Several times in the history of science and mathematics, the belief was expressed that absolute truth will eventually be reached through our grand systems of deduction and induction. An example is Laplace’s statement that, if he knew the position and velocity of every particle in the universe, he could precisely predict the future of the universe in complete detail. Similarly in political science, Rousseau thought that the General Will of a society was some precise quantity resulting from the addition of the individual wills of the citizens. And in mathematics, of course, everyone took it for granted that true theorems could always be deduced or proved from previous theorems plus a few axioms and assumptions; the proofs might be difficult, but there was full confidence that finding them was just a question of skill, luck and time.

Usually we do not even enunciate the assumption that we are approaching complete knowledge, bit by bit, and that eventually we will assemble the Whole Truth about Everything. Yet a few doubts seep in from time to time. Newton compared himself to a small boy playing with a few shells of knowledge on the beach, while a whole ocean of truths stretched before him undiscovered. Plato warned us that we observe only shadows on a cave wall, not the reality outside. Kant believed that most “facts” were only categories (artefacts) of the mind, and that we know nothing about things-in-themselves. Berkeley even doubted that the tree in the quad was still there when no one was looking at it. However, most practical working scientists dismissed the philosophers as scholastic nit-pickers, and even Newton’s statement (who WAS a practical scientist) was taken only to mean that it will take a little longer than we thought to collect all the shells in the ocean; but we had lots of time.

A few were more optimistic; a conference of physicists arount the year 1900 heard a declaration that physics was now practically a finished science; that only minor mopping-up operations were needed before the research labs could close down for ever. This proved to be a really major blooper: within a few years after that confident statement, Einstein introduced the theory of relativity, Planck the quantum theory, and physics was in the throes of a major revolution. Within a decade of that conference, physics was unrecognizable to the old-timers. So while we patiently add bits of knowledge to “normal” science, and think that we will soon be finished, we cannot always predict the coming in of the next paradigm shift, which comes in like a tidal wave, and we never know what it will bring. Kuhn made a “meta-science” of this theory of scientific revolutions.

But all this is only scratching the surface of the real limits to knowledge. What has come to light in the 20th century is the PROOF that our knowledge, in several fields and in several respects, must always remain incomplete. These are called “impossibility theorems”, and they call us to humility in even more fundamental ways than Copernicus, Darwin or Freud ever did. These scientific giants showed, respectively, that we are not the centre of the universe, of the living world, or even of our own mind. But the impossibility theorems show that our reason is not co-extensive with reality, that they touch at only some points, and that therefore whole realms of knowledge (we don’t know how extensive) are forever closed to us IN PRINCIPLE. Only three will be mentioned here: the Heisenberg uncertainty principle in physics, Goedel’s proof in mathematics, and Arrow’s theorem in political science.

Heisenberg’s uncertainty principle follows from Planck’s quantum theory and its offspring, the Schroedinger-De Broglie wave theory of electrons and other sub-atomic particles. Because such particles partake of the nature of both particles and waves (Bohr’s “complementarity”), no experiment can ever measure both their position and their velocity simultaneously and exactly; if we get an exact measurement of position, we know very little about velocity, and vice versa. This is because the measurement itself somehow determines whether the wave or the particle aspect will predominate, and they cannot both manifest themselves at the same time. Position is the property that goes with the particle aspect and velocity goes with the wave aspect. Heisenberg deduced the amount of unavoidable uncertainty, related to Planck’s constant h, from the wave theory. His uncertainty principle means that the limit to accurate measurement is fundamental and objective, not just temporarily due to imperfections of our methods of measurement; in other words, we cannot hope to remove it by future improvements in our measuring instruments.

The uncertainty principle is most evident in the sub-atomic realm, but it really exists at all scales, even the scale of ordinary everyday objects. But there, of course, the uncertainty is relatively so small that it lies well below the limits of our measuring instruments, and so the “objective” (unavoidable) uncertainty gets completely swamped by the “subjective” (in principle avoidable) uncertainty. In any case, Laplace’s ambition to predict the future of the universe in complete detail would be thwarted by Heisenberg’s uncertainty principle, because the positions and velocities of all particles can never be known simultaneously. We lack the starting point for Laplace’s deterministic system, and the rest of the process is not strictly deterministic either.

