Limits of Science?. John E. Beerbower
the probability that B was the winner (after the winner had been determined) did change with additional information.
Of course, we are now using probability in a somewhat different sense than we were using it when this discussion started. It is not the probability that something will happen in the future, like with the flip of a coin or roll of dice, but the probability of making the correct choice. The probability that C was the winner was either 0% or 100%, once the winner had been determined. The probability of selecting the actual winner changes with the increase of information.
This distinction is worth remembering. Later we will talk about theories that involve the probabilities that a particle is in a particular place or has particular characteristics and the probabilities that a particle will be in a particular place or will have certain characteristics. As we have just seen, those two similar-sounding propositions are actually rather different. The answers to both questions depend upon information. In both cases, one needs information relevant to the likelihoods of different future outcomes in order to calculate probabilities; but, in only one case, does information about what can be observed to have already happened become relevant.
The toss of a coin
There is one additional aspect of probabilities that I want to mention. There is a very great difference between the probability that out of 100 tosses of a coin, 50 will come up heads, and the probability that one will observe in the 100 tosses, 5 heads followed by 5 tails followed by 5 heads and so on (or that heads and tails will alternate perfectly). That statement is obvious. But, to generalize, any specific sequence of results will be highly improbable in advance. Yet, 100 tosses of a coin will necessarily result in some specific sequence of heads and tails, every time. Thus, whatever actually happens will have been highly unlikely to occur before it happens. Yet, it happens nonetheless, every time. That is just the way it is.
Usefulness Revisited
As we said above, mathematics reflects the formalization and extension of human logic or the process of rational thought. These extensions have become quite elaborate. Initially, mathematics reflected the analysis of “things” that people encounter is the physical world—“things” people experience through their senses. These “things” include numbers (as applied to quantities, counting and measurements) and shapes. Arithmetic and geometry became pretty sophisticated over time, revealing relationships and techniques that are not directly obvious.
However, mathematics evolved well beyond the practical or even the applicable.
Wigner, with some exaggeration, stated in his famous lecture that “Most advanced mathematical concepts, such as complex numbers, algebras, linear operations, Borel sets[,]…were so devised that they are apt subjects on which the mathematician can demonstrate his ingenuity and sense of formal beauty.” “The Unreasonable Effectiveness,” p.3. At least, we can say that mathematics is a human creation, that it has largely developed as an exercise in human reasoning and creativity and that it has often been pursued for the aesthetic and intellectual pleasures of the mathematician. See, e.g., Kline, Mathematics, pp.3–5; Barrow, Theories of Everything, pp.175–76; Wigner, “The Unreasonable Effectiveness,” p.7. As noted above, it is possible not only that mathematics is a creation of the mind, but of the human mind, with whatever lack of correspondence with or inability to see reality is inherent in the human mind.
These observations bring us back to the question of whether the human mind and its deductive theories can accurately reflect the structure of the physical world and, if so, why that is the case.20
As already suggested, science assumes that there are regularities in the physical world, since those regularities are what science attempts to understand. At the same time, it is clear that the world would not exist as it does, and we would not exist, if the natural world was not characterized by regularities—that the same things happen repeatedly, that certain phenomena regularly follow other phenomena, etc. As noted above, sometimes, such regularities are referred to as Laws of Nature. See Wigner, “The Unreasonable Effectiveness,” pp.4–6. Science has generally identified and expressed those Laws in the language of mathematics. Penrose explains that the use of mathematical models in science is necessary to obtain the precision required in the formulation of questions and “well-defined” answers, treating mathematics as a particularly precise or unambiguous form of communication. The Road to Reality, p.12.
Laws of Nature and ceteris paribus
As also noted above, scientists have generally assumed that such Laws consist of relationships that obtain regardless of time and place and, in fact, independently of a vast number of characteristics of any particular environment.21 Id. Remember the discussion of the ceteris paribus condition above. What needed to remain the same were only the characteristics relevant to or that had an effect upon the causal relationship being examined. Laws of Nature are relationships as to which the relevant ceteris paribus factors exclude much of the aspects of the natural world that our senses are capable of detecting. If that were not so, then our inherent limitations would prevent us from identifying those Laws. Simplification and selectivity are necessities for the human mind to cope with our exceedingly rich and complex physical environment.
I have discussed the ceteris paribus condition above. The terminology might sound a bit strange in this context. That is because the theories of the physical sciences are not normally set forth in such language. Instead, most of our theories in the physical sciences are presented (often implicitly) in terms of subsystems, in the context of which everything else is assumed to be irrelevant. This fact is apparent whenever one simply looks at a mathematical model of a theory; there are only a finite, even if sometimes very large (and sometimes very small), number of variables utilized in the model. Often, for essentially all practical purposes, everything else is deemed irrelevant (or, in some cases, can be controlled for). But, such an approach implicitly contains the ceteris paribus condition.
For example, the Newtonian Laws of Motion focus on the particular bodies of interest. In doing so, they effectively assume that, for example, the gravitational forces arising from all other matter in the Universe do not exist. Where those other forces are very, very small or do not have a systematic influence on the matter under examination (do not consistently bias the results one way or another), the assumption is satisfactory, because the resulting predictions are sufficiently accurate for the purposes at hand.
However, the model incorporating those Laws is simply not a full or, therefore, accurate representation of reality. Indeed, all theories and all Laws of Nature are inevitably only approximations, since they apply only to subsystems and exclude the influences of everything else in the Universe. Smolin, Time Reborn, pp.38–39.
Of course, many theories are also only approximations even within the subsystems to which they apply, because they do not fully or accurately capture even the relationships that exist within the subsystem. Over time we find that our approximations become better and also, often, more inclusive (such as General Relativity compared to Newtonian mechanics), but even if they come actually to reflect the real relationships within the subsystems, they are still approximations from the standpoint of the Universe as a whole. Thus, one could argue that such Laws cannot, by definition (since they apply to subsystems or parts of the whole), be truly universal or timeless. One could then conclude that this type of limitation to our scientific theories will necessarily continue to exist until we have a complete theory of everything, a theory that is able to incorporate everything in and about the Universe, including time. Id., p.40.
We shall return several times in the following pages to discuss issues that arise with respect to a potential theory of everything.For now, one might say that if nature did not obey certain simple Natural Laws, we would not be able to understand it. See Barrow, Theories of Everything, pp.10–11. Of course, such an observation proves very little: we think we understand some natural phenomena, so those phenomena must be obeying Natural Laws; but, if we do not really understand those phenomena, but just think that we do, we can