[FOM] How much of math is logic?
Robbie Lindauer
rlindauer at gmail.com
Wed Feb 28 17:22:10 EST 2007
Unfortunately, answering questions in line:
Joe Shipman wrote:
> I think you are stealing some bases here.
I rather thought the same of your post and was trying to say why.
> What I *am* proposing, first of all, is that the finite part of
> mathematics (equivalently, Peano Arithmetic, or the theory of
> hereditarily finite sets, etc.) can be derived from axioms which are
> "logical" in character.
Sure, it depends on what you call "logical in character". We have a
variety of logical systems of a variety of strengths, and depending
on whether you like one axiom or another you can say what things
you're happy with calling "logical". What we don't have is any
PRINCIPLED WAY OF DECIDING THE MATTER other than, I would say, a
"pure acceptable epistemology" along the lines of Constructivist
Intuitionism. And in that case, I think we're stuck with a rather
severe finitism along the lines of PRA. Since this is not considered
"mathematically acceptable" logic has to be "strengthened" by adding
more axioms, etc.
What I'm trying to say is that since there is no "one thing" called
"LOGIC" and since there is no universally acceptable way of deciding
what -should- count as "logical" that in our modern world we have
come to a comparative notion of "strength of a logical system" which
basically says "this is how much stuff you can prove in this, this is
how much stuff you could prove in that..."
This spectrum of "logically acceptable systems" has replaced the
univocal "logicism" philosophy that "mathematics is logic and logic
is mathematics" with a finer-grained way of saying what you mean.
Now you say "in Q you can prove P" "in ZFC you can prove a cognate of
P" but "You can not prove that cognate of P in FOL + PRA", for instance.
> To put it another way, I am trying to argue
> that the non-logical content of mathematics comes essentially from the
> treatment of infinite completed entities -- without an axiom of
> Infinity (in Raatikainen's second, stronger sense), one cannot get
> beyond PA, but that's still enough for an awful lot of math.
I disagree with this for reasons already expressed. The very notion
of "extension of a concept" is an existential notion - a claim about
the existence of a particular kind of thing, similarly with "set".
And the notion of "equipollence" of those "extensions" is an extra-
logical idea from a certain point of view. One introduces the notion
of "equipollence" and then one can decide whether or not one is happy
with that notion or not.
Again, there are a variety of logics with a variety of strengths.
And there is a clear way of discerning among them and depending on
which you choose to call "Logic" is how you decide how much of
mathematics can be derived from it.
> (Friedman ...
> Something
> like this has been done by Frege, Russell, and others, in a strict
> deductive way.
Yes, this is a well-worn subject. There are a variety of ways of
approaching it and no definitive way of deciding which axioms are
"purely logical" or not. If it was me "deciding" I would choose
something epistemologically conservative - Constructivist
Intuitionistic Transcendentalism :)
> The real reason logicism became passe was that the limits of finite
> mathematics became clear; which is why I am also asking for comment on
> the proposition, not that "math is logic", but that "math is logic
> PLUS
> the axiom of infinity". (Set theorists who go beyond ZFC and take
> offense at the word "the" may address the modified proposition
> "math is
> logic plus axioms of infinity".)
I think it became "passe" (nay "falsified") because it became clear
that there is no derivation of something strong enough to do all of
what "some people" want to call mathematics and there is no proof of
the axioms that "those people" like from prior logical notions not of
equal strength with mathematical system being proposed.
Given my previous remarks (there is no "purely logical" FOL proof of
the axioms of ZFC) it should be clear that infinity is just one
symptom. Each axiom of ZFC can be taken in parts or weaker or
stronger versions of them, ad infinitum, to form different logical
systems with different strengths. Whether any of them are "logical
in character" appears mostly to be a matter of choice given that we
aren't committed to some particular EPISTEMOLOGY OF LOGIC.
> My second proposal is based on the observation a that stronger form of
> second-order logic allows one to express the Continuum Hypothesis, so
> that determining logical validity in this strong setup is at least as
> hard as answering CH (conversely, if CH is actually indeterminate then
> this form of "logic" is illegitimate because there is a sentence whose
> validity status is undefined).
