Incompatible Parameters in Quantum Mechanics and Beyond
In The Quantum Synthesis of the World (from hereon, QS) I provided a general framework in which the oddity of quantum mechanics can be accounted for and its mystery resolved. The key to that development was acknowledging the fact that the problems that have plagued generations of theoreticians and practitioners in this area are not physical, or even empirical, but metaphysical, and must be addressed in proper metaphysical garb. Specifically, I adopted Kant’s metaphysics of experience, transcendental idealism (somewhat revised), where knowledge and its objects can only be understood to occur within contexts set up by acts of synthesis. Given a particular episode of knowledge and its particular bunch of objects, the relevant act of synthesis is external to them—on their own terms, nonobjective—and a quantum (not necessarily physical) theory is one that attempts to grab and incorporate that very act, becoming to this extent indefinite, nonempirical, and open to various empirical ways of settling the indefiniteness.
This is as far as QS went. I am no physicist, I said there, so I am not going to supply the mathematical and experimental work that is the physicists’ province: it is enough if I show them what sense that work makes, or why in some cases it does not appear to make much sense. And yet, without leaving the metaphysical realm, there is something I can add, and I will do it here; more precisely, I will show the metaphysical relevance of what experimental work in quantum mechanics has been pressing upon us—getting, I am afraid, but small recognition.
Once again, Kant will be my guide. The word “transcendental” which is his term of art refers back to the Middle Ages, where predicates like “one,” “true,” and “good” were called transcendental in the sense of transcategorial: they applied across the categories. You always need an extra dimension to study a transformation of something: a plane to study a transformation of a line, a three-dimensional space to study a transformation of a plane, and transcendental predicates to study a transformation of categorial space. Same thing with the matters that I am now concerned with: to study the transformation of physics brought about by quantum mechanics, we need to transcend physics into metaphysics. Too close focusing on quantum mechanics itself, or even physics, will make us miss the larger lessons we should be learning, and resign ourselves too quickly to commonplaces quantum mechanics is urging us to contest.
What I intend to deal with is the fact that quantum mechanics contains incompatible (in the relevant jargon, non-commutative) parameters. In QS I used, by way of illustration of my view, a particle’s position and velocity: each was treated by a separate theory, the result of its own act of synthesis, and a quantum theory (in my sense) incorporated both. That a single theory cannot treat positions and velocities together would have seemed then the sort of physical detail that a metaphysical story cannot tell; but such is actually not the case—there is a lot of metaphysics in that. And, before I bring it in, I find it useful, as I did in QS, to extend my view to empirical non-physical theories: the revolutionary impact of quantum mechanics is best appreciated by seeing how far it goes.
Suppose I am trying to explain some human behavior: that A gets into his car and drives to the airport. One way of doing it would be physiological, talking about neurons firing in A’s brain, his nervous system being properly stimulated, his limbs getting the proper signals and moving accordingly. But we could also do it psychologically, in terms of A’s desire to get to the airport, his decision to use the car, and his consequent plan to leave at a certain hour. Here I have to be careful, for to a Kantian like me (as indeed I mentioned in QS) a mind is not an object, nor are objects its states or objective its acts, since none of this is in space; so words like “desire,” “decision,” and “plan” would have to be cashed out in terms of spatiotemporal behavior that can in principle be publicly (spatially) observed—the desire to get to the airport, for example, in terms of A’s making a note to himself that he will meet a friend who is arriving from abroad, and clearing his morning of any other commitments. With this important qualification, what I need to emphasize is that predicates like “firing” (of neurons) an “desire” are incompatible. You can have (potentially) a complete physiological explanation of A’s behavior, and a complete psychological one; but the vocabulary of the one will rule out the vocabulary of the other. A will be a human being, and will have various other features, in both, as a result of previous acts of synthesis; but the acts of synthesis setting up the contexts of the physiological and the psychological theories that explain his behavior will be mutually exclusive.
Leibniz refuted physiological reductionism by saying that, if you could enlarge the brain enough to be able to walk around in it, you would not see an idea there. That refutation still holds; so, when people say nowadays that they have found the centers of fear or love in the brain, they are misspeaking. They are illegitimately using terms that belong in a different synthesis and a different theory: they should get on with their physiology and refrain from soliciting catchy headlines. Their only decent attitude is eliminativism: “desire,” “decision,” “plan,” and all other predicates of “folk psychology” should (for them) be gotten rid of. Otherwise, to say that you have found (say) the center of fear in the brain can only mean that you have found something in the brain which, properly stimulated, elicits physiological responses described in a different, incompatible language as fear (and, importantly, without being able to relate those responses to the rich psychological environment described by that other language).
So incompatible predicates are current in the empirical sciences, and signal exclusive divisions between theories, and their attendant syntheses. With that in mind, let us now address position and velocity in physics; should they also be incompatible? Are they not perfectly compatible in classical mechanics, where we can assign a definite position and a definite velocity to (say) a planet at the same time?
The simple answer to the last question is No, and the argument to buttress this answer is metaphysical: not based on making observations and taking measurements but on logical reasoning. A conceptual framework that assigned both definite positions and definite velocities to bodies would be incoherent, just as the naïve set theory of Frege and Cantor was. The author who established this result was Zeno of Elea, in one of his celebrated paradoxes.
An arrow is shot through the air. The arrow is moving, so it must have a velocity. But consider now its position p at a given instant t; does it have a velocity then? No, answers Zeno, because in t the arrow is at p, and what is at a place is not moving, and cannot have a velocity. So, we could add, either the arrow is regarded as moving, and then it has a velocity but no definite position, or it is regarded as located in a definite position, and then it will have no velocity.
But, someone will hurriedly (and somewhat disdainfully) counter, we know from classical physics that in t, while at p, the arrow does have a velocity: an instantaneous velocity. And here is where quantum mechanics calls the bluff of its classical counterpart, with appropriate experimental evidence: its empirical data blow up the bad metaphysics of old. For there is no instantaneous velocity. As Kant said, time is not made of instants (and space is not made of points) but of times (and space of spaces), and you can make those times (and spaces) arbitrarily small, but they will still be times, with a length during which the arrow can have a velocity but also during which it will occupy several positions, however close they might be. So-called instantaneous velocity is supposed to be the limit of this process of making the times (and spaces traveled) smaller and smaller; but a limit does not exist in reality; it is an ideal reference; and that it makes no practical difference, or causes no practical damage, to do as if this ideal reference were real is no metaphysical justification for doing so. In conclusion, as indeed quantum mechanics urges us to recognize, nothing can have both a position and a velocity, independently of quantum mechanics: these two predicates are incompatible.