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CARTS_AND_HORSES
September 25, 2019
Page numbers from a Gutenberg
Press "djvu.txt" file.
(I presume things like
"premisses" are Britishisms
and not just scanning errors.)
Bertrand Russell, "The Philosophy of Logical Atomism" (1918), p.129:
"When pure mathematics is organized as a deductive system--
i.e. as the set of all those propositions that can be
deduced from an assigned set of premisses-- it becomes
obvious that, if we are to believe in the truth of pure
mathematics, it cannot be solely because we believe in the
truth of the set of premisses. Some of the premisses are
much less obvious than some of their consequences, and are
believed chiefly because of their consequences."
And maybe that's the *real* Russell's Paradox. How can he be
aware of that and still feel assured there's some point to the
project?
"This will be found to be always the case when a science is
arranged as a deductive system. It is not the logically
simplest propositions of the system that are the most
obvious, or that provide the chief part of our reasons for
believing in the system. With the empirical sciences this is
evident."
Okay, so the idea here is the system of premises will be something
like a more compact representation of what we know, perhaps a
*more general* version of what we know?
Russell makes an analogy to physics which strikes me as the
best argument I've seen from him on this point...
"Electro-dynamics, for example, can be concentrated
into Maxwell’s equations, but these equations are believed
because of the observed truth of certain of their logical
consequences. Exactly the same thing happens in the pure
realm of logic; the logically first principles of logic--
at least some of them-- are to be believed, not on their
own account, but on account of their consequences."
And yet the analogy seems a bit strained to me. Something like
Maxwell's equations is regarded as true because it describes
known phenomena *and* could be used to predict the behavior of
systems that hadn't yet been studied.
For this to apply to this project of finding
"foundations" for math, you would have to be able
to use the foundations to derive some math you WHATS_ZFC_DONE_FOR_ME
didn't already know-- I don't think I've ever
heard of a case where that was done.
"The epistemological question: 'Why should I believe this
set of propositions?' is quite different from the logical
question: 'What is the smallest and logically simplest group
of propositions from which this set of propositions can be
deduced?' Our reasons for believing logic and pure
mathematics are, in part, only inductive and probable, "
Russell is a remarkably clear thinker, someone who tries hard to
avoid over-reaching, as I think these passages show...
How can he recognize this and not have some reservations about
the entire approach:
"in spite of the fact that, in their logical order, the
propositions of logic and pure mathematics follow from the
premisses of logic by pure deduction."
Nothing really *follows* from these things you're calling
"premises", because you're choosing the "premises" to get to
where you want to be.
In this piece, Russell addresses one of the more
difficult issues in this whole worship of the
idealism of "pure math": how do you *choose* the
initial premises?
He makes the point that there's a
certain empiricism in the choice, we
pick "fundamental principles" because
of their evident desireable outcomes,
which to my eye argues that the real And I think that's a variant
fundamentals are elsewhere. of one of the earliest
points I was making here
in "the doomfiles":
DESPERATE
CONSEQUENCES
What I'm looking for though is some reason to care
about finding a minimal set of principles...
"... every diminution in apparatus
diminishes the risk of error." -- p.123
I don't think I quite follow that...
risk of error in doing what?
If you could eliminate many complex chains of
reasoning by just presuming some of things we
really do believe are correct, wouldn't that
reduce the "risk of error"?
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