Gödel, Escher, Bach: An Eternal Golden Braid (20 page)

We can summarize our observations so far in terms of the word "consistency".

We began our discussion by manufacturing what appeared to be an inconsistent formal system-one which was internally inconsistent, as well as inconsistent with the external world. But a moment later we took it all back, when we realized our error: that we had chosen unfortunate interpretations for the symbols. By changing the interpretations, we regained consistency! It now becomes clear that
consistency is not a property of a formal
system per se, but depends on the interpretation which is proposed for it
. By the same token, inconsistency is not an intrinsic property of any formal system.

Varieties of Consistency

We have been speaking of "consistency" and "inconsistency" all along, without defining them. We have just relied on good old everyday notions. But now let us say exactly what is meant by
consistency
of a formal system (together with an interpretation): that every theorem, when interpreted, becomes a true statement. And we will say that inconsistency occurs when there is at least one false statement among the interpreted theorems.

This definition appears to be talking about inconsistency with the external world-what about internal inconsistencies? Presumably, a system would be internally inconsistent if it contained two or more theorems whose interpretations were incompatible with one another, and internally consistent if all interpreted theorems were compatible with one another. Consider, for example, a formal system which has only the following three theorems:
TbZ
,
ZbE
, and
EbT
. If
T
is interpreted as "the Tortoise",
Z
as

"Zeno",
E
as "Egbert", and x by as "x beats y in chess always", then we have the following interpreted theorems:

The Tortoise always beats Zeno at chess

Zeno always beats Egbert at chess.

Egbert always beats the Tortoise at chess.

The statements are not incompatible, although they describe a rather bizarre circle of chess players. Hence, under this interpretation, the form; system in which those three strings are theorems is internally consistent although, in point of fact, none of the three statements is true! Intern< consistency does not require all theorems to come out true, but merely that they come out
compatible
with one another.

Now suppose instead that x by is to be interpreted as "x was invented by y". Then we would have:

The Tortoise was invented by Zeno.

Zeno was invented by Egbert.

Egbert was invented by the Tortoise.

In this case, it doesn't matter whether the individual statements are true c false-and perhaps there is no way to know which ones are true, and which are not. What is nevertheless certain is that
not all three can be true at one
Thus, the interpretation makes the system internally inconsistent. The internal inconsistency depends not on the interpretations of the three capital letters, but only on that of b, and on the fact that the three capita are cyclically permuted around the occurrences of b. Thus, one can have internal inconsistency without having interpreted
all
of the symbols of the formal system.

(In this case it sufficed to interpret a single symbol.) By tl time sufficiently many symbols have been given interpretations, it may t clear that there is no way that the rest of them can be interpreted so that a theorems will come out true. But it is not just a question of truth-it is question of possibility. All three theorems would come out false if the capitals were interpreted as the names of real people-but that is not why we would call the system internally inconsistent; our grounds for doing s would be the circularity, combined with the interpretation of the letter I (By the way, you'll find more on this "authorship triangle"

in Chapter XX.;

Hypothetical Worlds and Consistency

We have given two ways of looking at consistency: the first says that system-plus-interpretation is
consistent with the external world
if every theorem comes out true when interpreted; the second says that a system-plus: interpretation is
internally
consistent
if all theorems come out mutually compatible when interpreted. Now there is a close relationship between these two types of consistency. In order to determine whether several statements at mutually compatible, you try to imagine a world in which all of them could be simultaneously true. Therefore, internal consistency depends upon consistency with the external world-only now, "the external world" allowed to be
any
imaginable world
, instead of the one we live in. But this is an extremely vague, unsatisfactory conclusion. What constitutes an “imaginable" world?

After all, it is possible to imagine a world in which three characters invent each other cyclically. Or is it? Is it possible to imagine a world in which there are square circles? Is a world imaginable in which Newton's laws, and not relativity, hold? Is it possible to imagine a world in which something can be simultaneously green and not green? Or a world in which animals exist which are not made of cells? In which Bach improvised an eight-part fugue on a theme of King Frederick the Great? In which mosquitoes are more intelligent than people? In which tortoises can play football-or talk? A tortoise talking football would be an anomaly, of course.

Some of these worlds seem more imaginable than others, since some seem to embody
logical
contradictions-for example, green and not green-while some of them seem, for want of a better word, "plausible" -- such as Bach improvising an eight-part fugue, or animals which are not made of cells. Or even, come to think of it, a world in which the laws of physics are different ... Roughly, then, it should be possible to establish different brands of consistency. For instance, the most lenient would be "logical consistency", putting no restraints on things at all, except those of logic. More specifically, a system-plus-interpretation would be
logically consistent
just as long as no two of its theorems, when interpreted as statements, directly contradict each other; and
mathematically consistent
just as long as interpreted theorems do not violate mathematics; and
physically consistent
just as long as all its interpreted theorems are compatible with physical law; then comes biological consistency, and so on. In a biologically consistent system, there could be a theorem whose interpretation is the statement "Shakespeare wrote an opera", but no theorem whose interpretation is the statement "Cell-less animals exist". Generally speaking, these fancier kinds of inconsistency are not studied, for the reason that they are very hard to disentangle from one another. What kind of inconsistency, for example, should one say is involved in the problem of the three characters who invent each other cyclically? Logical? Physical?

