Allan Randall

The Everett interpretation of quantum mechanics has come under fire for inadequately dealing with the Born rule. Numerous attempts have been made to derive this rule from the perspective of observers within the quantum wavefunction. These are not really analytic proofs, but are rather attempts to derive the Born rule as a synthetic a priori necessity, given the nature of human observers. I show why existing attempts are unsuccessful or only partly successful, and postulate that Solomonoff's algorithmic approach to the interpretation of probability theory could clarify the problems with these approaches. The Sleeping Beauty probability puzzle is used as a springboard from which to deduce an objectivist, yet synthetic a priori framework for quantum probabilities, that properly frames the role of self-location and self-selection principles in probability theory. I call this framework "algorithmic synthetic unity". I offer no new formal proof of the Born rule, largely because I feel that existing proofs are already adequate, and as close to being a formal proof as one should expect or want. Gleason's one unjustified assumption--known as noncontextuality--is, I will argue, completely benign when considered within the algorithmic framework that I propose. I will also argue that, to the extent the Born rule can be derived within ASU, there is no reason to suppose that we could not also derive all the other fundamental postulates of quantum theory, as well. There is nothing special here about the Born rule, and I suggest that a completely successful Born rule proof might only be possible once all the other postulates become part of the derivation. As a start towards this end, I show how we can already derive the essential content of the fundamental postulates of quantum mechanics, at least in outline, and especially if we allow some educated and well-motivated guesswork along the way. The result is some steps towards a coherent and consistent algorithmic interpretation of quantum mechanics.

Those who interpret quantum mechanics literally are forced to follow some variant of Everett's relative state formulation (or "many worlds" interpretation). It is generally assumed that this is a rather bizarre result that many physicists (especially cosmologists) have been forced into because of the evidence. I look at the history of philosophy, however, reveals that rationalism has always flirted with this very idea, from Parmenides to Leibniz to modern times. I will survey some of the philosophical history, and show how the so-called paradox of quantum superposition can be considered a consequence of basic rationalist assumptions such as the principle of sufficient reason and the identity of indiscernibles.

An overview of Bradley's metaphysics and epistemology, which had much of the basic structure of quantum mechanics, but was all but ignored in the years following the formal quantum theories discovered by Heisenberg and Schrödinger. Bradley's version of absolute idealism was infected with the mentalism that was generally associated with idealism in the late nineteenth century. I develop his ideas from a standpoint somewhat more friendly to modern formal methods, although this is not much of a stretch, as Bradley had already taken absolute idealism strongly in that direction, if not all the way.

Starting with the Descartes' cogito, "I think, therefore I am"--and taking an uncompromisingly rational, rigorously phenomenological approach--I attempt to derive the basic principles of recursion theory (the backbone of all mathematics and logic), and from that the principles of feedback control theory (the backbone of all biology), leading to the basic ideas of quantum mechanics (the backbone of all physics). What is derived is not the full quantum theory, but a basic framework--derived from a priori principles along with common everyday experience--of how the universe of everyday experience should work if it operates according to rational principles. We find, to our surprise, that the resulting system has all the most puzzling features of quantum physics that make physicists scratch their heads. Far from being "bizarre" and "weird", as is usually thought, the strangest paradoxes of quantum theory turn out to be just what one ought to expect of a rational universe. It is the classical, pre-quantum universe of the nineteenth century that has irrational, mystical components. The quantum-mechanics-like theory that is developed is, furthermore, most compatible with the strictest, most uncompromisingly rationalist of the standard interpretations of quantum mechanics, those which add no ad hoc elements to the theory, and which generally trace their history to the relative state formulation of Everett (also called the "many worlds" interpretation). These interpretations take the universe to be quite literally describable as a quantum wavefunction. As with any project this far-reaching in scope, I confess I have had to make some working assumptions along the way. I have attempted to isolate these, and clearly label them as points of possible future revision--they are marked in the text with an asterisk (*).

Plato's theory of forms is developed and compared to the modern theory of recursion. I show how Plato's theory, as it applies to mathematical objects, is essentially a primitve version of modern recursion theory, which has all the essential elements of the ancient theory. However, Plato himself thought there was more than mathematics to his forms. He believed that form had a noncomposite, unanalyzable component. So, while recursion theory provides an adequate formalization of Plato's theory, it cannot be considered identical to it. I argue--drawing from material in the "Meno," "Phaedo," and "Republic"--that Plato's arguments for noncomposite form are largely fallacious. Plato would, I believe, have taken the computational version developed here seriously, since mathematics was his primary source for clear examples of forms (one could argue that it was his only source short of dipping into mysticism). In fact, there is a long-standing oral tradition that Plato developed a more formal version of his theory in lecture notes for courses he taught at the Academy (the university he founded in Athens). Any such notes, if they existed at all, have been lost. But if such a formal version did exist, it is tantalizing to wonder to what extent Plato may have anticipated modern theoretical computer science and metamathematics.

