Harvey Friedman on Cantorian pseudomathematics
david petry
The following is a quote from Harvey Friedman. I believe that it
captures the essence of what I have been arguing in these newsgroups in
my essays on "Cantorian pseudomathematics". The full article may be
found at:
http://www.cs.nyu.edu/pipermail/fom/2006-January/009526.html
***Start quote***
An absolutely crucial issue for the foundations of mathematics is
whether there is any significant use of set theoretic methods for "real
mathematics". [...] In particular, mathematics is really Pi01. YES,
Pi01. When it is Pi02 or Pi03, it BEGS to become Pi01 by placing upper
bounds on the existential quantifiers. Look at Fields medals, prize
winning work, million dollar problems, etcetera. Sure, sometimes there
are real numbers and continuous functions, but usually it is clear by
approximation that what is going on is very finitary. E.g., finite
simplicial complexes. And when that is not clear, time and time again
when the problems are solved, there is a Pi01 essence that is the hard
part that easily implies the full result. So the CONTENT is Pi01. And
even when that is perhaps debatable, the cases people are really
interested in push us down to Pi01.
***End quote***
The "Pi01 mathematics" is essentially (though not exactly) what I have
been calling "the science of phenomena observable in the world of
computation".
http://groups.google.com/group/sci.math/msg/8245894cf9c14ac6?dmode=source&hl=en
abo
A few comments:
1/ I'd agree with you that Cantorian set theory is dissociated from
reality.
2/ Friedman is fond of calling ZFC the "gold standard" of mathematics,
so I don't think he would embrace your (or my) dismissal of Cantor.
3/ Not that I understand "the science of phenomena observable in the
world of computation", but I think the intuition lurking in this quote
by Friedman is more that mathematics doesn't need the successor axiom.
Pi01 statements can all be reformulated in terms of assertions where
the only numbers asserted to exist are those less than numbers already
assumed to exist. For instance,
(x)(there exists y < f(x)) P
can be reformulated as
(x)(z)(z = f(x) => (there exists y < z) P).
So the assertion that the content of mathematics is essentially Pi01 is
just that one doesn't really need the number series to go on and on -
all one needs are the initial segments.
Daryl McCullough
>The following is a quote from Harvey Friedman. I believe that it
>captures the essence of what I have been arguing in these newsgroups in
>my essays on "Cantorian pseudomathematics".
I don't see where Harvey says: "the Cantorian pseudomathematicians are
defending a religion, and they really can't see what monsters they have
become. What the Cantorians are doing is nothing less than a crime against
humanity."
--
Daryl McCullough
Ithaca, NY
--
NewsGuy.Com 30Gb $9.95 Carry Forward and On Demand Bandwidth
Daryl McCullough
> The following is a quote from Harvey Friedman. I believe that it
> captures the essence of what I have been arguing in these newsgroups
in
> my essays on "Cantorian pseudomathematics".
>
Ross A. Finlayson
Harvey's pretty cool. He doesn't often reply to e-mail and he's a
moderator of that list. So, insularized is perhaps an adequate term.
Maybe you should actually write to him in case he hasn't read your post
and invite him to comment. He has an e-mail address to which you can
write which he presumably reads.
Harvey is kind of like the standard-bearer of FOM, "old school",
probably met Goedel and so forth, Cohen, Rosser, all now passe, altough
Cohen and Rosser might have something to say about it. It was
interesting to read about the AMS meeting, about infinity and FOM, at
the end of last year, there was some friction. I don't much care,
finding it all quite irrelevant to FOM.
Perhaps he would care to discuss that. I encourage it.
There are a few places wher I find it useful to quote Harvey as well,
and you know what I say.
I don't necessarily interpret you two the same way, Dave. Also, I
don't think Harvey does either, Dave.
