[OPE-L:1439] Re: Math and numbers

Gilbert Skillman (gskillman@mail.wesleyan.edu)
Mon, 11 Mar 1996 14:47:14 -0800

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As seen in my following remarks, I feel that Alan and I are talking
at cross purposes. He seems to suggest that I was defending a
certain paradigm or analytical approach, which was certainly not the
case; rather I was just pointing out that numerical examples, to be
meaningful, must have some context, if only implicit. The process of
modelling only makes one make implicit assumptions explicit. As in
everything, of course, this can be done clumsily or incompletely, but
that really is beside the point.

Alan replies to my comment:

> 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."

as follows:

> 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.

I don't think that this comparison is relevant to what I said. The conditions
which support the arithmetical properties of numbers are not at issue
here. Rather, my point is that numerical examples are based
necessarily on some implicit*economic* model which is in the example-
giver's mind. Alan, for example, has a particular vision of the
economic world to which Marx speaks in Ch. 5 when he offers numerical
examples with respect to issues raised in that chapter. Paul, in a
separate post, lists some conditions that might be presumed in such
examples. My point is that the modelling process simply forces one
to make implicit assumptions explicit.

> 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.

At mimimum, enough assumptions have to be adduced to "close" the
model. In any case, this is necessarily more assumptions than are
made explicit in offering numerical examples, without other explicit
context. In other words, Alan's critique here is absolute rather than
comparative, as in the sense of my original remarks.

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

I see no reason why simultaneous models must presume any of these
things, as a general rule. But I wasn't writing to defend
simultaneous models, so I don't see what this has to do with what I

> 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.

That's legitimate; but such examples are based on an alternative
model, some of which is not specified. The alternative to this is
if, as in Alan's examples below, numerical examples are concocted
with respect to an *already given* model, in which case the unaltered
assumptions have already been made explicit.

> I am becoming alarmed that we are still
> awaiting a reply to these examples, some of which have been posted
> three times now.

Imagine my reaction, then, when innumerable posts of mine have failed
to elicit general agreement about the manifest invalidity of Marx's
conclusion to Ch. 5, which for example has nothing to do with Marx's
notion of the value of money or whether new value can be created
solely in the process of exchange (of course it can't, by Marx's

> 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
> models.

What I wrote contradicted neither of these claims. Really, the point
was a simple one: Alan championed numerical examples over the
process of formal model development, to which I responded that the
numerical examples (if they are to be considered relevant to
anything!) must have reference to some vision of how the world works,
which has not been spelled out.

> 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.

This has nothing to do with my point.

> 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 has nothing to do with my point. For example, I wasn't defining
"simultaneous constructions".

> 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.

If you mean "test" in the sense of "check deductive validity",
careful deductive reasoning based on a model's postulates renders such
"tests" redundant, if possibly comforting to those who don't follow the
formal argument. If you mean "test" in the sense of empirical test,
this has nothing to do with my point. If you mean "test" in the
sense of "evaluate the generality or plausibility of given
assumptions", I agree with this assessment, but it has nothing to do
with my point, since the numerical examples presume the given model

> But the reverse does not apply. You don't test numbers on models.

I didn't say you should. But numbers without (possibly implicit)
context have no meaning whatsoever.

> 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'.

Such obscurantism is bad, but it has nothing to do with what I said.

> 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.

Star Chamber? What?

In puzzlement, Gil