Ambiguity at the Roots of Precision
School of Natural Sciences
Institute for Advanced Study
Princeton, NJ 08540, USAA hundred years ago, scientists expected to be able to give an ever more precise description of the world, based on mathematics and physics. In the first few decades of the twentieth century, this hope was shattered. In physics, both relativity theory and quantum mechanics specify well-defined limits to the accuracy of measurements and hence the accuracy of descriptions of a physical system. In mathematics, Gödel showed that there are limits to logic, and in computer science Turing proved that we cannot determine in general whether a given computer program will halt or not. Digging ever deeper, we find that the roots of our scientific world view are planted in the soil of ambiguity. In order to study reality, we cannot avoid imposing our own limitations, our own choices of how to measure and interpret reality. As long as we remain aware of the choice we have, we can enjoy a form of freedom from identification.1. The Dream of a Complete Description
During the eighteenth and nineteenth century, starting with the Enlightenment, scientists expected that their discoveries would enable us to describe the world in ever greater detail, with ever greater precision. That expectation seemed reasonable, especially at the end of the nineteenth and the beginning of the twentieth century. Many physicists felt that they more or less understood the world, with the great advances in electromagnetism and thermodynamics that had been accomplished in the last few decades before the turn of the century. It came as a great shock that this optimistic program of an ever-more-detailed exploration of the natural world turned out to be impossible to realize.
The first surprise came in physics. Einstein's relativity theory showed that space and time are interwoven in such a way that there is an intermediate area in spacetime, in between the future and the past, of which we cannot obtain any knowledge in the present. This is the `elsewhere', the area outside the light cone associated with the present. Whether any event in the elsewhere should be considered to occur earlier or later than the present is ambiguous; depending on the velocity and direction of the movement of the observer in the present, either is possible.
An even larger surprise came with the discovery of quantum mechanics. It quickly became clear that there were fundamental limits to the accuracy of measurements, even in the here and now. The famous uncertainty relation in quantum mechanics tells us that we cannot measure both the position and the velocity of an electron, for example, with arbitrary accuracy. The more accurate we determine the position of an electron, the more ambiguous the velocity, and the more accurate we determine the velocity of an electron, the more ambiguous the position. The very idea of a complete description turned out to be a fantasy, something that could never be realized in practice.
Until then, physicists had firmly believed that Nature is deterministic. If you repeat an experiment in exactly the same way, of course you would get the exact same result. Indeed, everything that had been discovered during the first three centuries after Galileo did obey this requirement, to within the experimental precision. However, quantum mechanics turned out to be different. We can prepare two radioactive atoms in exactly the same way, and yet they will decay at completely different times, perhaps years or centuries or more apart, depending on their half-life. With the discovery of quantum mechanics, it has become clear that Nature is non-deterministic at heart. The determinacy that seems so obvious to us is only an approximation, on scales of space and time that are large compared to the atomic domain.
It was not only in physics, that the dream of a complete description was shattered. Even in the more idealized world of mathematics, the notion of a purely logical full description of a mathematical system had to be given up for all but the simplest systems. Since mathematics underlies all of natural science, and seems to provide it with rigor and precision, let us have a closer look at the ambiguity that was discovered in mathematics, roughly seventy years ago.
2. The Firm Ground of Mathematics
Mathematics has turned out to be an excellent tool to analyze the physical processes in the world around us. But mathematics can do more than analyze the world, or even model imaginary worlds. By its very nature as an abstract system, it also has the possibility to analyze itself.
Mathematical analysis of nature is always ultimately a haphazard enterprise. Even those laws of nature that seem most fundamental at some point in time can later be found to be only approximately true, to be superseded by the more penetrating insight of later generations of physicists. For example, Einstein's general relativity theory of gravity supersedes the Newtonian theory of gravity, although under every-day circumstances the two theories make very nearly the same prediction. Similarly, particle physicists may soon find a theory unifying gravity and the other fundamental forces into an even more fundamental theory. Such a theory will then supersede general relativity.
