Re: Testable Predictions by HdB
- From: "Jesse F. Hughes" <jesse@xxxxxxxxxxxxx>
- Date: Wed, 28 Sep 2005 20:32:10 +0200
Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> writes:
> Jesse F. Hughes wrote:
>
>> You have some theorems of the form "For all x, P(x)" where P is
>> quantifier free, right?
>
> Right (I suppose).
>
>> You suggest that these theorems generate testable predictions, namely
>> P(b) for each number b.
>
> Maybe I suggested that. But since modern philosophy doesn't understand
> common speech very well, we have to be _extremely_ careful. Therefore:
>
> The numbers in the Theory are real numbers: r \in R .
> The numbers in the Tests are: floating point numbers.
> But:
> Floating point numbers <> Real numbers
>
> For example. Real numbers always obey the triangle inequality, while
> floating point numbers often do not. See Dik T. Winter's hobby horse:
>
> http://groups.google.nl/groups?q=sci.math+%22Dik+T.+Winter%22+inequality&hl=nl
>
> We conclude that, strictly speaking, the "numbers" of the Theory are
> NOT the "numbers" of the Tests. This makes the Testable Predictions
> somewhat more complicated: they say that P'(b') for each "number" b',
> whih is thus _not_ the same as P(b) for each number b.
>
> Of course, there exists some relationship between the Theoretical P(b)
> and the Testable P'(b'). They are not completely ambiguous properties.
> But this relationship is highly non-trivial, if you want to care about
> the details. (Most of us, Applied, don't care too much) A problem is,
> furthermore, that P'(b') is not so much accessible to rigourousness.
>
> The relationship between P(b) and P'(b') is what I call the "pathway"
> from Heaven (i.e. mathematical theorem) back to Earth (i.e. testable
> predictions).
You mean that the fact that you test a theorem about real numbers by
doing a calculation involving floating point numbers is a good thing?
Why on earth is that a good thing? The theorem is literally not about
floating point numbers (although it may generate some consequences for
floating point calculations).
And if the test comes out positive, how does that verify the theorem?
And if the test comes out negative, how does that refute the theorem?
Clearly, there are situations in which the calculation will give wrong
results. How did I test the theorem?
[...]
>> There are two responses to this picture, if I have it right. First,
>> if I have a proof of "For all x, P(x)", I cannot see how checking P(b)
>> for values of b add any certainty at all. After all, checking P(b)
>> depends on the same axioms and rules of inference that proving the
>> universal did, so why would I find the universal more credible after
>> checking a few values?
>
> As I have said, this is not true. You are checking P'(b') instead of
> P(b). The latter is trivial, indeed, while the former is not. Worse,
> there exist propositions P(b) in contemporary mathematics which don't
> have any counterpart P'(b') at all in the world of computation. Just
> take the transfinite ordinals or cardinals, as an outstanding example.
>
> And you are trying to accuse me of exactly the opposite of what I want
> to demonstrate: it's the theory that predicts the outcomes of a test,
> not the test that predicts the outcomes of the theory.
I never said anything like that.
> But the test assures that the theory is not something out of the
> blue sky. And the theory assures that knowledge will not be limited
> to "a few values" of the tests.
The theorem is about real numbers. I fail to see how a computer
program involving floating point numbers is a test of the theorem in
any meaningful sense at all.
>> I can't see any value to these so-called testable predictions. If
>> your computations are calculated according to the axioms of a theory,
>> then it cannot be the case that they contradict a universal theorem
>> (assuming consistency). If they do contradict a theorem, then either
>> the theory is inconsistent (which is a discovery, but not a refutation
>> of the theorem) or the computations are wrong.
>
> Use your imagination. Suppose that this theory of Fibonacci
> Iterations is becoming employed for the purpose of designing a
> computer program - maybe this is not too far from the truth. In that
> program, division by zero has to be avoided at all costs, to ensure
> the "robustness", which is a highly desirable property for number
> crunching computer code. Now the Theory says that division by zero
> can only occur if iterates start within the interval [ 1 , 2 ]
> . This can be implemented in the code by forcing a _Halt_ if the
> inputs give rise to such a first iterate. Why would that be just a
> "silly computation", without an additional value?
What you just described is not a test of the theorem at all. Rather,
it is a use of the theorem to ensure that a certain condition is
avoided.
--
Jesse F. Hughes.
Me: It's very sad when one's husband or wife dies.
Quincy (Age 4 1/2): Yeah. You might want to tell them something and
you just can't. [Long pause] Like "Take out the trash."
.
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