Abstract
<p class="1Body">Unlike mathematics, in which the notion of truth might be abstract, in physics, the emphasis must be placed on algorithmic procedures for obtaining numerical results subject to the experimental verifiability. For, a physical science is exactly that: algorithmic procedures (built on a certain mathematical formalism) for obtaining verifiable conclusions from a set of basic hypotheses. By admitting non-constructivist statements, a physical theory loses its concrete applicability and thus verifiability of its predictions. Accordingly, the requirement of constructivism must be indispensable to any physical theory. Nevertheless, in at least some physical theories, and especially in quantum mechanics, one can find examples of non-constructive statements. The present paper demonstrates a couple of such examples dealing with macroscopic quantum states (i.e., with the applicability of the standard quantum formalism to macroscopic systems). As it is shown, in these examples the proofs of the existence of macroscopic quantum states are based on logical principles allowing one to decide the truth of predicates over an infinite number of things.</p>
Highlights
The Measurement Paradox in Quantum MechanicsAs it is known, originally the quantum formalism was only designed to explain phenomena occurring in the region of single electrons, photons and atoms
The present paper demonstrates a couple of such examples dealing with macroscopic quantum states
Unlike classical analysis that allows the abstraction of actual infinity and proof-independent truth commitments, a physical science must be enclosed within bounds of potential realizability and place the emphasis on hands-on provability of existence of any of its theoretical objects
Summary
Originally the quantum formalism was only designed to explain phenomena occurring in the region of single electrons, photons (i.e., the photoelectric effect) and atoms. It is possible to show that quantum states of an arbitrary macroscopic system cannot be effectively calculable, and their existence cannot be proved (verified) with any algorithmic procedure for obtaining a conclusion from the set of fundamental hypotheses of the quantum formalism With such a goal in mind, in this paper, we will give examples demonstrating that the application of the quantum formalism to at least some macroscopic systems implies principles that are known to be nonconstructive (that is, uncomputable). Readers interested in the constructive trend in mathematics can be referred to the corresponding literature; see, for example, an overview of constructivism given in both Troelstra and van Dalen(1988) and Beeson (1985)
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