Taking it to the limits

John Barrow's Impossibility is a sweeping and ambitious book that explores a wide range of mathematical, natural and social phenomena. The book focuses on types of impossibility, ranging from whether there exist limits to what human beings can discover about nature (impossibility of learning about certain aspects of the origins of the universe or other natural phenomena), to the computational intractability of certain classes of mathematical problems (impossibility of finding solutions to well-posed mathematical problems given time constraints), to the incompleteness of first-order arithmetic (impossibility of proving statements that are nevertheless true in certain formal systems), to the non-existence of any way to aggregate individual preferences through voting such that certain desired criteria are preserved (impossibility of constructing a mathematical function with certain properties). No simple definition can capture the various types of impossibility discussed. The book can be read either as an exposition of the ways in which limits to knowledge, logical impossibilities and the like are endemic to human inquiry or as a survey of many recent scientific developments.

Despite or perhaps because of its ambitions, Impossibility is ultimately a failure both as an exposition of modern scientific developments and in terms of its overarching themes. By considering such a vast range of topics, Barrow's interesting discussions of cosmology - his area of expertise - are forced to coexist with a hotchpotch of superficial, often inaccurate descriptions of areas in which the author is presumably a relative layman. Further, the book fails to produce a coherent argument that impossibility is an interesting meta-theme in science and mathematics.

The failures of the book in terms of scientific exposition are most easily seen in its discussions of impossibility in socio-economic contexts. In this the author's claims are often incorrect and quite generally reflect a deep misunderstanding of the social science that he discusses. Three examples are particularly egregious. The first concerns the effect of public-opinion polls on voting. An old question in political science asks whether such a poll could ever be accurate, because knowledge of public-opinion polls can influence individual preferences over candidates.

Formally, the question is whether the feedback effect of a poll on public opinion ensures that the poll will, once announced, no longer truly reflect public preferences. That there is no logical reason why the influence of polls on public opinion should lead to inaccuracy was demonstrated through a simple mathematical analysis by Herbert Simon many years ago. Barrow goes to great lengths to dismiss Simon's argument on the grounds that it made a mathematical assumption that does not hold. This completely misses the point: Simon's research was designed to show that there is no logical inconsistency between the availability of polling information and polling accuracy. This consistency (possibility in Barrow's language) is unaffected by the modelling assumption Barrow chooses to criticise.

Further, Barrow appears unaware that Simon's 1954 analysis is the precursor of a vast modern literature in economics on "rational expectations", in which a standard question is whether, in an economic system in which individuals' expectations affect behaviour, it is possible for expectations to be self-consistent, ie for belief-driven behaviour to be such that the beliefs are correct. Barrow's vague allusions to arguments of philosophers such as Karl Popper or Donald Mackay about free will and their implications for socio-economic phenomena are no substitute for formal analysis of socio-economic systems; even an elementary analysis would have indicated that these philosophical claims do not augment, let alone invalidate, what social science has to say about self-consistent beliefs.

Another example of misunderstanding concerns the Arrow impossibility theorem. This states, in essence, that under certain conditions of rationality and equality, it is impossible to guarantee that a ranking of societal preferences will correspond to rankings of individual preferences. Unfortunately, Barrow asserts the real-world consequence of this theorem with a description of the 1980 Senate election in New York, in which the presence of a third candidate caused the Democratic nominee to lose even though public opinion polls indicated that she would have beaten either candidate head to head. This example has nothing to do with Arrow's theorem.

A final example is Barrow's remark that "the simple models of idealised equilibrium economies that seem to pervade the study of economics all fail to capture the essence of self-organisation". This claim, as far as I can tell, is vacuous. There are many models in economics that can generate fluctuations at any and all frequencies, non-linear behaviour, spatial structure and uncertainty. (Whether these models are empirically accurate or conceptually useful is a distinct question.) And there is no established empirical fact about the economy that corresponds to "self-organisation". So on what basis is Barrow in a position to make such dismissive comments?

These examples of poor understanding of social science are linked to a more general failing, which is the tendency to accept at face value (and sometimes out of context) the claims of any paper, published or yet to be peer-reviewed, that furthers the author's arguments. In this, the book does a disservice to the lay-reader because it fails to clarify the distinction between theories that are now part of received scientific understanding as opposed to those theories that are best thought of as conjectures about the world and have yet to be corroborated.