Whereas mathematical structures built to explain nature are always in danger of losing their foothold, it would seem that we are on firmer ground when we apply mathematics to itself. Once we have defined a mathematical system, firmly based on a small number of axioms, we are dealing with an idealized universe, unperturbed by whatever is going on in the real world. Mathematics appears to provide a rigorous form of knowledge, one of the few areas in which black-and-white thinking still seems to make sense. With all the uncertainties these days about how to behave in a politically correct fashion, at least we can still rest assured of the fact that in arithmetic the statement 2+2=4 is correct and the statement 2+2=3 is false. Black and white, pure and simple, no doubt about it. No room for anything grey or doubtful here, or so it would seem.
Of course, it would be nice to take this conclusion and prove it absolutely rigorously, once and for all, using the full rigor of mathematics to prove that mathematics provides us with a black-and-white world. And indeed, in the nineteen thirties, a program to do just that was in full swing, led by the German mathematician Hilbert. And it was around this time that the analysis of math by math provided the world with a shocking surprise: mathematics turned out not to be black-and-white at all, but full of grey areas, almost everywhere.
Forget about the more complicated forms of mathematics, anything to do with integrals or differential equations, for example. Take only that part of mathematics that contains the notion of integers: zero, one, two, three, etc. We can even leave out fractions or negative numbers. Just take the integers together with the simple operations of addition and multiplication. Even this small part of mathematics is incomplete, in the sense that there are grey areas that can neither be proved nor disproved. This was the shocking and totally unexpected result, announced by Gödel in 1931.
3. How the Firm Ground was Shaken.
At the core of Gödel's `unprovability' proof, a paradox was used, written completely in the language of arithmetic itself. The details of the proof are far too technical to discuss here, but the underlying paradox can be exhibited in plain words, in the following way. If someone would say ``I lie'', would that person speak the truth or not? Clearly, we are in quandary when we try to find an answer. If true, the statement must be false; and if false, the statement must be true. Well, which is it? We cannot say. Neither is correct. Simple as it is, this statement jumps the bounds of a straightforward black and white yes-no classification.
Gödel constructed a similar paradox within arithmetic. Instead of ``what I say is false'' he constructed the following sentence (encoded in arithmetic, i.e. as a number, using a clever coding scheme): ``what I say is unprovable''. Let us look at this sentence. Imagine that you could prove it to be true. By doing so, you would have contradicted the content of the message, since you have just shown that the sentence was provable. So the content, and thereby the message itself, would be false. But this would be a terrible situation: you would have just proved a false statement to be true! In this case, arithmetic would contain an inherent contradiction. And since we believe arithmetic to be consistent, we would like to conclude that this statement cannot be proved to be true.
How about the converse? Can we hope to prove this statement to be false? If so, it would not be true that ``what I say is unprovable''. Therefore, we have to conclude that instead ``what I just said is not unprovable, but provable''. In other words, it is provable that ``what I say is unprovable''. Again, we are back at a contradiction, the very same contradiction we encountered before!
So, whether ``what I say is unprovable'' is right or wrong, in both case we would have an inconsistency. The only way to salvage arithmetic is to conclude that we cannot decide either way. In other words, the sentence ``what I say is unprovable'' cannot be proved to be right or wrong, and must forever remain in a `grey area', on the borderline between true and false. The borderline has turned into a border area.
But, lo and behold: we've just found the answer! If ``what I say is unprovable'' cannot be proved to be right, we have to conclude that the statement is correct. If we can neither prove nor disprove it, we at least cannot prove it, so, yes, what was said was unprovable all right, and, yes, the statement turned out to be correct. How strange!
The catch is this: what seems so clear and simple now, all of a sudden, only seems clear and simple because we are using a natural language, English in this case, to make statements about mathematics. Gödel showed that a translation of the sentence ``what I say is unprovable'' into arithmetic cannot be proved to be true within arithmetic itself. One needs a higher-order formal system, a form of meta-arithmetic, looking down on arithmetic from above, so to speak, and not bound by its internal quandaries. Only from outside the system of arithmetic is it possible to realize that a sentence such as ``what I say is unprovable'' is indeed a correct sentence.
Thus already in mathematics, a description of an object or system cannot be separated from our choice of how we take that object or system to be. Our choice of framework necessarily comes in between the reality and our description. This simple notion, familiar in daily life, flies in the face of the nineteenth century hope to find a complete description of the world.
