Re: Rigorous proof of natural numbers' properties (by Edmund Landau).

From: Jonathan (jbloom_at_hotmail.com)
Date: 07/01/04 

Date: Thu, 1 Jul 2004 21:21:25 +0000 (UTC)

Professor Ullrich:

Here is what I think you are saying about Landau, in slightly
different words. Please correct me if I am wrong.

1. Landau's proof is essentially valid.

2. However, Landau, in the context in which he is writing his book,
has a somewhat different attitude from the modern mathematician about
what it means to define an operation or to prove that an operation
exists and is unique.

3. To the modern mathematician, an operation is a particular set.
To prove the existence of an operation with certain properties, we
give an example of a set that has those properties. To prove
uniqueness, we show that if two sets have the
required properties, they are equal.

4. Landau (in the context of this book at least) is not viewing set
theory as the underlying substrate for mathematics. He views the
natural numbers as the primitive elements, and operations are rules
that give a unique natural number for any two natural numbers. This
definition of an operation, while less exact than the modern one
because it leaves vague what a "rule" is, is highly traditional.

5. In the context of this view, Landau's proof is valid. It provides
a definition for addition of any two natural numbers (viewed naively
as a rule yielding a number given any pair of numbers), shows that
addition thus defined has two properties, and then proves that this
is the unique operation (rule) with these two properties. All
further properties of addition proved in his book are based only on
the two properties.

6. The modern objections to Landau have little to do with the flow
of his logic, which is impeccable. They have more to do with the
ontology of the objects he is talking about. You do need more
logical apparatus to prove the existence of a function, and that is
what ZFC set theory is all about, and what a more modern proof like
Enderton's provides. But Landau's approach suffices to define
addition when addition is viewed naively as a rule that produces a
natural number given any two natural numbers, rather than as a set
with particular properties.

Any thoughts?

[By the way, it is clear from the introduction to Landau's book that
Landau grappled with the issue of how addition and multiplication
should be introduced. Apparently, the proof that was then traditional
was guilty of begging the question, and Landau replaced it with the
one he actually uses, which was developed by one of his assistants.
If anyone has more knowledge of the struggle that Landau describes in
his introduction about defining addition and multiplication, that
would be a useful addition to the discussion.]

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