Ross
Jesse F. Hughes
> The following is a quote from Harvey Friedman. I believe that it
> captures the essence of what I have been arguing in these newsgroups in
> my essays on "Cantorian pseudomathematics". The full article may be
> found at:
>
> http://www.cs.nyu.edu/pipermail/fom/2006-January/009526.html
>
> ***Start quote***
>
> An absolutely crucial issue for the foundations of mathematics is
> whether there is any significant use of set theoretic methods for "real
> mathematics". [...] In particular, mathematics is really Pi01. YES,
> Pi01. When it is Pi02 or Pi03, it BEGS to become Pi01 by placing upper
> bounds on the existential quantifiers. Look at Fields medals, prize
> winning work, million dollar problems, etcetera. Sure, sometimes there
> are real numbers and continuous functions, but usually it is clear by
> approximation that what is going on is very finitary. E.g., finite
> simplicial complexes. And when that is not clear, time and time again
> when the problems are solved, there is a Pi01 essence that is the hard
> part that easily implies the full result. So the CONTENT is Pi01. And
> even when that is perhaps debatable, the cases people are really
> interested in push us down to Pi01.
>
> ***End quote***
This is not at all similar to your crusade. Friedman is saying that,
as a matter of fact, interesting bits of mathematics are in Pi01.
You are saying as a matter of principle, a mathematical statement is
*meaningless* unless it is "computationally testable" (which now means
Pi01).
Two utterly different claims.
--
Jesse F. Hughes
"That's the base tautological space where by tautological space I mean
a region of truth." -- James S. Harris does philosophy of mathematics.
JSH is a renaissance man.
Han de Bruijn
> "david petry" <david_lawr...@yahoo.com> writes:
>
>>The following is a quote from Harvey Friedman. I believe that it
>>captures the essence of what I have been arguing in these newsgroups in
>>my essays on "Cantorian pseudomathematics". The full article may be
>>found at:
>>
>>http://www.cs.nyu.edu/pipermail/fom/2006-January/009526.html
[ ... snip ... ]
> This is not at all similar to your crusade.
Maybe I've missed something, but for one time I agree with Jesse.
Han de Bruijn
david petry
Jesse F. Hughes wrote:
> "david petry" <david_lawr...@yahoo.com> writes:
>
> > The following is a quote from Harvey Friedman. I believe that it
> > captures the essence of what I have been arguing in these newsgroups in
> > my essays on "Cantorian pseudomathematics". The full article may be
> > found at:
> >
> > http://www.cs.nyu.edu/pipermail/fom/2006-January/009526.html
> >
> > ***Start quote***
> >
> > An absolutely crucial issue for the foundations of mathematics is
> > whether there is any significant use of set theoretic methods for "real
> > mathematics". [...] In particular, mathematics is really Pi01. YES,
> > Pi01. When it is Pi02 or Pi03, it BEGS to become Pi01 by placing upper
> > bounds on the existential quantifiers. Look at Fields medals, prize
> > winning work, million dollar problems, etcetera. Sure, sometimes there
> > are real numbers and continuous functions, but usually it is clear by
> > approximation that what is going on is very finitary. E.g., finite
> > simplicial complexes. And when that is not clear, time and time again
> > when the problems are solved, there is a Pi01 essence that is the hard
> > part that easily implies the full result. So the CONTENT is Pi01. And
> > even when that is perhaps debatable, the cases people are really
> > interested in push us down to Pi01.
> >
> > ***End quote***
>
>
> This is not at all similar to your crusade. Friedman is saying that,
> as a matter of fact, interesting bits of mathematics are in Pi01.
You seem to have completely missed what he actually said. He said
"mathematics is really Pi01"
david petry
abo wrote:
> 3/ Not that I understand "the science of phenomena observable in the
> world of computation",
The idea is that we can think of the computer as the mathematicians'
microscope, and then mathematics is the study of what the mathematician
sees when he looks through that microscope. All scientifically
applicable mathematics (what Friedman calls "real" mathematics) fits
within that paradigm.
Gerry Myerson
"david petry" <david_lawr...@yahoo.com> wrote:
> abo wrote:
>
> > 3/ Not that I understand "the science of phenomena observable in the
> > world of computation",
>
> The idea is that we can think of the computer as the mathematicians'
> microscope, and then mathematics is the study of what the mathematician
> sees when he looks through that microscope.
We can do that, but why would we want to?
Does the biologist limit herself to what she can see
through her microscope?
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
david petry
Gerry Myerson wrote:
> In article <1142980934.5...@g10g2000cwb.googlegroups.com>,
> "david petry" <david_lawr...@yahoo.com> wrote:
>
> > abo wrote:
> >
> > > 3/ Not that I understand "the science of phenomena observable in the
> > > world of computation",
> >
> > The idea is that we can think of the computer as the mathematicians'
> > microscope, and then mathematics is the study of what the mathematician
> > sees when he looks through that microscope.
