Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

the danger of speculating about infinity 173In considering a novel hypothetical theory that is being proposed, one should thereforedistinguish between two different levels. On the one hand, we must ask whetherthe novel theory is “physically correct” or not. A consistent theory may very well turnout to be physically wrong: the only reliable judge we know is empirical evidence.The strength of natural sciences has always been the use of empirical evidence as ameans to discard theoretical hypothesis. From this point of view, we must take mostof modern physics speculations with great care: nothing tells us that there really aremultiple universes, strings, loops, or the like, and we should definitely not make harddeductions on the basis of the reality of these objects.On the other hand, any new theoretical construction is also a possible new modelfor thinking about the world, and the very fact that a model of this sort is possible hasdeep consequences for our understanding of the universe. In particular, the existence ofa coherent conceptual scheme can disprove facile deductions based on what we know,or what we think we know.Back to our problem, the possibility that space is not infinitely divisible, realizedby loop quantum gravity, bears strongly on all discussions on infinity in the small.The possibility that the universe is boundless but finite, implied by cosmology, bearsstrongly on all discussions on infinity in the large.These possibilities indicate that the difficulty with infinity might very well be in ourerroneous formulation of the problem, or in our erroneous hidden assumptions, and notin the nature of things. Irrespective of which quantum gravity will ultimately turn out tobe confirmed by experiments, the very existence of the conceptual possibility that spaceis not infinitely divisible and not infinitely extended should deter us from taking ourown “natural” intuitions about infinity as solid. Science has constantly taught us thatwhat appears “obvious” or “intuitive” to us is often just a prejudice or, more precisely,an unwarranted extrapolation.7.4 The Danger of Speculating about InfinityThe two cases illustrated in the preceding section cover the two major examples ofinfinities discussed in the history of thinking: infinity toward the small, namely, infinitedivisibility of space; and infinity toward the large, namely, infinite extension of space. Inboth cases, the discussion shows that the traditional arguments put forward in supportof the existence of these infinities can very well be wrong. In fact, they are wrongbecause they implicitly rely on intuitions and ideas that we have about physical space.To extend these intuitions and ideas too far is an illegitimate extrapolation.And what if infinity was always nothing else than the place where our illegitimateextrapolations go wrong? My own impression is that it could very well be so. The twocases illustrated, indeed, undercut one of the sources of our confusion with infinity.They warn us that the source of the problems raised by our intuition about infinitymight be in the limitation of our intuition, not in the nature of things.I certainly do not want to suggest that these examples provide a general solution tothe problem of the infinite, and I am not claiming to be a depository of the correspondingknowledge. Precisely to the contrary, I would like to express a warning against facileargumentations, based on our own certitudes or intuitions, that might be disproved bythe acquisition of a slightly more extended knowledge.