[Agda] Classical Mathematics for a Constructive World
m.escardo at cs.bham.ac.uk
Tue Nov 27 23:20:06 CET 2012
There is also Peter Hancock's question. Well, probably not a question.
But I lost your message, Peter, and I can't find it.
My quibble with the so-called Church's Thesis in constructive
mathematics is that ... I never need it when I prove a theorem. Knowing
that my input is computable doesn't help me in my calculations. I have
no need to inspect the source code of the input, or of merely knowing
that the source code exists.
Moreover, I would think that many inputs come without evidence of
Example 1. Suppose I program a certified web server in Agda. My input is
a stream, and so is my output. If I assumed CT, the protocol would have
to work like this: please, before sending me the stream, send me its
Godel number, or promise that it secretly exists, and send me evidence
of your promise.
Example 2. Suppose I program a certified flight control system for
airports. Again my input and output are streams. Assuming that the
inputs come with their Godel numbers, there is no need for radars or
other tracking devices. I can compute what will happen from the Godel
number, and I can effectively predict which planes will be where and
when by just computing. Wouldn't that be marvelous?
Example 3. Suppose I program a certified nuclear power station control
system. One of my stream inputs comes from a Geiger counter. I ask the
physicists and engineers to design and build an improved Geiger counter
that first gives the Godel number of the sequence that it produces.
What do you need to know that your input if computable for? And what if
your input is not (known to be) computable?
Should you implement a validation procedure that rejects non-computable
More information about the Agda