RE: [OPE] Dialectics for the New Century

wpc@dcs.gla.ac.uk

 Jerry

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Computation refers to "any type of information processing 
that can be represented mathematically." 

Formal logic is synonymous with mathematical logic: 
http://en.wikipedia.org/wiki/Formal_logic

Computation is not "more abstract than" mathematics:
rather, it is a a subject which _is_ mathematics and,
hence, utilizes formal logic.

Also, the very architecture of the computer is dependent
on the binary system - by definition, an expression of 
mathematical / formal logic. 

Paul

I think you are verging on Platonic idealism here, in which the ideal
mathematics underpins the physical reality of the computational system.
Computation is always everywhere a material process governed by material
laws, it is these material laws, the laws of physics, that give rise to
the possibility of mathematics.

"Calculi are rules for the manipulation of strings of symbols and these
rules will not do any

calculations unless there is some material apparatus to interpret them.
Leave a book on the

l-calculus on a shelf along with a sheet of paper containing a formula
in the l-calculus and

nothing will happen. Bring along a mathematician, give them the book and
the formula

and, given enough paper and pencil the ensemble can compute.
Alternatively, feed the

l-calculus formula into a computer with a lisp interpreter and it will
evaluate." (Are There New Models of Computation? Reply to Wegner and
Eberbach, Paul Cockshott and Greg Michaelson,, The Computer Journal 2007
50(2):232-247; doi:10.1093/comjnl/bxl062)

"And conversely, when we interpret

Turing's theorem as a statement about what can and cannot be computed in

physical fact, we are adopting some of his tacit assumptions about
physical

reality or equivalently about the laws of physics.

So where does mathematical effectiveness come from? It is not simply a

miracle, "a wonderful gift which we neither understand nor deserve" [17]

- at least, no more so than our ability to discover empirical knowledge,

for our knowledge of mathematics and logic is inextricably entangled
with

our knowledge of physical reality: every mathematical proof depends for

its acceptance upon our agreement about the rules that govern the
behavior

of physical objects such as computers or our brains. Hence when we im-

prove our knowledge about physical reality, we may also gain new means
of

improving our knowledge of logic, mathematics and formal constructs. It

seems that we have no choice but to recognize the dependence of our
math-

ematical knowledge (though not, we stress, of mathematical truth itself)
on

physics, and that being so, it is time to abandon the classical view of
com-

putation as a purely logical notion independent of that of computation
as a

physical process. In the following we discuss how the discovery of
quantum

mechanics in particular has changed our understanding of the nature of

computation."( MACHINES, LOGIC AND QUANTUM PHYSICS

DAVID DEUTSCH, ARTUR EKERT, AND ROSSELLA LUPACCHINI)

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