defined. So you just have to think of
I
as being some specific one of the many possible supernatural numbers which form
TNT
-proof-pairs with the arithmoquinification of
u.

Supernatural Theorems Have Infinitely Long Derivations.

We haven't yet faced head on what it means to throw -
G
in as an axiom. We have said it but not stressed it. The point is that -G asserts that
G
has a proof. How can a system survive, when one of its axioms asserts that its own negation has a proof? We must be in hot water now! Well, it is not so bad as you might think. As long as we only construct
finite
proofs, we will never prove
G
Therefore, no calamitous collision between
G
and its negative ~
G
will ever take place. The supernatural number
–I
won’t cause any disaster.

However, we will have to get used to the idea that ---
G
is now the one which asserts a truth ("
G
has a proof "), while
G
asserts a falsity ("
G
has no proof"). In standard number theory it is the other way around-but then, in standard number theory there aren't any supernatural numbers. Notice that a supernatural theorem of
TNT
-namely
G
-may assert a falsity, but all natural theorems still assert truths.

Supernatural Addition and Multiplication

There is one extremely curious and unexpected fact about supernaturals which I would like to tell you, without proof. (I don't know the proof either.) This fact is reminiscent of the Heisenberg uncertainty principle in quantum mechanics. It turns out that you can

"index" the supernaturals in a simple and natural way by associating with each supernatural number a trio of ordinary integers (including negative ones). Thus, our original supernatural number,
I
, might have the index set (9,-8,3), and its successor,
I
+ 1, might have the index set (9,-8,4). Now there is no unique way to index the supernaturals; different methods offer different advantages and disadvantages. Under some indexing schemes, it is very easy to calculate the index triplet for the sum of two supernaturals, given the indices of the two numbers to be added. Under other indexing schemes, it is very easy to calculate the index triplet for the
product
of two supernaturals, given the indices of the two numbers to be multiplied. But under
no
indexing scheme is it possible to calculate both. More precisely, if the sum's index can be calculated by a recursive function, then the product's index will not be a recursive function; and conversely, if the product's index is a recursive function, then the sum's index will not be. Therefore, supernatural schoolchildren who learn their supernatural plus-tables will have to be excused if they do not know their supernatural times-tables-and vice versa! You cannot know both at the same time.

Supernaturals Are Useful ...

One can go beyond the number theory of supernaturals, and consider supernatural fractions (ratios of two supernaturals), supernatural real numbers, and so on. In fact, the calculus can be put on a new footing, using the notion of supernatural real numbers.

Infinitesimals such as dx and dy, those old bugaboos of mathematicians, can be completely justified, by considering them to be reciprocals of infinitely large real numbers! Some theorems in advanced analysis can be proven more intuitively with the aid of "nonstandard analysis".

But Are They Real?

Nonstandard number theory is a disorienting thing when you first meet up with it. But, then, non-Euclidean geometry is also a disorienting subject. In

both instances, one is powerfully driven to ask, "But which one of these two rival theories is correct? Which
is the truth
?" In a certain sense, there is no answer to such a question.

(And vet, in another sense-to be discussed later-there is an answer.) The reason that there is no answer to the question is that the two rival theories, although they employ the same terms, do not talk about the same concepts. Therefore, they are only superficially rivals, just like Euclidean and non-Euclidean geometries. In geometry, the words "point", "line", and so on are undefined terms, and their meanings are determined by the axiomatic system within which they are used.

Likewise for number theory. When we decided to formalize
TNT
. we preselected the terms we would use as interpretation words-for instance, words such as "number",

"plus", "times", and so on. By taking the step of formalization, we were committing ourselves to accepting whatever passive meanings these terms might take on. But just like Saccheri-we didn't anticipate any surprises. We thought we knew what the true, the real, the only theory of natural numbers was. We didn't know that there would be some questions about numbers which
TNT
would leave open, and which could therefore be answered ad libitum by extensions of
TNT
heading off in different directions. Thus, there is no basis on which to say that number theory "really" is this way or that, just as one would be loath to say that the square root of -1 "really" exists, or "really" does not.

Bifurcations in Geometry, and Physicists

There is one argument which can be, and perhaps ought to be, raised against the preceding. Suppose experiments in the real, physical world can be explained more economically in terms of one particular version of geometry than in terms of any other.