Wittgenstein's philosophies, from both the Tractatus and the Philosophical Investigations, are explained and developed. Wittgenstein uses a primitive version of recursion theory to develop his attempt at a purely logical metaphysics in the Tractatus. However, due to his implicit materialist assumptions, he could not make the system completely logical, and built in a mystical division of possible worlds into the true and the false. This incoherence eventually lead him to reject logic as a method for doing metaphysics, and indeed to reject metaphysics entirely. I argue that his move from the Tractatus to the Investigations was valid, but only given his materialist assumptions. If he had been willing to drop this unnecessary baggage, recursion would have played a very different role in his system, since he would then have had no need to separate static objects from processes, which he saw as purely mental. F.H. Bradley developed such a nonmaterialist metaphysics in the nineteenth century, but was crippled by a mentalism that Wittgenstein was free of. The anti-mentalism and anti-materialism that Wittgenstein considered as given were not so obvious to his predecessor, Russell, who revolted against Bradley's idealism in much the same way Wittgenstein ended up revolting against Russell's logical atomism. In my view, none of these positions was the right approach, which would require nonmentalism and nonmaterialism. But for some reason, these things (which seem to go together quite naturally to me) have been widely considered to be incompatible. Bradley was appropriately a non-materialist, but suffered from mentalism. Russell and the early Wittgenstein were appropriately nonmentalists, but suffered from materialism. The later Wittgenstein was, I would argue, still an ardent materialist and anti-mentalist, in spite of the fact that he had long since realized the contradictions to which materialism leads; he just had not recognized that it was his materialist assumptions that had lead him there, since these assumptions were so firmly engrained in his thinking as to be invisible..

The many worlds anthropic principle is explored here from the a priori perspective of rationalist metaphysics, within the framework of modal logic. It is shown how the apparent contradictions of quantum superposition can be thought of in terms of different levels of world models. The framework of modal logic is used, but given the rationalist assumption that all possible worlds exist. There is thus no absolute distinction between possibility and necessity. To take the point of view of a conscious being in a world, however, is to adopt some such distinction--something we must do in order to do physics. This paper is intended to lay the groundwork for future attempts to develop theories of what is necessarily true in a world with conscious entities. It also contains some tentative speculations on the difficult issue of death and quantum mechanics.

A survey of Kant's views on space, time, geometry and the synthetic nature of mathematics. I concentrate mostly on geometry, but comment briefly on the syntheticity of logic and arithmetic as well. I believe the view of many that Kant's system denied the possibility of non-Euclidean geometries is clearly mistaken, as Kant himself used a non-Euclidean geometry (spherical geometry, used in his day for navigational purposes) in order to explain his idea, which amounts to an anticipation of the later discovery of the general concept of non- Euclidean geometries. Kant's view of geometry and arithmetic as synthetic was, I believe, essentially correct, in that geometry and arithmetic are both synthetic a priori if considered as branches of mathematics independent of the rest of mathematics. However, the view that somehow logic is analytic, while mathematics is synthetic for Kantian reasons, is mistaken. All three disciplines—logic, arithmetic and geometry—are synthetic as disciplines independent from one another. However, they have a common basis, recursion theory, which I prefer to identify with mathematics as a whole. As a result, I do not say, as is often considered to be the Kantian view, that mathematics is synthetic while logic is analytic. Rather, I prefer to say that mathematics is analytic, while logic is synthetic. This is perfectly consistent with Kant's system, since it was arithmetic and geometry individually that he argued were synthetic. What Kant called the analytic is recursion theory, which could be considered as a basic formulation of mathematics or logic—or better, both mathematics and logic could be recognized as essentially the same discipline. However, if "logic" is taken to mean "predicate logic", as is often the case in modern times, then it is mathematics that is closer to Kant's analytic, not logic. Such ambiguities, of course, can be avoided by simply associating Kant's analytic with recursion theory, and avoiding the controversies as to what counts as mathematics or logic..

While the essays on this web site, taken together, explain most of the essentials of my metaphysical system, some material is not covered, and the different essays take quite different approaches. The essays were mostly written for undergraduate and graduate courses in philosophy at the University of Toronto and York University. Thus, each essay is slanted to the issues that were addressed in whatever course it was written for. However, I hope soon to pull all this material together into a cohesive, single work that will explain my system more fully and with more focus, which will serve as my Ph.D. dissertation. In the meantime this brief overview gives the broad picture of what my philosophical system is, and how the various essays on these pages are inter-related. For those who do not wish to read through this entire page, the Quantum Phenomenology essay is probably the one you should read if you only have time to read one, as it is the closest to being comprehensive.