unlimited iterations (which FlooP allows). The Church-Turing Thesis is not
a provable fact in the sense of a Theorem of mathematics-it is a hypothesis
about the processes which human brains use.The Public-Processes VersionNow some people might feel that this version asserts too much. These
people might put their objections as follows: "Someone such as the Crab
might exist-someone with an almost mystical insight into mathematics, but
who is just as much in the dark about his own peculiar abilities as anyone
else-and perhaps that person's mental mechanisms carry out operations
which have no counterpart in FlooP." The idea is that perhaps we have a
subconscious potential for doing things which transcend the conscious
processes-things which are somehow inexpressible in terms of the
elementary FlooP operations. For these objectors, we shall give a weaker
version of the Thesis: one which di&tinguishes between public and private
mental processes:CHURCH-TURING THESIS, PUBLIC-PROCESSES VERSION: Suppose there is a
method which a sentient being follows in order to sort numbers into
two classes. Suppose further that this method always yields an answer
within a finite amount of time, and that it always gives the same answer
for a given number. Proviso: Suppose also that this method can be
communicated reliably from one sentient being to another by means of
language. Then: Some terminating FlooP program (i.e., general recur-
sive function) exists which gives exactly the same answers as the sen-
tient beings' method does.This says that public methods are subject to "FlooPification", but asserts
nothing about private methods. It does not say that they are un-FlooP-able,
but it at least leaves the door open.Srinivasa RamanujanAs evidence against any stronger version of the Church-Turing Thesis, let
us consider the case of the famous Indian mathematician of the first
quarter of the twentieth century, Srinivasa Ramanujan (1887-1920).
Ramanujan (Fig. 105) came from Tamilnadu, the southernmost part of
India, and studied mathematics a little in high school. One day, someone
who recognized Ramanujan's talent for math presented him with a copy of
a slightly out-of-date textbook on analysis, which Ramanujan devoured
(figuratively speaking). He then began making his own forays into the
world of analysis, and by the time he was twenty-three, he had made a
number of discoveries which he considered worthwhile. He did not know
to whom to turn, but somehow was told about a professor of mathematics
in faraway England, named G. H. Hardy. Ramanujan compiled his best(^562) Church, Turing, Tarski, and Others