[OPE-L:1410] Math and numbers

Alan Freeman (100042.617@compuserve.com)
Sun, 10 Mar 1996 17:44:37 -0800

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Gil [1397] says

"in using numbers one is making implicit reference to an
underlying set of postulates; the process of modelling simply
forces one to make these assumptions explicit. And jolly
good, too."

This does not sound quite right.

The statement that 2+2=4 can be supported with a large number
of different axiomatisations of number theory.

Since there is no single set of 'implicit' underlying
postulates behind the statement that 2+2=4, you cannot claim
that such an underlying set of postulates exists. After all,
people counted long before they invented number theory.

Second, constructing a model does not - unfortunately - force
anyone to make their assumptions clear. It only requires them
to expose enough assumptions to make the model function. Many
additional postulates can be absent, particularly those needed
to apply the model to reality. Sometimes not even the model
makers realise they are there.

For example all simultaneous models implicitly assume universal
market clearing, constant prices and no technical change but
very few simultaneists admit this.

Usually the only way we can bring out these implicit postulates
is to confront a model with numerical examples which break its
restrictions, as I do. I am becoming alarmed that we are still
awaiting a reply to these examples, some of which have been posted
three times now.

Therefore, your statement to me sounds back to front. The
problem is surely not to use models to impose assumptions on
data; it is to use the data to reveal the assumptions behind

That is why I deliberately construct numerical examples in
which prices change while production is in progress, precisely
because an *implicit* assumption of the simultaneist
construction is that this cannot happen. When we apply the
simultaneist construction to these examples, it breaks down and
the hidden assumptions become clear. Without the numbers,
this assumption would never become apparent.

Those who defend the simultaneous construction systematically
refuse to confront the numerical examples which illustrate their
contradictions. Instead they take refuge in the internal perfection
of their models.

This is unacceptable.

A good logician constantly tries to break her or his model by
throwing numbers at it which violate its implicit assumptions,
to see what happens. You test models on numbers.

But the reverse does not apply. You don't test numbers on models.
Such an idea quickly leads in my experience to the idea that data
or examples somehow 'don't count' unless the model behind them
is explicit. This is just obscurantism; another way of saying
'don't blind me with facts'.

If an apple falls, an apple falls, and if two numbers add up then
two numbers add up, and these are facts to be explained, without
asking the apple or the numbers to account for their actions in
some Star Chamber of acceptable postulates.