>
> We can do that, but why would we want to?
It gives us a better definition for mathematics than "what
mathematicians do", and it gives us a good way to keep out
pseudoscience.
> Does the biologist limit herself to what she can see
> through her microscope?
Yes.
Gerry Myerson
"david petry" <david_lawr...@yahoo.com> wrote:
> Gerry Myerson wrote:
> > In article <1142980934.5...@g10g2000cwb.googlegroups.com>,
> > "david petry" <david_lawr...@yahoo.com> wrote:
> >
> > > The idea is that we can think of the computer as the mathematicians'
> > > microscope, and then mathematics is the study of what the mathematician
> > > sees when he looks through that microscope.
> >
> > We can do that, but why would we want to?
>
> It gives us a better definition for mathematics than "what
> mathematicians do", and it gives us a good way to keep out
> pseudoscience.
Pseudoscience has never been much of a problem in mathematics,
as far as I can tell.
> > Does the biologist limit herself to what she can see
> > through her microscope?
>
> Yes.
So she never talks about oxygen and methane and uracil
and other molecules that are too small to be seen in her microscope?
She never talks about lions and tigers and bears (oh, my!)
because they are too big to fit under her microscope?
Biologists must be a lot more limited than I thought they were.
Dave Rusin
Gerry Myerson <ge...@maths.mq.edi.ai.i2u4email> wrote:
>>> Does the biologist limit herself to what she can see
>>> through her microscope?
>>
>> Yes.
>
>So she never talks about oxygen and methane and uracil
>and other molecules that are too small to be seen in her microscope?
>She never talks about lions and tigers and bears (oh, my!)
>because they are too big to fit under her microscope?
>Biologists must be a lot more limited than I thought they were.
Maybe today, not originally:
http://www.darylscience.com/graphics/FarSideMicroscope.gif
abo
david petry wrote:
> abo wrote:
>
> > 3/ Not that I understand "the science of phenomena observable in the
> > world of computation",
>
> The idea is that we can think of the computer as the mathematicians'
> microscope,
I'm afraid this (and the next clause, below) is too reliant on metaphor
to be useful to me.
> and then mathematics is the study of what the mathematician
> sees when he looks through that microscope.
As written, I think this is patently false. It sounds like your claim
is that mathematics today, sociologically speaking, is like that, when
there is plenty of mathematics which needs ZFC. It would seem you need
to write, "Mathematics should..." or (as you do in your next line)
"Scientifically applicable mathematics is..." or something like "The
real content of mathematics is..." But even with these qualifications,
I really can't evaluate your claim, because it is metaphorical.
> All scientifically
> applicable mathematics (what Friedman calls "real" mathematics) fits
> within that paradigm.
Maybe. Again, I think you should be changing "fits" to "can be made to
fit".
Jesse F. Hughes
But what he means is that the higher levels of the hierarchy are
unnecessary for doing most interesting mathematics (outside of set
theory).
What he did *not* say is that any non-Pi01 statement is meaningless
and that any theory that involves non-Pi01 statements is mere
religion.
It is not at all the same claim as yours. It is at best superficially
similar.
--
Jesse F. Hughes
"Most people don't even know what a rootkit is, so why should they
care about it."
-- Thomas Hesse, sony executive defends DRM-by-rootkit.
Jesse F. Hughes
> Gerry Myerson wrote:
>> In article <1142980934.5...@g10g2000cwb.googlegroups.com>,
>> "david petry" <david_lawr...@yahoo.com> wrote:
>>
>> > abo wrote:
>> >
>> > > 3/ Not that I understand "the science of phenomena observable in the
>> > > world of computation",
>> >
>> > The idea is that we can think of the computer as the mathematicians'
>> > microscope, and then mathematics is the study of what the mathematician
>> > sees when he looks through that microscope.
>>
>> We can do that, but why would we want to?
>
> It gives us a better definition for mathematics than "what
> mathematicians do", and it gives us a good way to keep out
> pseudoscience.
And now we see the difference between what Friedman wrote and your
opinion.
Has Friedman ever called set theory "pseudoscience"?
