by Henk Barendregt

Scope of computing

Some questions can be solved by computing. "What is the area of a circle with radius 4m?"

Answer: pi.4² m² = 50.2654824 m².

Also many qualitative questions may be answered by computing. "Are points (x₁,y₁), (x₂,y₂) and (x₃,y₃) in R² collinear,

i.e. do they lie on a straight line?"

Answer: if and only if (iff) (x₁-x₃)(y₂-y₃)=(x₂-x₃)(y₁-y₃).

Leibniz (1646-1716) expressed the hope that all well stated questions could be answered by computing (calculemus). He wanted to

1. create a precise language (lingua universalis) for the statement of the questions;

2. construct a machine that could answer all the questions posed in the language.

The first question Leibniz wanted to ask to such a machine is said to be "Does God exist?".

Quite daring around 1700 to ask this question to a machine!

For the statement of mathematical questions such a universal language has been constructed.

Set theory formulated in first order logic is an example.

Type theory is an alternative, which is better for computations.

Many mathematical questions can be solved by computer algebra systems like

the general systems Mathematica and Maple or some more specialised ones like Gap, Pari and Magma.

So there exist interesting approximations towards Leibniz's ideal.

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