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I gave a lot of thought to what version of this problem I could acquaint
you with so that, without resorting to mathematical somersaults, the abyss
dividing the synthetic problem from the analytic one would become clear.
I finally allowed myself a compromise, in that I won't demonstrate how the
analytical problem is unsolvable
in principle
but rather only an easier
version, namely, that all the taxes in the world and all the time available in
our universe would by no means be sufficient to solve the analytical
problem for even relatively simple “non-trivial machines”: the problem is
“transcomputational,” our ignorance is fundamental.
This abysmal ignorance, this complete, fundamental ignorance is some-
thing that I've still never really seen presented at full strength, and that is
what I'd like to present to you today, so that we can get some insight into
the question of how, in the face of such fundamental ignorance, we can
concern ourselves with our problems? In the second chapter then I'd like
to sketch the development of recursive functors. I'll make it as easy and
playful as possible, so you can enjoy following these trains of thought. And
in the third chapter I'd like to speak about compositions, compositions of
functors, of compositions of systems.
First Chapter: Machines
I'll begin the first chapter by recapitulating a language that was actually
introduced by Alan Turing, an English mathematician, in order to leave the
long-windedness of deductively logical, argument to a machine, a concep-
tual machine, that would then turn all the wheels and buttons, so that one
only has to observe it: if one enters the problem on one side of the machine,
then the solution emerges on the other side. Once this machine has been
established, we have a language that can very easily jump from one well-
defined expression to another, and if you then want to know how this
machine works, you can always take it apart. Therefore: machine language.
This language is already current among you, but permit me, despite that, to
briefly repeat it, for, as I said, I'll need to concepts in a few minutes.
I come to my proposition:
01. Trivial machines: (i) synthetically determined; (ii) independent of the past; (iii)
analytically determinable; (iv) predictable.
A trivial machine is defined by the fact that it always bravely does the very
same thing that it originally did. If for example the machine says it adds 2
to every number you give it, then if you give it a 5, out comes a 7, if you
give it a 10, out comes a 12, and if you put this machine on the shelf for a
million years, come back, and give it a 5, out will come a 7, give it a 9, out
will come an 11. That's what's so nice about a trivial machine.
But you don't have to input numbers. You could also input other forms.
For example, the medieval logicians input logical propositions. The classi-
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