Alternatives to axiomatic systems
JB
possible to use another way of reasoning
than on the basis of axiomas in order to avoid the incompleteness-problems
inherent to aximatic reasoning.
I am very curious to know the different opinions on this subject!
Jan Bours
jfhm-bours
AT
David R. MacIver
To the best of my knowledge, there is nothing else which has been
explored on a formal basis. Axiomatic reasoning is fairly central to
mathematics after all.
The main alternative I suppose is intuitive mathematics; you work from
what 'looks true' to prove your results. In practice this is mostly
equivalent to axiomatic mathematics though - Generally what you think
looks true is going to lie in some axiomatic theory. Also, a lot of
mathematicians work in intuitive mathematics while officially declaring
themselves to be working in an axiomatic system, which is fine - it's
rarely worth checking through a precise formal proof in the system, as
long as you can convince yourself and others that you could turn it into
a formal proof if you were so inclined to do so (which I find I rarely
am, on the grounds that it would be really really boring :).
The major problems with this are as follows:
1. Two different mathematicians could disagree as to what 'looks true'.
The most common examples of this are statements like the axiom of choice
and the continuum hypothesis. In practice this tends not to be a problem
most of the time, but it could in principle be.
2. Often what looks true doesn't make a great deal of sense when you
think about it harder. For example you are presumably familiar with
Russel's paradox? i.e. Consider the set { x : x is not an element of x
}. So how do we know our intuition is consistent (this is of course not
avoided entirely by axiomatic mathematics, but it minimises the problem)
3. You often end up with statements that you have no intuition about and
can't be reduced to anything you do have a better intuition of. This
typically just means you avoid the question, but that can be a problem
if it's something you want to use. (Of course this can happen in an
axiomatic theory as well, but at least there you have an actual notion
of undecidability and the possibility of adding in your statement as an
axiom).
All this aside, I would argue that there is no incompleteness problem
inherent to axiomatic reasoning.
Before you start shouting, I'm not disputing the incompleteness
theorems. They are nice, elegant and very interesting results.
It is, however, not a problem.
Consider first the second incompletness theorem: If a system is
consistent, it cannot prove it's own consistency. I really can't see how
this is a problem - if a system is inconsistent then it can prove it's
own consistency; therefore a proof within a system of it's own
consistency seems inherently worthless. You're always going to need to
refer to some bigger system (e.g. intuitive reasoning)... and how do you
know that's consistent? Consistency is an annoying problem to have worry
about, but it's always going to be a problem regardless of what approach
you take. So it's interesting, but not really terribly relevant.
The first incompleteness theorem is much more interesting: It says there
are always undecidable results (assuming your theory is sufficiently
interesting). To my mind this is far from a problem: It basically says
that no matter how much you know, there's always more mathematics that
you can discover. In an infinity of mathematics there's bound to be
various interesting things we don't know about, so the first
incompleteness theorem says nothing more than that there will always be
new and interesting mathematics to explore, no matter how much we
already know! This is hardly a problem...
David
(E-mail address spam-blocked in the obvious way)
Zdislav V. Kovarik
Yes: truth by decree; in one system (which I remember only too
well), the ultimate test of validity was whether a statement
serves the working class. It worked, with the help of a strong
police force, for some time.
Another, less violent but sillier, is truth by ballot (theorem
proved by a majority of at least fifty per cent plus one vote).
Cheers, ZVK(Slavek).
cdj
lol - Kovarik's Theorem: Nothing good results from sentences that
begin with "If by 'truth' you mean......"
:)
> > I was wondering, in the light of Kurt G dels findings, if it would be
> > possible to use another way of reasoning
> > than on the basis of axiomas in order to avoid the incompleteness-problems
> > inherent to aximatic reasoning.
