On the Limits to Scientific Knowledge

Earlier in this century, work in mathematical logic and computing turned up the astonishing fact that not every true fact about numbers can be obtained by following a set of rules, in essence, a computer program. These findings place limits on our ability to know in the world of mathematics and logic. Are there similar limits to our ability to answer questions about natural and human affairs? Can we ever hope to understand things like stock price movements or traffic jams or national economies by invoking the tools of scientific argumentation? Or are there some questions that can be shown to be logically, not just practically, beyond the boundaries of science?
This question seems espescially important nowadays, as we see an accelerating movement away from rational analysis of human problems to answers provided by fundamentalist religions, political demagogues, and those that can only be described as magicans and occultists.

Science, as a reality-generating mechanism, is simply a procedure for producing a set of rules for answering questions posed about the universe. These are not just any old rules, however, but have a specific properties - explicitness, public availability, bias-free, reliable - and are generated by a particular procedure, the scientific method. So when we say we have a scientific answer to a question, it means we have what amounts to a computer program that produces the answer to the question as its output.

Now how would one actually establish that some particular question about the real world, not the mathematical, is truly scientifically unanswerable? One way would be to argue by analogy with the limitative results in mathematics, which state that the world of numbers cannot be both consistent and complete. Translating these terms into everyday language, consistency would mean that there are no true paradoxes in nature; water does flow both uphill and downhill at the same time; a particle cannot move both left and right simultaneously. In general, when we encounter what appears to be such a paradox - such as "jets" that seemed to be moving away from quasars at faster-than-light speeds - subsequent investigation has provided a resolution.

By the same token, completeness means simply that every event has a cause. Universes don't just spontaneously appear out of the vacuum and automobile accidents don't just come out of nowhere. Events have traceable causes - although at a given point in time it may be very unclear to us what the chain of causation for a particular event actually is. It is my belief that unlike the world of mathematical objects, nature is both consistent and complete; there are no mutually contradictory events and things don't just happen. Since this is a pretty bold claim, at least in some circles, let me give some arguments in its support.

The Uncomputable

In the language of computing, incompleteness translates into the existence of uncomputable quantities. Such quantities, in turn, arise from playing fast and loose with the notion of the infinite. So one way in which the natural and mathematical worlds could part company would be for there to be no actual infinities in nature. In a logical system in which there are only a finite number of questions that can be stated of finite length, then incompleteness cannot arise. The empirical evidence in favor of such a finite universe is pretty solid. For instance, all the numbers that have ever been written down, uttered in speech, or in any other way expressed by human beings is certainly finite. And, in fact, the uncomputable numbers that theoreticians assure us exist, can never actually be displayed - if they could, then they would cease to be uncomputable since we would then have a rule for characterizing each digit of the number. Of course, this fact does not prove that infinities do not exist in the universe. But our finite minds have never seen one - and probably couldn't comprehend one if we did.
In this connection, don't be misled by statements about large numbers, such as 1080, the number of protons in the known universe or other such immense quantities. While a number like 1080 certainly commands some respect, it is as far away from infinity as 1 or 2.

The stability of the solar system

In order to bring the issues of limits to scientific knowledge into sharper focus, let's look at a well-known question from the of physics that suggests some of the difficulty in coming to terms with limits. This is the classical problem of the stability of the solar system.

Certainly the most famous question of classical celestial mechanics is the N-Body Problem, which comes in many forms. One version involves N point masses moving in accordance with Newton's laws of gravitational attraction. Mathematically, the trajectories of the particles are given by the solution of the set of differential equations:


Here mi is the mass and r i is the position vector of the ith particle, while rij=ri- rj is the euclidean distance between the ith and jth particles. The quantity is the self-potential (the negative of the potential energy of the particle system).

If we let be the set of all points in R3n where the above set of equations is not defined, then one way of expressing the N-Body Problem is to ask if there is some set of initial positions and velocities of the particles, such that r(t) as t ? One way for this to happen would be for two particles to collide, in which case ri= rj for some i and j. Another way would be to have a noncollision singularity, in which r(t) approaches without actually approaching any point in this set. In such a case, some particle in the system would acquire an unbounded velocity, and hence ``fly off'' out of the system. In the special case when N=10, these situations represent a mathematical formulation of the question of whether or not our solar system is stable.

In a 1988 doctoral dissertation, Jeff Xia of Northwestern University used earlier work by Don Saari to give a definitive answer to the second possibility, constructing a 5-body system for which one of the bodies does indeed acquire an unbounded velocity. Figure~1 above shows the general idea underlying Xia's construction, which involves two binary systems and a fifth body that shuttles back-and-forth between them. By arranging things just right, the single body can be made to move faster and faster between the binaries until at some finite time it acquires as large a velocity as you please. Of course, this result says nothing about the specific case of our solar system. But it does suggest that perhaps the solar system is not stable, and more importantly offers new tools with which to further investigate the matter.


What Xia's results underscore is the point that there is a vast difference between a physical phenomenon such as planetary motion, and a mathematical picture of that phenomenon. And when it comes to limits, it's the real-world process that we're interested in, not the mathematical model.

Computability is relative

A second point to bear in mind is that computability is a relative, not an absolute, notion. Quantities are computable only relative to a given model of what it means to "carry out a computation.'' The default model used in science is one developed by the British computer scientist, Alan Turing, in the mid 1930s. But it is very far from the only possibility, and today we see great attention being paid to other models, such as those based on the biological properties of DNA or on the properties of quantum phenomena. Each such model generates its own class of computable quantities, and it's an open question whether what is uncomputable in the Turing sense may become computable in the framework of one of these other models.

Finally, there is the issue of deduction. All the incompleteness results from mathematics and computing rest on the use of deductive modes of reasoning, essentially the following of a prescribed set of deductive rules from general premises to specific conclusions. But deduction is far from the only mode of reasoning at our disposal. There is also induction, in which we argue from specific instances to general conclusions, as well as other less familiar forms of argument such as abduction. When using these modes of reasoning, it's not even clear what incompleteness would actually mean, let alone whether the logical system possesses it or not.

What this all adds up to is that claims about the limits to scientific knowledge based on incompleteness arguments in mathematics are of pretty dubious merit, at best. Finiteness, non-Turing modes of computation, and alternate modes of reasoning all lend weight to the idea that any question we might care to address to the universe can be answered by some set of scientific rules, including those normally linked to notions of human creativity.

Literature

First published in Complexity Vol. 3/No. 1, September/October 1997. Copyright by
John L. Casti.