If experiment, observation and measurement fail at the limit, perhaps because the observer enters into the situation and disturbs nature by the very act of measurement, are we any safer in the deductive realm, where pure reason unassisted by the senses discovers truth with unerring certainty through proofs? The answer is “no”, there is leakage there too, and we don’t know how large.

Goedel’s proof is quite difficult to grasp, but essentially it shows that in any deductive system there are true statements which are not theorems, i.e. they cannot be proved. According to Douglas Hofstadter (GEB, p. 272), this comes from a statement which includes self-reference. Statement G says: “G is not a theorem of the system.” If this is true, then G is not a theorem. This is in the same class of paradoxes as a Cretan saying “All Cretans are liars.” It can be easily seen that we can neither accept this as true nor as false without running into a contradiction. It is undecidable.

Are such deductively unprovable statements exceptional or common? We cannot even determine that. It could be an unimportant quirk or it could be a vast realm of reality. GEB has a picture in which proved theorems are black islands in a white field in the lower half and non-theorem truths are white islands in a black field in the upper half. This implies that there are equal numbers of proved and unproved truths. But as far as I can see, Hofstadter is only guessing. The original Goedel non-theorem truth was rather difficult to find.

So it seems that we cannot encompass the whole truth by either inductive or deductive methods, not in science nor in mathematics. Would the social sciences do better? Even in such a well-defined field as the theory of voting, or, more generally, making collective decisions, Arrow’s theorem shows that this is impossible to do without violating at least one of the 5 fairly modest common-sense postulates.

There is no problem if voters or decision-makers are presented with only two alternatives. But when there are three or more choices, we can get intransitivity and “voter’s paradox”, in which A beats B and B beats C but C beats A. (See my essay on “Intransitivity”/intransitivity for examples.)

Collective decision-making depends on somehow aggregating the individual utility preferences into a group utility preference. Individual utility preferences can have intransitivities too, but the main problem is the question whether utilities can be compared inter-personally. Are utilities only on an ordinal scale (put into a rank-ordering), or on an interval scale (assigning numbers or “weights” to each preference)? That is, do I know only that I prefer candidate A to candidate B, and B to C, or could I actually say that A is 3 times preferable to B and B 1.5 times preferable to C? Different individuals may do this differently, and to aggregate this to a group decision becomes extremely difficult. Ordinary voting requires only ordinal preference ranking, which is one of its drawbacks; it does not indicate the depth of feeling with which some voters may prefer or reject some options or candidates. One may be 100 times or a 1000 times preferable, or some totally unacceptable; this strong preference is not distinguished in ordinary voting from a nearly random choice by the uninformed or uninvolved.

In a formal sense, Kenneth Arrow set out the following postulates that would have to be satisfied by a rational voting system: (1) Symmetry if roles are interchanged. (2) Invariance under positive linear transformation. (3) Independence from irrelevant alternatives. (The relative positions of A and B should not be changed if C is also running.) (4) If all are for it, it should be accepted. – Surprisingly, he found that all 4 postulates could be satisfied only if we accept the dictatorship of one person.
Or, formulated alternatively, if a 5th reasonable postulate requires that there should be no dictator, then there is no way at all to satisfy all 5 requirements.

Is democracy then doomed? Only if we insist on having a perfect world or none at all. Such absolutism is self-defeating. We need to be maximizers, not absolutists, i.e. do the best we can with what we have in an imperfect world. That means compromising on some of the postulates or requirements, or making trade-offs between them. We can then manage reasonably well with democracy, as we did before Arrow revealed the awful truth to us. “Practical reason” often finds a way when “pure reason” gets bogged down.

We do manage in our humble ways, but the impossibility theorems should make us aware of our limits, and not get too carried away with pride.