The incompleteness phenomenon (better than "problems") doesn't have
anything essential to do with axiomatic systems per se (although
that's of course the road Godel took). There are many other roads to
the same result, which is partly the point of Godel's 2nd
incompleteness theorem. Any system, axiomatic or otherwise, which can
express a little bit of arithmetic, or a little bit of recursive
functions or..... is either inconsistent or incomplete. There are all
sorts of non-axiomatic systems to which this applies - the key is the
expressive power of the system, not the manner in which that
expressive capability is implemented. Gentzen-type inferential systems
are often axiom-free, for example. And his consistency-of-arithmetic
proof not withstanding, Godel's theorem applies equally well to his
systems....
cdj
Theo
> possible to use another way of reasoning
> than on the basis of axiomas in order to avoid the incompleteness-problems
> inherent to aximatic reasoning.
Here is an off the wall idea. Why not have a quantum-like rules
whereby the result is unkown and contains a supposition of all
possible outcomes until the result collapses into a state giving an
observable outcome. The entire system would be based on the
wavefunctions rather than just functions.
Yes rather off the wall. I think I will go to sleep now.
Shmuel (Seymour J.) Metz
on 12/04/2003
at 10:45 AM, "Zdislav V. Kovarik" <kov...@mcmail.cis.mcmaster.ca>
said:
>Yes: truth by decree; in one system (which I remember only too well),
>the ultimate test of validity was whether a statement serves the
>working class.
ITYM whether it served the party; the opinions of the working class
were not solicited. From each according to his abilities, to each
according to his party rank.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
Unsolicited bulk E-mail will be subject to legal action. I reserve
the right to publicly post or ridicule any abusive E-mail.
Reply to domain Patriot dot net user shmuel+news to contact me. Do
not reply to spam...@library.lspace.org
KRamsay
In article <bqnhss$mg7$1...@news4.tilbu1.nb.home.nl>, "JB" <jfhm_...@planet.nl>
writes:
|I was wondering, in the light of Kurt Gödels findings, if it would be
|possible to use another way of reasoning
|than on the basis of axiomas in order to avoid the incompleteness-problems
|inherent to aximatic reasoning.
The incompleteness phenomenon is very broad indeed.
There are ways of broadening formal reasoning such as permitting tentative
conclusions which might be revised in the light of future evidence. The
results which are concluded as true "in the limit", i.e. without ever being
reversed, can be a more complicated set than the set of theorems of a
formal system.
But even with this kind of broadening, one still has incompleteness.
One way of generalizing incompleteness leads to Tarski's theorem on the
undefinability of truth. In a language such as the first-order language
of arithmetic, there does not exist a predicate T with the property that
T(n) is true if and only if n is a number encoding (by Goedel coding, say)
a true statement of first-order arithmetic. The proof shows how, given a
definable predicate P, to produce a counterexample to the idea that P is
the same as truth: a statement which is true if P is false of its encoding
and false if P is true of its encoding.
It's a generalization because when we define a formal system, it defines
a predicate P(n) meaning that the statement encoded by n is provable in
the system. But since T(n) can't be P(n), provability in the system can't
be the same thing as truth. Also, more general things that you might try
to use instead of a formal system fail, as long as they're also definable
in the system. For example, this notion of "provability in the limit" can
also be defined using number theoretic encodings, so it fails to be truth
too.
Keith Ramsay
Jeffrey H
>
>
>On Thu, 4 Dec 2003, JB wrote:
>
>> I was wondering, in the light of Kurt G=F6dels findings, if it would be
>> possible to use another way of reasoning
>> than on the basis of axiomas in order to avoid the incompleteness-problem=
>s
>> inherent to aximatic reasoning.
>>
>> I am very curious to know the different opinions on this subject!
>>
>> Jan Bours
>>
>> jfhm-bours
>>
>> AT
>>
>> home.nl
>
>Yes: truth by decree; in one system (which I remember only too
>well), the ultimate test of validity was whether a statement
>serves the working class. It worked, with the help of a strong
>police force, for some time.
>
>Another, less violent but sillier, is truth by ballot (theorem
>proved by a majority of at least fifty per cent plus one vote).
>
>Cheers, ZVK(Slavek).
>
Perhaps math cannot be understood as a discipline unless part of
something else, but politics takes us a bit too far. How about
history? Can you imagine math without Euclid? It would be very hard
to think of western math without axioms.
About methods of proof, Aristotle did not call on any axioms to prove
that no rational number squared equals two, but he did need to appeal
to people's sense that some things have already been agreed on, such
as obvious things about even and odd numbers.
