--- On Thu, 16/12/10, Robert Paul <rpaul@xxxxxxxx> wrote:
Newton's laws of motion work perfectly well within a limited domainÿÿgetting spacecraft into
orbit around the earth, etc.ÿÿyet we now think we know that they do not obtain at the
sub-atomic level; whether they obtain in the remoter (a not entirely unbiased word) regions of space
we aren't sure. Still, electromagnetic waves reach us from regions beyond Newton's ken, and we assume
that whatever stuff is out there there is subatomic activity in it. Newton knew that there were no
perfectly spherical objects or frictionless surfaces on earth, yet his laws are still good (wherever
they are good) despite that. The laws of falling bodies were derived from observations of how
comparatively rough-hewn objects behaved under certain conditions.>
This, so far, seems entirely consistent with the view that Newton's laws remain
a good approximation within certain limits and still work well within those
limits if considered as instruments of prediction but are nevertheless false,
and _falsified_, as universal laws. This view is Popper's view.
Popper admits theories are instruments and may be considered as such [as we
might admit knives are weapons] but for him they are not merely instruments and
to be considered as such [knives are not just weapons]. As an instrument, for
most practical purposes Newton's laws may be most useful, and indeed more
useful than Einstein's theories because they are more manageable. This does not
make them true.
I had written:-
"As to why these laws must apply throughout the universe, the simple answer is that they would lack the
character of universal laws if they did not. A more sophisticated answer is that if, say, we found that there
was a portion of the universe where the 'invariant' did not apply [where perhaps it appeared that
"E=mc4" and "2+2=35"], the invariant would be falsified as a universal law; we would have
to abandon it as a law altogether or explain why the structural invariant was limited in its application so
that those limitations themselves were explained in invariant terms."
The implication is clearly that where Newton's theories break down we want to
explain this using a deeper theory that contains Newton's theories as a
remarkably successful yet nevertheless false approximation.
Robert goes on to say, among other things, that:-
"This doesn't mean that when we speak Euclidean we're making contingent statements
about purely mathematical forms. Statements 'inside' mathematics are tautologies, no
matter whether one uses a base-ten numbering system or some other, and if one uses some
other, if A is using our usual decimal system and Sally some other system, the results of
counting a finite number of objects ought to agree."
This raises the question of whether "Statements 'inside' mathematics are tautologies". For example, does "7 + 5 = 12" by virtue of the meaning of "7", "5" and "12" and
"+" and "="? So that when we know the meaning of these terms we know the truth of "7 + 5 = 12" as a tautology. If this is Robert's view, then I hope to say more on this subsequently. But of course to
say "Statements 'inside' mathematics are tautologies" does not explain what definition of tautology is being offered and it may be Robert does not use tautology so that "7 + 5 = 12" is tautologically true by
virtue of the meaning of "7", "5" and "12" and "+" and "=".
Robert concludes on this interesting point:-
"*****
There are five dots just above this sentence. There's no way to radicalize
mathematical schemes so that there are more or fewer than five. So, a different
mathematical scheme may use different terms and sometimes different operators, but
whatever the result of counting or enumerating these dots yields 'mathematically'
(whatever that might mean) 'in a different system,' its expression must be
equivalent to five. (Taking a stand.)"
Perhaps I have misunderstood, but I am not sure the conclusion "its expression must be equivalent to five" strictly follows. What might
strictly follow is that "its expression will not be inconsistent with the expression 'five'". I say this because we could have a
mathematical scheme consisting only of 'one' and 'more than one'. In this scheme the "five dots" could only be expressed as "dots of
'more than one'". Here 'more than one' is not _equivalent_ to "five", and need not be; but we may say that if "dots of 'more than
one'" is true then it must not be inconsistent with the truth that there are "five dots".
Donal
London
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