Biological? Literary?

Usually, the borderline between uninteresting and interesting is drawn between physical consistency and mathematical consistency. (Of course, it is the mathematicians and logicians who do the drawing-hardly an impartial crew . . .) This means that the kinds of inconsistency which "count", for formal systems, are just the logical and mathematical kinds. According to this convention, then, we haven't yet found an interpretation which makes the trio of theorems
TbZ, ZbE, EbT
inconsistent. We can do so by interpreting b as "is bigger than". What about
T
and
Z
and
E
? They can be interpreted as natural numbers-for example,
Z
as 0,
T
as 2, and
E
as 11. Notice that two theorems come out true this way, one false. If, instead, we had interpreted Z as 3, there would have been two falsehoods and only one truth. But either way, we'd have had inconsistency. In fact, the values assigned to
T, Z,
and
E
are irrelevant, as long as it is understood that they are restricted to natural numbers. Once again we see a case where only
some
of the interpretation is needed, in order to recognize internal inconsistency.

Embedding of One Formal System In Another

The preceding example, in which some symbols could have interpretations while others didn't, is reminiscent of doing geometry in natural languag4 using some words as undefined terms. In such a case, words are divide into two classes: those whose meaning is fixed and immutable, and, those whose meaning is to be adjusted until the system is consistent (these are th undefined terms). Doing geometry in this way requires that meanings have already been established for words in the first class, somewhere outside c geometry. Those words form a rigid skeleton, giving an underlying structure to the system; filling in that skeleton comes other material, which ca vary (Euclidean or non-Euclidean geometry).

Formal systems are often built up in just this type of sequential, c hierarchical, manner. For example, Formal System I may be devised, wit rules and axioms that give certain intended passive meanings to its symbol Then Formal System I is incorporated fully into a larger system with more symbols-Formal System II. Since Formal System I's axioms and rules at part of Formal System II, the passive meanings of Formal System I symbols remain valid; they form an immutable skeleton which then plays large role in the determination of the passive meanings of the new symbols of Formal System II. The second system may in turn play the role of skeleton with respect to a third system, and so on. It is also possible-an geometry is a good example of this-to have a system (e.g., absolute geometry) which
partly
pins down the passive meanings of its undefined terms, and which can be supplemented by extra rules or axioms, which then
further
restrict the passive meanings of the undefined terms. This the case with Euclidean versus non-Euclidean geometry.

Layers of Stability in Visual Perception

In a similar, hierarchical way, we acquire new knowledge, new vocabulary or perceive unfamiliar objects. It is particularly interesting in the case understanding drawings by Escher, such as Relativity (Fig. 22), in which there occur blatantly impossible images. You might think that we won seek to reinterpret the picture over and over again until we came to interpretation of its parts which was free of contradictions-but we dot do that at all. We sit there amused and puzzled by staircases which go eve which way, and by people going in inconsistent directions on a sing staircase. Those staircases are "islands of certainty" upon which we base of interpretation of the overall picture. Having once identified them, we try extend our understanding, by seeking to establish the relationship which they bear to one another. At that stage, we encounter trouble. But if i attempted to backtrack-that is, to question the "islands of certainty"-s would also encounter trouble, of another sort. There's no way of backtracking and

"undeciding" that they are staircases. They are not fishes, or whip or hands-they are just staircases. (There is, actually, one other on t-i leave all the lines of the picture totally uninterpreted, like the "meaningless

FIGURE 22. Relativity, by M. C. Escher (lithograph, 1953).

symbols" of a formal system. This ultimate escape route is an example of a "U-mode"

response-a Zen attitude towards symbolism.)

So we are forced, by the hierarchical nature of our perceptive processes, to see either a crazy world or just a bunch of pointless lines. A similar analysis could be made of dozens of Escher pictures, which rely heavily upon the recognition of certain basic forms, which are then put together in nonstandard ways; and by the time the observer sees the paradox on a high level, it is too late-he can't go back and change his mind about how to interpret the lower-level objects. The difference between an Escher drawing and non-Euclidean geometry is that in the latter, comprehensible interpretations can be found for the undefined terms, resulting in a com

prehensible total system, whereas for the former, the end result is not reconcilable with one's conception of the world, no matter how long or stares at the pictures. Of course, one can still manufacture hypothetic worlds, in which Escherian events can happen ... but in such worlds, t1 laws of biology, physics, mathematics, or even logic will be violated on or level, while simultaneously being obeyed on another, which makes the: extremely weird worlds. (An example of this is in
Waterfall
(Fig. 5), whet normal gravitation applies to the moving water, but where the nature space violates the laws of physics.)
Is Mathematics the Same in Every Conceivable World?