Every method of proof must require something to seem obvious. If it
is codified and made into axioms, then we get a powerful system like
Euclid's. But it has seemed very unlikely, to me, that Godel has
appealed to anything of the kind, since he requires a generalized
notion of infinity which is not intuitive, as Euclid's, and in fact
may or not be based on the definitions based on taking limits used in
analysis. It is hard to imagine an "infinite set." An infinite
geometrical set, yes. An infinite set of continuous functions, yes.
But infinite "sets"? There is no intuitive notion here at all, since
a "set" is anything where you can swap members and still call it the
same "set." How in the world can human imagination grasp such an idea
when the kinds of members are not specified?
So I don't think this Godelism has touched the heart of math, the way
that Euclid, Descartes, or Newton did.
Herman Jurjus
"Jeffrey H" <aptsol...@mindspring.com> wrote in message news:3fe33519...@news.east.earthlink.net...
[snip]
Can you imagine math without Euclid? It would be very hard
> to think of western math without axioms.
Even more: mathematical reasoning is exact, and involves _steps_.
So, no matter what you do, these steps can be analyzed and then
formalized as an axiom system.
However, there are forms of reasoning that are not based on steps
or are not 'exact' enough to be so easily formalized.
(For example: everyday life reasoning). Attempts to formalize these
are sometimes done using fixpoint constructions, and other tricks.
You might be interested to Google for non-monotonic logic.
Especially the argumentation-theoretic approach (Phan Minh Dung,
Henry Prakken) and inheritance networks (John Horty).
Hope this brings you further.
Good luck and cheers,
Herman Jurjus
KRamsay
In article <3fe33519...@news.east.earthlink.net>,
aptsol...@mindspring.com (Jeffrey H) writes:
|Every method of proof must require something to seem obvious. If it
|is codified and made into axioms, then we get a powerful system like
|Euclid's. But it has seemed very unlikely, to me, that Godel has
|appealed to anything of the kind, since he requires a generalized
|notion of infinity which is not intuitive, as Euclid's, and in fact
|may or not be based on the definitions based on taking limits used in
|analysis.
Huh? Goedel deliberately made his most famous results rely only
upon finite combinatorial reasoning. Where do you get this stuff
about a generalized notion of infinity? Do you mean to refer to
Cantor and not Goedel?
|It is hard to imagine an "infinite set." An infinite
|geometrical set, yes. An infinite set of continuous functions, yes.
|But infinite "sets"? There is no intuitive notion here at all, since
|a "set" is anything where you can swap members and still call it the
|same "set." How in the world can human imagination grasp such an idea
|when the kinds of members are not specified?
Does the idea of visualizing an arbitrary "pair of objects" seem funny
to you? In a way it is kind of funny, since it's such a vague description.
An arbitrary set of things is not so visualizable for the same reason.
The case of an arbitrary set of some previously known kind of thing
is perhaps the most important one. One assumes something equivalent
to the existence of a set consisting of all natural numbers. Then one
considers sets of natural numbers, sets of sets of natural numbers,
and so on. The sets which can be obtained after finitely many such
steps are the ones most typical of ordinarily mathematics. In set theory
one keeps going, taking the sets of elements which are allowed to be
at any arbitrary finite number of levels above the natural numbers to
form the elements of a new level of the hierarchy. I'm describing the
hierarchy taking the natural numbers to be atoms here; it's common
of course to encode the natural numbers so that everything fits into
the hierarchy of "pure" sets, with no atoms, starting with the empty
set as the member of the first level of the hierarchy.
Each step involves considering only sets whose elements are in
previous steps of the hierarchy. So in a sense we're relying on just
being able to understand the notion of "set of X's" once we understand
what X's are.
Set theorists permit the levels of the cumulative hierarchy to correspond
to any arbitrary ordinal number, which is the order type of a well-ordered
arbitrary set. Here is where things get somewhat more slippery. If you
feel a little unease at such a construction, then I suppose you're not
alone in that.