Then it might make sense to say that that geometry is "true". From the point of view of a physicist who wants to use the "correct" geometry, then it makes some sense to distinguish between the "true" geometry, and other geometries. But this cannot be taken too simplistically. Physicists are always dealing with approximations and idealizations of situations. For instance, my own Ph.D. work, mentioned in Chapter V, was based on an extreme idealization of the problem of a crystal in a magnetic field. The mathematics which emerged was of a high degree of beauty and symmetry. Despite-or rather, because of-the artificiality of the model, some fundamental features emerged conspicuously in the graph. These features then suggest some guesses about the kinds of things that might happen in more realistic situations. But without the simplifying assumptions which produced my graph, there could never be such insights. One can see this kind of thing over and over again in physics, where a physicist uses a "nonreal" situation to learn about deeply hidden features of reality. Therefore, one should be extremely cautious in saying that the brand of geometry which physicists might wish to use would represent “the true geometry", for in fact, physicists will always use a variety of different geometries, choosing in any given situation the one that seems simplest and most convenient.

Furthermore-and perhaps this is even more to the point-physicists do not study just the 3-D space we live in. There are whole families of "abstract spaces" within which physical calculations take place, spaces which have totally different geometrical properties from the physical space within which we live. Who is to say, then, that "the true geometry" is defined by the space in which Uranus and Neptune orbit around the sun? There is "Hilbert space", where quantum-mechanical wave functions undulate; there is "momentum space", where Fourier components dwell; there is "reciprocal space", where wave-vectors cavort; there is "phase space", where many-particle configurations swish; and so on. There is absolutely no reason that the geometries of all these spaces should be the same; in fact, they couldn't possibly be the same! So it is essential and vital for physicists that different and "rival" geometries should exist.

Bifurcations in Number Theory, and Bankers

So much for geometry. What about number theory? Is it also essential and vital that different number theories should coexist with each other? If you asked a bank officer, my guess is that you would get an expression of horror and disbelief. How could 2 and 2 add up to anything but 4? And moreover, if 2 and 2 did not make 4, wouldn't world economies collapse immediately under the unbearable uncertainty opened up by that fact? Not really. First of all, nonstandard number theory doesn't threaten the age-old idea that 2 plus 2 equals 4. It differs from ordinary number theory only in the way it deals with the concept of the infinite. After all,
every theorem of
TNT
remains a theorem in any
extension of
TNT
! So bankers need not despair of the chaos that will arrive when nonstandard number theory takes over.

And anyway, entertaining fears about old facts being changed betrays a misunderstanding of the relationship between mathematics and the real world.

Mathematics only tells you answers to questions in the real world after you have taken the one vital step of choosing which kind of mathematics to apply. Even if there were a rival number theory which used the symbols `2', `3', and `+', and in which a theorem said

"2 + 2 = 3", there would be little reason for bankers to choose to use that theory! For that theory does not fit the way money works. You fit your mathematics to the world, and not the other way around. For instance, we don't apply number theory to cloud systems, because the very concept of whole numbers hardly fits. There can be one cloud and another cloud, and they will come together and instead of there being two clouds, there will still only be one. This doesn't prove that 1 plus 1 equals 1; it just proves that our number theoretical concept of “one” is not applicable in its full power to cloud counting.

Bifurcations in Number Theory, and Metamathematicians

So bankers, cloud-counters, and most of the rest of us need not worry ,about the advent of supernatural numbers: they won't affect our everyday perception of the world in the slightest. The only people who might actually be a little worried are people whose endeavors depend in some crucial way on the nature of infinite entities. There aren't too many such people around-but mathematical logicians are members of this category. How can the existence of a bifurcation in number theory affect them Well, number theory plays two roles in logic: (1) when axiomatized, it is an object of study; and (2) when used informally, it is an indispensable tool with which formal systems can be investigated.

This is the use-mention distinction once again, in fact: in role (1), number theory is mentioned, in role (2) it is used.

Now mathematicians have judged that number theory is applicable to the study of formal systems even if not to cloud-counting, just as bankers have judged that the arithmetic of real numbers is applicable to their transactions. This is an extramathematical judgement, and shows that the thought processes involved in doing mathematics, just like those in other areas, involve "tangled hierarchies" in which thoughts on one level can affect thoughts on any other level. Levels are not cleanly separated, as the formalist version of what mathematics is would have one believe.

The formalist philosophy claims that mathematicians only deal with abstract symbols, and that they couldn't care less whether those symbols have any applications to or connections with reality. But that is quite a distorted picture. Nowhere is this clearer than in metamathematics. If the theory of numbers is itself used as an aid in gaining factual knowledge about formal systems, then mathematicians are tacitly showing that they believe these ethereal things called "natural numbers" are actually part of reality not just figments of the imagination. This is why I parenthetically remarked earlier that, in a certain sense, there is an answer to the question of which version of number theory is