awex@wxlvax.UUCP (Alan Wexelblat) (01/11/84)
{Yes, I know this ought to go in net.philosophy, but y'all are debating
about what is true here, so I thought I'd post it here.}
There are two things to think about when talking about truth. One is the
distinction between *a priori* truth and *a posteriori* truth. Things
that are *a priori* true, are true "by definition." Things that are
*a posteriori* true are true in virtue of what we have discovered.
The second dimension is the distiction between *analytic* truth and
*synthetic* truth. Something is analytically true if it can be
rigorously proven (within a given system, as Godel showed). Something is
synthetically true if it is supported by an acceptable majority of the
experimental evidence.
Now, things like mathematics and logic are *a priori analytically* true.
For example, 2 + 2 = 4 is true by virtue of the menaings of the symbols
'2','4','+', and '='. Similarly, True and False is False, by the rules
of logic.
On the other hand, all of science is *a posteriori synthetically* true.
Which is to say that it's not possible to PROVE any scientific
conclusion. All we can say is that all evidence so far supports it.
Therefore, greg, it is incorrect to lump science and mathematics
together in talking about truth.
One interesting note is that geometry is the only thing that is
*a posterioir analytically* true, in that it is ture by virtue of the
way that the world if, but we can still construct proofs about it. This
is the case for both Euclidean and non-Euclidean geometries; the only
difference is what you assume the universe to be like.
--Alan Wexelblat (the vanishing philosopher)
...decvax!ittvax!wxlvax!awexgds@mit-eddie.UUCP (Greg Skinner) (01/11/84)
Note, Alan, in my previous argument, I stated that there was no such thing as absolute proof, only proof subject to known information. Therfore, the acceptance of the Bohr atom model would fall under that category. (jmp to philosphy) However, a statement such as 2 + 2 = 4 is not true by virtue of the symbology itself, but by the physical equivalents of the symbols. The statement 1 + 1 = 3 would not be false if one could logically construct a world in which a whole theory of numbers followed from this, but in our world such a statement has no physical equivalent. (rtn to religion) -- --greg ...decvax!genrad!mit-eddie!gds (uucp) Gds@XX (arpa)
tjt@kobold.UUCP (01/12/84)
Greg Skinner (mit-eddie!gds) says:
However, a statement such as 2 + 2 = 4 is not true by virtue of the
symbology itself, but by the physical equivalents of the symbols.
The statement 1 + 1 = 3 would not be false if one could logically
construct a world in which a whole theory of numbers followed from this,
but in our world such a statement has no physical equivalent.
This closely echos Alan Wexelblat (wxlvax!awex):
One interesting note is that geometry is the only thing that is
*a posterioir analytically* true, in that it is ture by virtue of the
way that the world if, but we can still construct proofs about it. This
is the case for both Euclidean and non-Euclidean geometries; the only
difference is what you assume the universe to be like.
I would take a third approach and claim that in the context of a
*mathematical theory*, making 2 + 2 = 4 or 1 + 1 = 3 are equally valid,
as are both Euclidean and non-Euclidean geometries. However, theories
where 2 + 2 = 4 are more interesting than those where 1 + 1 = 3 because
they appear to more closely model our own world. The *theory* is is
always "a priori analytically* true while a *model* is always *a
posteriori synthetically* true. A theory is a game that can do
whatever you want it to, but a model maps concepts of the theory onto
objects and actions in the real world, and must agree with experimental
evidence.
Therefore, when a theory fails to model new experimental evidence, the
theory has to be revised. Note that any *new* theory is constrained to
model the old evidence as well as the new evidence.
Occasionally, a new theory and a new model will change the
interpretation of old evidence. This is acceptable because of some
uncertainty in any evidence. For example, the negative resistance of a
tunnel diode was apparent in some earlier devices, but was ignored as
noise since the negative resistance was not very pronounced *and* the
existing theory did not predict that behavior. Later on, the negative
resistance was recognized since the theory was extended to include
negative resistance (I'm sure to have made some blunders here -- I'm a
computer scientist, not a solid state physicist).
Going back to Greg's earlier article:
Ancient man had evidence that the sun rose and shined every day, until
the first eclipse came, then the evidence was false.
In this case, the *theory* is that the sun rose and shined every day.
The *evidence* for this is (presumably) that it always had in the past,
or more precisely: it came up yesterday, and the day before, and the
day before that and ... If you number the days this becomes: the sun
rose and shined on day 1 and on day 2, day 3, ... After the eclipse on
day N, the preceding evidence is still true: the sun rose and shined on
days 1 to N-1, but not on day N (or at least, something peculiar
happened on day N). This requires a more complicated theory to model
the existing evidence, but hardly invalidates that evidence, and (in
this case) only partially invalidates the theory.
--
Tom Teixeira, Massachusetts Computer Corporation. Westford MA
...!{ihnp4,harpo,decvax}!masscomp!tjt (617) 692-6200 x275emjej@uokvax.UUCP (01/21/84)
#R:wxlvax:-21400:uokvax:8300027:000:801 uokvax!emjej Jan 18 00:44:00 1984 Sorry, but the claim that 2 + 2 = 4 is true "by virtue of the physical equivalents of the symbols" is at best ambiguous. 2 + 2 = 4 is a derivable string given the meanings one usually assigns to those symbols in such systems as Russell and Whitehead, Hofstadter's TNT, etc. In that sense, it is "true." It has proven applicable to physical situations involving discrete objects whose interactions are ignorable (two marbles plus two marbles works reasonably well, but two suns plus two planets may give you two suns and a large cloud of gas), and in that sense one might call 2 + 2 = 4 "true" also. It's entirely possible that one might construct a system in which 1 + 1 = 3 is a derivable string, but I do hope that whoever does it uses some different symbols, to avoid confusion. James Jones