Laszlo E. Szabo

In 1935, Einstein, Podolsky, and Rosen (EPR) published an important paper in which they claimed that the whole formalism of quantum mechanics together with what they called a “Reality Criterion” imply that quantum mechanics cannot be complete. That is, there must exist some elements of reality that are not described by quantum mechanics. They concluded that there must be a more complete description of physical reality involving some hidden variables that can characterize the state of affairs in the world in more detail than the quantum mechanical state. This conclusion leads to paradoxical results. As Bell proved in 1964, under some further but quite plausible assumptions, this conclusion that there are hidden variables implies that, in some spin-correlation experiments, the measured quantum mechanical probabilities should satisfy particular inequalities (Bell-type inequalities). The paradox consists in the fact that quantum probabilities do not satisfy these inequalities. And this paradoxical fact has been confirmed by several laboratory experiments since the 1970s. Some researchers have interpreted this result as showing that quantum mechanics is telling us nature is non-local, that is, that particles can affect each other across great distances in a time too brief for the effect to have been due to ordinary causal interaction. Others object to this interpretation, and the problem is still open and hotly debated among both physicists and philosophers. It has motivated a wide range of research from the most fundamental quantum mechanical experiments through foundations of probability theory to the theory of stochastic causality as well as the metaphysics of free will.

It is widely believed that the principal difference between Einstein's special relativity and its contemporary rival Lorentz-type theories was that while the Lorentz-type theories were also capable of “explaining away” the null result of the Michelson-Morley experiment and other experimental findings by means of the distortions of moving measuring-rods and moving clocks, special relativity revealed more fundamental new facts about the geometry of space-time behind these phenomena. I shall argue that special relativity tells us nothing new about the geometry of space-time, in comparison with the pre-relativistic Galileo-invariant conceptions; it simply calls something else "space-time", and this something else has different properties. All statements of special relativity about those features of reality that correspond to the original meaning of the terms "space" and "time" are identical with the corresponding traditional pre-relativistic statements. It will be also argued that special relativity and Lorentz theory are completely identical in both senses, as theories about space-time and as theories about the behavior of moving physical objects.

Realists, Platonists and intuitionists jointly believe that mathematical concepts and propositions have meanings, and when we formalize the language of mathematics, these meanings are meant to be reflected in a more precise and more concise form. According to the formalist understanding of mathematics (at least, according to the radical version of formalism I am proposing here) the truth, on the contrary, is that a mathematical object has no meaning; we have marks and rules governing how these marks can be combined. That’s all.