Suppose we decide to use only ordinals that are the
order type of well-orderings of sets in an already constructed level of
the hierarchy. So we take the natural numbers as given, and the
well-orderings of the natural numbers are the countable ordinals, which
have an order type of b1 = aleph-1. The first aleph-1 levels of the
hierarchy contain much bigger sets than the natural numbers, and the
well-orderings of sets on those levels go up to some bigger ordinal,
call it b2. So then I consider the sets on levels of the hierarchy (ranks)
up to b2. I can keep going inductively, defining b_{k+1} to be the sup
of the ordinals that have the same order type as some set in the sets
of rank < b_k.
Now the sequence b1<b2<b3<... has a limit, b_omega. Or so we say.
(And prove, assuming the usual axioms of set theory.) The ordinal
b_omega is something we take as pretty definite. Probably
it has a name I just have forgotten. The odd thing is, b_omega is not
the order type of a set of rank <b_omega. If we had a set X of rank
<b_omega, then X would be <b_k for some k, since b_omega is just
the limit of the b_k. But then by the construction of b_{k+1}, we have
that the rank of any well-ordering of X is <b_{k+1}<b_omega. So when
we prove that there exists such an ordinal b_omega, we're relying on
something a little more subtle than the original step-by-step construction
of a hierarchy I just outlined, using only ordinals that have the order
type of previously constructed levels.
Set theorists have the intuition that the ordinals in a sense keep going
"very far", and that the axioms we choose to describe the entire
cumulative hierarchy should reflect that, not imposing arbitrary limitations
on how high the hierarchy goes. The ones who believe in the hierarchy
generally believe in large ordinals that are even more resistant to being
constructed "from below", ones for which the sets of ranks below them
look increasingly like the entire hierarchy. The ordinal b_omega can at
least be constructed as the union of a collection of ordinals below it,
where the collection has countable many elements (which is less than
b_omega). For example, it's generally considered safe to assume that
there are ordinals B akin to b_omega in not being the order type of rank
<B, but which are not the union of a set of previous ordinals which has
fewer elements than B either.
That's set theory for you. Luckily for you, if this is not your cup of tea,
is that hardly anybody not doing set theory needs any of the more
tenuous seeming parts of the hierarchy. Goedel made a point of using
just finitistic reasoning in his most famous results, and normal mathematics
can be thought of as dealing just with the first few ranks above the natural
numbers (which if you represent the natural numbers as sets, is a few
ranks on top of all the finite ranks, which come first). I'm hopeful that
this will not seem unreasonably abstract to you, as indeed IMHO it's
not unreasonably abstract.
Keith Ramsay
Shmuel (Seymour J.) Metz
at 05:26 PM, aptsol...@mindspring.com (Jeffrey H) said:
>Can you imagine math without Euclid?
Yes.
>It would be very hard to think of western math without axioms.
But very easy to think of axioms without Euclid.
>since he requires a generalized notion of infinity which is not
>intuitive, as Euclid's,
What generalized notion did Gödel require? And what was Euclid's
notion of infinity?
>and in fact may or not be based on the definitions based on taking limits
>used in analysis.
What definitions are those? Why would they be relevant to Gödel's
incompleteness results?
>It is hard to imagine an "infinite set." An infinite
>geometrical set, yes. An infinite set of continuous functions, yes.
>But infinite "sets"?
How does one imagine an infinite geometrical set that is not an
infinite set?
>There is no intuitive notion here at all,
Incorrect; I'm far from the only one to find it intuitive.
>since a "set" is anything where you can swap members and still
>call it the same "set."
What gives you the idea that sets have anything to do with swapping?
Also, two sets with different members are *not* the same set.
>How in the world can human imagination grasp such an idea when the
>kinds of members are not specified?
How can the human imagination grasp an opera in a foreign language?
Perhaps your imagination is deficient; that doesn't mean that
everyone's imagination is.
>So I don't think this Godelism has touched the heart of math,
But others, better educated, do think that it does.
>the way that Euclid, Descartes, or Newton did.
And what way was that? None of them touched the heart of Mathematics
the way that, e.g., Galois, Gauss, Gödel, Hilbert, Klein and Reimann
did.