Ingvar and Chris, (01)
I admit that I should qualify that statement: (02)
JFS> I certainly admit that the definitions of the physical units
> normally uses theoretical terminology of the time when the
> definition was stated. But I would claim that those definitions
> can be mapped to experimental methods that do not depend on the
> theory. (03)
The first point, which I very strongly believe, is that our normal
use of language, including scientific language, depends critically
on our assumed concepts. For scientific purposes, that means the
currently established science of the day. But I added that our
basic units of measure were well established in the 19th century. (04)
The current level of precision requires far more sophisticated
technology, but the basic ideas in the UoM ontology should be
compatible with as wide a range of scientific theories as possible.
Since Newtonian mechanics is still very widely used, that implies
that the UoM ontology should not depend on further developments. (05)
In particular, the seven base units of the MKS system of units
can be understood in terms of late 19th century technology:
meter, kilogram, second, ampere, kelvin, mole, and candela. (06)
C. S. Peirce, for example, was the first to propose the use of a
wavelength of light as a basis for linear measurements, and he
designed the equipment to use it (in the 1880s) for measuring
the lengths of the pendulums he designed for measuring gravity. (07)
IJ> All experimental methods that measure forces rely on Newton's
> second law, F = m x a. (08)
I'll certainly accept that. (09)
IJ> The Kelvin (ratio) scale for temperature is dependent on the theory
> of statistical thermodynamics. It has an absolute zero point that
> presupposes the view that temperature at bottom is caused by, or
> even identical to, kinetic energy. (010)
Based on his "air thermometer," Amontons estimated the temperature at
which air pressure would be zero at the equivalent of -240 Celsius.
That was in 1702. In 1779, Lambert improved that value to -270C,
which he called "absolute zero." The notion of entropy and very
good approximations to absolute zero were firmly established on
the basis of traditional thermodynamics in the late 19th century. (011)
Although Boltzmann developed statistical mechanics in the 19th
century, he fought an uphill battle against people like Ernst Mach,
who refused to admit the existence of atoms. Einstein finally
overcame the last opposition to the atomic hypothesis with his
paper on Brownian motion in 1905. (012)
The chemists were far ahead of the physicists in developing atomic
theory. In 1865, Loschmidt had an estimate of Avogadro's number
that was off by only a factor of 2. In any case, the chemists had
an equivalent notion to the mole long before anyone knew how many
molecules it contained. (013)
JFS>> 'infinite length' means a length that is sufficiently long
>> that no further increase has a measurable effect. (014)
IJ> To me, 'infinite length' MEANS infinite length... Every
> experimenter knows how to handle things like these when doing
> experiments. No redefinitions are needed. (015)
I certainly agree with you about the meaning of the word 'infinity'.
But what I said is that the way experimenters handle such things is
to do the experimental equivalent of what mathematicians do: (016)
Measure a limiting sequence of values at points that approach
the the limit as far as possible and estimate where the limit
would be if the process could be continued to the limit. (017)
That is essentially what Amontons did in 1702 and Lambert in 1779. (018)
IJ> One of the classical criticisms of operationalism and pragmatism
> is that such attempted definitions are trying to do the impossible. (019)
The positivist and behaviorist operationalism was very different
from Peirce's pragmatism. CSP was highly critical of both Ernst Mach
and Karl Pearson. (020)
CP> I believe many mathematicians would argue that non-constructive
> proofs are very useful. (021)
I agree that they're useful as a last resort, but whenever possible
mathematicians keep looking for constructive replacements. (022)
CP> Is your point that when one gives an intensional definition,
> for those with a large enough (semantic) extension, only God can
> see the whole extension? Or is it that one cannot practically
> give an extensional definition for these? (023)
From Aristotle to the present, fundamental definitions are stated
by intensions (a statement or method for identifying instances).
The method of definition by necessary and sufficient conditions
is certainly intensional. But Aristotle also established the
practice of defining biological species by prototype: give a
precise description of a typical specimen, and declare "this
specimen and anything that resembles it more than others of
its genus belong to this species." (024)
Both of those methods are definitions by intension. (025)
CP> ... we are agreed that we can give intensional definitions of
objects with extensional criteria of identity, aren't we? (026)
I'm not sure what you mean. If you define an arbitrary set by
extension (an exhaustive list of all its members), it may be
extremely difficult to find an intensional definition that
selects precisely the same entities. (027)
There are, of course, some brute force methods for creating an
intensional definition. For example, consider the definition (028)
S = {2, 137, 28902, 4350926} (029)
I can define S intensionally as the set of integers for which
the following polynomial is zero: (030)
f(x) = (x - 2)(x - 137)(x - 28902)(x - 4350926) (031)
But that definition is longer than listing the extension. (032)
John (033)
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