One example of this failure is the discussion of self-organised criticality, which refers to a class of models of the evolution of certain types of non-linear and interdependent behaviour; it was introduced to study phenomena such as earthquakes, and it has recently been employed in theoretical models of species extinction, traffic and economic fluctuations. Barrow writes of claims of the empirical importance of self-organised criticality without any attention to the fact that in the biological and social contexts there is no evidence that the substantive phenomena under study are actually characterised by it. There is a vast difference between conjecturing that a particular mathematical model explains a particular empirical phenomenon and corroborating that it actually does so, but this distinction is lost in Barrow's exposition. This point is not merely theoretical. David Raup has recently shown that models of self-organised criticality do not satisfactorily explain extinction data. In fact, the major limitation of complexity theory is the failure of specific instantiations of the theory such as self-organised criticality to come to grips with empirical phenomena in a way that is persuasive.

But there is a more general failure in the book. It provides little coherence in its lumping together of different types of impossibility. Arrow's impossibility theorem, the "liar's paradox" and the implications of the inflationary model of the big bang for limitations on the observability of the universe may all reflect some aspect of the meanings of the word "impossibility", but this does not mean it is an interesting organisational concept. After all, the sentence "Robert is both dead and alive" and the claim that string theory is untestable because technological constraints will always preclude generating sufficient energy to produce decisive experiments may both embody some connotation of impossibility, but the connotations are wholly different once one peels away the common word "impossibility" to ask what is actually going on. Nothing in Impossibility indicates how one connotation of impossibility elucidates another.

Beyond this, there is the question of the coherence of the analysis of impossibility itself. Much of the book addresses the question of whether we are at the "end of science", by which writers typically mean that either the rate or the magnitude of scientific advances will slow or cease in the foreseeable future. Barrow reviews many of these claims, ranging from the practical (the costs of experimentation are or will become too high to permit falsification of theories) to the fundamental (our scientific picture of the world is sufficiently accurate that there will be no new revolutions). While he is circumspect about whether he regards these claims as correctly showing that there are limits to science, he does not address whether such claims can be meaningfully assessed.

Within economics, there is a distinction introduced by the economist Frank Knight between risk and uncertainty. Risk characterises those environments in which outcomes are not predictable yet in which probabilities can be assigned to the various possible outcomes. Uncertainty is used to describe environments in which one is unable even to assign probabilities to possible outcomes. This distinction has proven useful in understanding technical change, where innovators face a situation in which it is impossible meaningfully to assign probabilities to various outcomes of research. The same issues arise in the context of scientific advances. Scientific advances such as relativity and quantum mechanics have rested on a reformulation of the vocabulary and syntax by which we describe the world, and hence cannot be subjected to an assessment as to their likelihood.

Similar considerations apply to Barrow's suggestion that the physical and cognitive limitations of Homo sapiens may pose a barrier to some kinds of scientific understanding. Such speculation is intrinsically unpersuasive - these alleged limitations have not prevented the development of such counter-intuitive (relative to human experience) ideas as those that undergird quantum mechanics. Even granting that these limitations currently exist, how can one assess the impact of future technological developments, of which the computer revolution is the most obvious example, in relaxing those limitations in the future?

In some respects, the meaninglessness of speculations about the future of science is apparent in Barrow's discussion of evolution and natural selection. As he correctly points out, the theory of evolution does not permit detailed prediction as to what sorts of species one should expect to see over long-time horizons. The process is too sensitive to random perturbations and occurs in too large a space of possibilities to permit prediction. These same factors suggest that efforts to predict the future of science will be equally unsuccessful.

While many of the types of impossibility that Barrow describes are of great intellectual importance within the fields where they arose, this does not mean that impossibility is of particular interest as an overarching perspective on different areas of study. The lessons of Impossibility about impossibility may very well be twofold. First, the natural and social sciences have become too complicated to permit any one person to arrive at a deep enough understanding of them to produce sweeping interdisciplinary surveys that do justice to the fields involved. Second, determination of many conjectured types of impossibility may itself be beyond the limits of human inquiry.

Steven Durlauf is professor of economics, University of Wisconsin at Madison, United States.