In "higher order set theor(y/ies)", one can, in a way, decide the
continuum hypothesis (e.g. by Fiat). The question will be the
ultimate "logicalness" that is, acceptability on purely logical
grounds, of the resulting axiom-group.
> I therefore
> asked whether there is any question in ordinary mathematics that would
> not be "settled" by an oracle for "second-order validity" in the
> standard semantics for second-order logic.
Not knowing what an "oracle" would be, I have no way of answering
what it could or could not settle. Surely "oracle" is not a "logical
concept" is it? (I think I argued this point specifically with
Friedmann here before.) But what I'm trying to impress upon you is
that there are a variety of options in "second order logic". For
instance, the second-order logic of sets and classes, or HOST, etc.
> This is logicism in a different sense, because we no longer have a
> complete deductive calculus. The point here is to address
> philosophical
> questions about meaningfulness -- whether mathematics requires an
> external "subject matter" or whether it is already implicit in a pure
> "theory of concepts".
That has been decided by showing the relative strengths of different
kinds of deductive systems. There is a "continuum" of answers here
and none of them decide the whole thing.
That is, if "mathematics" is "the stuff to which Godel's Theorem
applies" for instance, then we KNOW that there are more axioms than
we have necessary for its construction -NO MATTER WHAT- axioms we
start with (assuming they're strong enough to generate the
incompleteness theorem) and that consequently, no single (finite)
group of FOL or SOL axioms is sufficient for its complete description.
> I will now try to ask some more precisely focused questions:
>
> 1) Do objections to set theory as "non-logical" even apply to the
> extremely weak axioms for sets needed to develop the usual theory of
> hereditarily finite sets, which is naturally bi-interpretable with PA?
The introduction of the notion of "set" is a non-logical component of
set-theory. It's now-standard definition is ZF(C). ZFC is
demonstrably stronger than FOL and depending on which kinds of axioms
you add to FOL to get SOL, it may or may not be stronger than those.
> 2) If the answer to 1) is positive, is there any way whatsoever in
> which we can talk about "classes" or "concepts" that deserves to be
> called "logical"? (Because the properties of classes or concepts
> needed
> to build up something equivalent to PA are also very weak.)
I think this is logically obvious:
If for all x , either Px or !Px, then ES: S = (x, x', x'',
x''', ...): Px -> x in S
You may want to introduce another symbol to get PRA:
|S| "the cardinality of S" but that would be to introduce the notion
"number of things in S"
defining this (cardinality) along Fregean lines is unacceptable to me
for reasons already stated, namely that the decision that :
(1, (1, 2, 3, ...) 4) is "3" things and not "infinitely many things"
or "one thing"
seems arbitrary until you've decided what you're counting and how
you're counting them. And that would be to have a prior notion of
"set" and "number".
If you restrict your Fregean objects to "pure individuals" then you
simultaneously admit that "sets" are not pure individuals and that
introducing them is yet another abstraction.
> 3) Why might ZFC be regarded as "odoriferous"?
Is "ordinal" a property that one could use in FOL to define a second-
order class of things? If not, what kind of thing is it?
I, personally, regard it as odoriferous because it's
epistemologically unacceptable to add axioms to a system that
effectively decide these matters by fiat. Other people, no doubt,
consider it bad because it's too weak. I also find the epistemic
problem of dealing with infinite objects too fishy and that deciding
these matters by additions of axioms to deal with them is a fishy
procedure. How can we really KNOW anything by giving more axioms?
And if we can't PROVE the axioms, then what is our motivation for
accepting them?
These and other questions related to it have been dealt with in this
forum.
> 4) If second-order logic is really "set theory in disguise", exactly
> how much set theory does it disguise? Has anyone ever claimed that all
> of ZFC could be recovered from a deductive calculus for second-order
> logic?
Well, you can derive Russel's paradox in (some varieties of) SOL
without the proper restrictions, e.g. type theory or ZF, etc. Thus
one needs to introduce the restrictive axioms preventing such
activities and THEN one has (effectively) full-blown set theory again.
Best Wishes,
Robbie Lindauer
robblin at thetip.org
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