Thinking and Computation - Thinking as Computation - page 12

1.3.1 Leibniz and his idea
The idea that thinking can be seen as a kind of computation is one of the rare ideas in
Western culture that does not go back to the ancient Greeks. The first person to take
this idea seriously was the German philosopher Gottfried Leibniz (1646-1716).
Leibniz was an amazing thinker. Among many other ideas and discoveries, he
invented the calculus at the same time as Isaac Newton did. While Newton was inter-
ested in problems in physics and chemistry, Leibniz was more interested in symbols
and symbol manipulation. He only started doing mathematics seriously later in life.
But intrigued by how symbols standing for variables and constants could be shuf-
fled around to solve equations in algebra, he wondered whether there were symbolic
solutions to problems involving tangents and areas. And the infinitesimal calculus
(derivatives and integrals) came out of this.
When it came to arithmetic, Leibniz observed that it was sufficient to manipulate
symbols on a piece of paper according to certain rules to be able to draw conclusions
about otherwise abstract numbers. A number (like fourteen, say) might be a com-
pletely abstract notion, but the symbols used to represent it (like the symbols 14 or
XIV or 1110 ) are much more tangible: we can write them down, look at them, move
them around. We can determine if a certain relation holds among these numbers (for
example, determining whether a number is the sum of two others) just by manipu-
lating the symbols. (The symbols 1110 represent the number fourteen in the binary
number system invented by Leibniz that is used by digital computers.)
His idea then was this: Ideas , that is, the objects of ordinary thought, are like num-
bers. It will be sufficient to manipulate symbols standing for them according to certain
rules. The ideas may be abstract, but the symbols are concrete. One will be able to go
from one idea to the next just by doing symbolic manipulation.
In other words, he drew the following analogy:
The rules of arithmetic allow us to deal with abstract numbers in terms of con-
crete symbols. The manipulation of those symbols mirrors the relations among
the numbers being represented.
The rules of logic allow us to deal with abstract ideas in terms of concrete sym-
bols. The manipulation of those symbols mirrors the relations among the ideas
being represented.
What a truly remarkable idea! It says that although the objects of human thought are
formless and abstract, we can still deal with them concretely as a kind of arithmetic,
by representing them symbolically and operating on the symbols.