# Formal Systems and Machines:An Isomorphism

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10 Οκτ 2013 (πριν από 4 χρόνια και 9 μήνες)

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Formal Systems and Machines:

An Isomorphism

Peter Suber
,
Philosophy Department
,
Earlham College

Forma
l systems and digital computers are isomorphic. A computer is an
instantiation of a formal system, and a formal system is an idealization of a
computer. We already know the defining elements of a formal system; here's
how they map onto their computer count
erparts.

1.

The Formal Language
. In most computers today each memory address
contains one byte (eight bits). Each byte is a series of eight 1's and 0's.
Interpreted as numerals in base 2, these bytes code the 256 natural numbers
which in base 10 we repres
ent as 0, 1, 2,...255.

But the 1's and 0's are symbols of a
formal

language. Hence this numerical
interpretation is just one among many. Interpreting these bytes under ASCII
conventions, they code the Roman alphabet, upper and lower case, the 10
numerals
of base 10, the punctuation marks used in English, and other useful
(and useless) marks.

The
alphabet

of the formal language of the computer consists of the 1's and
0's. (Of course, this is just shorthand for speaking of "ons" and "offs", or pulses
above
and below a critical voltage threshold.)

The
grammar

that determines which strings of these symbols are wffs consists
of one simple rule: any string of eight bits is a wff. Hence each string or
numeral from 00000000 to 11111111 is a wff.

For rigor we cou
ld say there is a second rule of the grammar: only bytes are
wffs. The two rules together say that all and only bytes are wffs.

2.

Axioms
. The input to the machine plays the role of the axioms. Whether the
input comes from the user interactively or from a

file, it consists of a set of
bytes, or wffs of the formal language. Just as a formal system can have zero
axioms, a program can take zero input.

3.

Transformation rules
. The program running on the machine plays the role
of the transformation rules. The
program takes the input and transforms it
according to the rules coded in the program, and returns some output. Output
bytes generated by the program from the input bytes correspond to theorems.

To run a program on a computer is to generate output from th
e input by means
of the rules contained in the program. Input and output are expressed in bytes
of bits. In logical terms, this is to generate theorems from axioms by means of
rules of inference when the axioms and theorems are expressed as wffs of
symbols

of the formal language of the system.

Formal system

Digital computer

alphabet of the
language

1's and 0's (bits)

wffs

bytes (strings of 8
bits)

grammar of the
language

one rule: all and only
bytes are wffs

axioms

input

transformation rules

program

theorems

output

proof

computation

The consequences of this isomorphism are important for theory.

"Proof" corresponds to "computation". Hence demonstrable limits on provability
will translate into demonstrable limits on computability, and
vice versa
. We'
ll
what is the machine equivalent of Gödel's first incompleteness theorem? What
is the logic
-
systems equivalent of Turing's proof of the incomputability of the
halting problem?

A
nything that can be simulated by a computer can be simulated by a formal
system, and
vice versa
. Hence the attempt to write formal systems with
"intended interpretations" (e.g. plane geometry, set theory, predicate logic,
-
casting..
.) is analogous to the attempt to write
programs that simulate determinate aspects of reality. From this point of view
you can look forward to Gödel's incompleteness theorem and the Löwenheim
-
Skolem theorem: the first showed that formalizations or simulati
ons of a
certain strength always capture less than their intended interpretation, and the
second showed they are importantly ambiguous and always capture more than
their intended interpretation.

If computers can think, then their machine
-
states can be iso
morphic in relevant
ways with brain
-
states. If so, then these machine
-

and brain
-
states will be
isomorphic in relevant ways with theorems in some formal system. In this sense
"artificial intelligence" is a logical problem, even if "intelligence" is not log
ical.
If the "mind" is reducible to the brain and its states, then our mental life is the
output of hardware running software, or equivalently, the instantiation of a
formal system.
Which

formal system?

What is the significance of the fact that computers
can be programmed to
generate theorems in a formal system? Can formal systems be about
themselves? Can machines?