  Gaisi Takeuti introduced Boolean valued analysis around 1974 to provide
systematic applications of Boolean valued models of set theory to analysis.
Later, his methods were further developed by his followers, leading to solving
several open problems in analysis and algebra. Using the methods of Boolean
valued analysis, he further stepped forward to construct set theory based on
quantum logic, as the first step to construct "quantum mathematics", a
mathematics based on quantum logic. While it is known that the distributive law
does not apply to quantum logic, and the equality axiom turns out not to hold
in quantum set theory, he showed that the real numbers in quantum set theory
are in one-to-one correspondence with the self-adjoint operators on a Hilbert
space, or equivalently the physical quantities of the corresponding quantum
system. As quantum logic is intrinsic and empirical, the results of the quantum
set theory can be experimentally verified by quantum mechanics. In this paper,
we analyze Takeuti's mathematical world view underlying his program from two
perspectives: set theoretical foundations of modern mathematics and extending
the notion of sets to multi-valued logic. We outlook the present status of his
program, and envisage the further development of the program, by which we would
be able to take a huge step forward toward unraveling the mysteries of quantum
mechanics that have persisted for many years.