4. The Map is not the Territory is not Reality
The description of an object is of course very different from the object itself. A map of a city contains information encoded in lines of ink of a piece of paper, or perhaps a pattern of pixels lit up in a hand-held computer. The city itself contains streets and buildings made of stone and metal and glass. But not only the map, also the city encodes information. At every street corner there are signs with the name of the street. Without those signs, maps would be much more difficult to use, since it would be hard to find the correspondence between the lines on the map and actual streets. On a more fundamental level, an interpretation of a city in terms of streets and buildings and blocks and wards is something in which a certain amount of description has already crept in. A cat or a rat might have a very different view of the same city, making completely different distinctions, carving up the same reality in different ways.
In a more natural setting, such as a forest, different species of animals divide the same areas into different territories. Squirrels and birds are not in each others way, and can by and large ignore each other's view of the world. Although they are aware of each other's presence, the boundaries of their respective territories have nothing to do with each other. In general, reality and territory and map are three different things. In order to describe reality, by making a map, we first have to take a specific stance. This stance taking is often described in military terms: as a conquering, a colonizing, a turning into a territory.
Looking at a world map, we immediately see the choice that we have in mapping reality. Some maps describe the political division of the world in terms of nations; other maps describe divisions in terms of mountains and other natural features. The middle step between reality and description resides in the `as': we can see the surface of the Earth as made up by different countries, or as made up by mountains and rivers, or as divided in regions of high and low population density, and so on. Somewhat similar notions of a middle ground have recently been discussed by various authors, such as Latour, Smith, and Gallison [see the references below].
Even in everyday life, this middle ground can be found everywhere. The same apple can be seen by a farmer as something that has economic value, by a chemist as something that can be analyzed into its chemical components, by a painter as something that can be painted, and by anyone as something that can be tasted and eaten. Seen as a part of reality, the apple is the same apple. However, the painter turns the apple into a different territory than a farmer or a chemist does. The `as' that is neither part of reality, nor yet part of the description, is the stepping stone that allows a description to be applied to a part of reality.
It is fascinating that this tripartition in map, territory and reality is more than only a poetic distinction. In physics, a single elementary particle such as an electron is treated in a similar way. An electron as such cannot be mapped completely, since each description is only partial, by necessity. As was mentioned already above, choosing to measure the position accurately leaves less room for measurements of the velocity of the electron and vice versa. The choice of what to measure corresponds to the establishment of a territory. Once we have thus `tamed' the electron, we can map its velocity or position or other properties. However, the latter description is intrinsically influenced by the choice of territory. The described value of the electron's position, for example, cannot by considered to be part of an objective reality, independent of the choice of measurement.
5. Freedom from Identification
Once we realize that there is a tripartite structure to the world we are dealing with, that of reality, territory, and map, we realize that we have far greater freedom. On all levels, whether we deal with countries, apples, or electrons, we do not have to take any description as being exhaustive. Yes, a description can be very accurate, but such precision is always a precision relative to a given framework, reflecting a particular territory. Seen `as' an object in that territory, the description can be right or wrong, and more accurate or less accurate. However, the question whether the object is `really' that way, without any ambiguity, cannot be answered. The object itself, in so far as it even makes sense to talk about it, is inexhaustible: we cannot exclude the possibility that new angles will reveal new aspects of that object, seen in a new light, as part of a new territory.
It is this freedom that lies at the heart of reality, beyond any attempt to colonize what appears in restricted ways, that grounds any form of creativity. What sets us free, in any situation, is the intrinsic ambiguity in our choice of how to focus on an object, from which angle to look at it, or even how to parse a piece of reality into different objects. In order to deal with reality, we have to make specific identifications before we can describe it. After having made those identifications, the temptation is large to project those choices back onto reality itself, and to think only in terms of map versus territory. But whenever we shift our focus to the other distinction, between territory and reality, we can regain our original freedom, which transcends territories.
Smith, B.C. 1996, On the Origin of Objects (Cambridge, MA: M.I.T. Press)
Latour, B. 1988, Science in Action (Cambridge, MA: Harvard University Press)
Galison, P. 1997, Image and Logic (Chicago: University of Chicago Press)
Note: During the conference, while the author presented this paper, it was pointed out to him that the poet Tanikawa Shuntaro, who also participated in the conference, had written a poem on apples that conveyed a similar sentiment to what was presented in the present paper.
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