The term of “mathematical
hope” (espérance) had been introduced by Pierre-Simon de Laplace (1812), and
now it is replaced by “mathematical expectation” to designate the average of a
statistical quantity where its values are weighted by their corresponding
probabilities to occur.
One should restore the former
in the distinction between “hope”, where the probability is subjective, and
“expectation”, which would use objective probability, though their mathematical
formulas coincide.
That distinction generates two
different interpretations of the notion of quantity in quantum mechanics, which
can be thought both as “expectation” and as “hope” in the same formalism of
Hilbert space. It can be visualized in a series of alleged “paradoxes” such as
“Schrödinger’s cat”, “Wigner’s friend”, that of Einstein – Podolsky – Rosen, or
the many-worlds interpretation of quantum mechanics: all confronting our
everyday experience with quantum mechanics transferred in its terms: Subjective
and objective probability, hope and expectancy can and usually differ from each
other in the former, and the latter shows that they should always coincide,
generating comic discrepancies with common sense.
Furthermore the constant
coincidence of the mathematical “hope” and “expectation” and thus that of
subjective and objective probability in quantum mechanics requires the axiom of
choice to guarantee it in general. In particular this hints that ‘choice’ can
be defined mathematically as the relation and even ratio of the corresponding
“hope” and “expectation”, and quantum information as the quantity of those
quantum choices termed quantum bits (qubits).
There exist even discrepancies
between the mathematical hope and expectation in a quantum system just as in
our life. Einstein, Pododlsky, and Rosen emphasized it to found the ostensible
incompleteness of quantum mechanics. In fact those discrepancies are due to
entanglement, a fundamentally new phenomenon in quantum mechanics.
The distinction between “hope”
and “expectation” can be continued into other areas of philosophy.
Key
words: mathematical hope, mathematical expectation, subjective probability,
objecive probability, quantum mechanics, quantum information, entanglement
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