Re: Are there things that don't evolve?

I am responding to both of Phil's recent posts, copied below.

You have come up with a perverse silent assumption to buttress your
position. Only if what is now valid becomes invalid do we have the
required shift relative to Zermelo and Quine. But the development of
logic does not entail such an inversion. Since logic is not that familiar
to most, let me give a different illustration. Newton's gravitational
discoveries were not eliminated by Einstein's general relativity, which
in turn will not be upended by M theory or whatever unified field theory
eventually holds sway.

I noted that there are three Aristotelian calculi. What is valid in the
simplest, which is what passes for Aristotelian logic or syllogistic, is
also valid in the two advanced modal calculi. Because of differences in
form, what is valid in syllogistic is not valid in the common
propositional or sentential calculus. Also, there are valid deductions in
the modal logics which are not valid in simple syllogistic. That does not
make them invalid, just recognizes that they is not testable by the
different set of rules.

There are arithmetics in which 4+4=4, xor 4+4=3, xor 4+4=2, xor 4=4=1
holds, and 4+4=8 does not. These and many other sets of numbers can be
generated by modifying Peano's postulates. They do what numbers are
expected to do formally, but not for balancing checkbooks, except by
determining that the arithmetic in the checkbook entries is correct.

The syllogism that you sent to Jim is not a syllogism. There is no way
that it can be formulated to fit the correct Aristotelian or degenerate
Renaissance versions of the rules. It is informally valid. However, there
have been a number of studies on decision making which indicate that the
processes people actually use are often faulty. Indeed, you illustrate
this in your attempt to refute my claim.
Dave

On Sun, 26 Mar 2006 01:47:39 EST Philtill@aol.com writes:
In a message dated 3/26/2006 12:00:55 AM Eastern Standard Time,
dfsiemensjr@juno.com writes:
You have just dogmatically disenfranchised a group of mathematicians, the
strict constructionists. Also, the three Aristotelian calculi do not
match the logics that have developed from Frege's work. I recall a remark
that C. L. Dodgson had come up with a calculus that differed from both
Aristotle and PM, which remained undeveloped. I have not dug into that.

I note that there are various conjectures floating around mathematics.
Are these discovered axioms, invented assumptions, wicked perversions, or
something else?

I think we can safely say that nothing that is a constant evolves. This
must apply as surely to pi as to G. If they change, they can hardly be
constants.
Dave

Hi Dave,

I think you're reading too much into what I said, or maybe not enough.
Let me just take the issue you raised about conjectures as illustrative
of the whole:

it is conjectured that if Zermello's set theory is consistent, then this
implies that Quine's set theory is also consistent.

Well, if mathematical logic were evolving, then it would be impossible
even to meaningfully raise such a conjecture, much less hope to prove it.
 If it were proven today using today's logic, but then logic evolved,
then tomorrow it would be unproven again. If this could actually happen
then mathematics would cease to exist as a discipline and everyone would
stop making conjectures. but it doesn't, and we don't, because of our
deep-seated, universal belief that there really is essence to logic and
that it does not evolve.

I hope that clarifies what I intended. Of course folks can build
different axiomatic systems or calculi or logics and then discuss
consistencies using the different rules, but the ability to even engage
one another in these discussions implies that there is something that
doesn't evolve, and that is what I'm talking about.

God bless!
Phil Metzger

In a message dated 3/26/2006 1:10:27 AM Eastern Standard Time,
jarmstro@qwest.net writes:
1. I think it may said that mathematics evolves. New techniques and tools
emergy, conjectures are proven or disproven, philosophical insight
development continues. So the discipline is evolving. Perhaps there is an
underlying God-view mathematics that is complete and not subject to
change, but that is not mathematics as we know it.
Hi, Jim.

I do agree with this. That's what I meant when I said that the "body" of
mathematics does evolve. But mathematicians of all ages and stripes
share in a common logical ability that allows us to discuss math with
each other. We can all understand the axioms invented by each other. We
couldn't do this if there weren't a common, fundamental, unevolving
essence of logic in reality.

Consider this syllogism as an example:

All A are in B
All B are in C
Therefore all A are in C

Does the logic of that syllogism evolve, or does it stay the same?

That's what I was talking about.
God bless!
Phil
Received on Sun Mar 26 22:55:49 2006