--
Jesse F. Hughes
"That's cool for us in Alabama. 'Cause you know, it's either this or
the monster truck rally." -- An Alabaman expresses appreciation for
local repertory theater on NPR
Ross A. Finlayson
Hi Jesse, how's it going,
I wrote to Harvey and told him we were discussing his words here, and
invited him to comment. I mentioned that Herb posts to sci.logic, and
he's cowritten a paper with Herb, Enderton, to try and imply that his
reputation would be ruined by posting to sci.logic, entering the
vicious fishbowl, the premier open and UN-moderated mathematical logic
discussion group.
I think it worked, but he hasn't really had much time to respond.
Just kidding, Harvey, Dr. Friedman, however I do agree with some of
your points, you seem to be talking about a maximal element of the
ordinals in the quote above. You think it guarantees those conjectures
there that have million dollar prizes from the Clay Institute. Harvey
is coming around.
Good luck with that. I happen to agree.
Ross
Herman Jurjus
The way i understood Friedman (not this particular quote, but some of
the other things he said), he claims that large cardinal axioms are
(mathematically) relevant, -because- they influence what Pi01 sentences
one can prove.
So, he does claim that Pi01 sentences are the concrete things by which
one can/should judge theories for their real, mathematical relevance.
(And this is in line with what David Petry keeps saying).
However, Friedman claims to have proofs that large cardinal axioms -do-
make a difference about which Pi01 sentences can be proved. And
precisely for that reason, he strongly -defends- large cardinal axioms
as very meaningful and relevant also for the practicing mathematician.
At least that's how i understand it.
--
Cheers,
Herman Jurjus
Jesse F. Hughes
[...]
> However, Friedman claims to have proofs that large cardinal axioms
> -do- make a difference about which Pi01 sentences can be proved. And
> precisely for that reason, he strongly -defends- large cardinal
> axioms as very meaningful and relevant also for the practicing
> mathematician.
Thanks for the clarification. I certainly don't keep up with
Friedman's work.
Clearly, what you say here makes a very strong difference between
Friedman's philosophy and Petry's, well, whatever.
--
"I liked the world a lot better over ten years ago. I believed in a
lot more things. Hell, most people believed in a lot more things.
Back then the United States was still, well, known as most people used
to know the United States." -- James S. Harris in a nostalgic mood
Jesse F. Hughes
> I wrote to Harvey and told him we were discussing his words here,
> and invited him to comment.
I can't imagine that he would care to join this thread.
--
Jesse F. Hughes
"Through implied infinite iterations of transitive logical implication
over the set."
-- Sometimes, Tony Orlow is a poet. Usually when he's doing math.
Kent Paul Dolan
wrote:
> So she never talks about oxygen and methane and
> uracil and other molecules that are too small to
> be seen in her microscope? She never talks about
> lions and tigers and bears (oh, my!) because they
> are too big to fit under her microscope?
> Biologists must be a lot more limited than I
> thought they were.
May I presume that the lack of an "oh, then I was in
error" response here from David, despite plenty of
time for it to have been posted, places him firmly
in the ranks of the invincibly ignorant, those
incapable either of confessing error or of learning
based on factual input that their preconceived and
dearly held notions are simply _wrong_?
I ask because this thread sounds like it is already
one in an unbounded series of similar threads
involving the same initial author, and similar to
neverending discussions involving similarly
immovable authors, which achieve zero progress
to the individual involved or to the science of
math, despite massive input of received wisdom to
show why some "contrary to the knowledge base of the
dicipline" idea' fixee' (or however that should be
spelled in 7 bit ASCII) is and will forever remain
unacceptable.
No matter how long I stay away, I always see this
same dance going on when I return. I suppose I
should attempt to learn from my own observations
how thin a gruel sci.math is and will remain, and
learn to content myself with the available level of
nourishment per effort exerted to read the newsgroup.
Sigh.
xanthian.
Why is it that people so often insist that merely
because their minds are insufficient to wrap around
a difficult concept, that the concept must therefore
be _wrong_? Is it so hard to accept: "this is beyond
my grasp, so I leave it to my betters to handle"?
--
Posted via Mailgate.ORG Server - http://www.Mailgate.ORG
Han de Bruijn
>
> Why is it that people so often insist that merely
> because their minds are insufficient to wrap around
> a difficult concept, that the concept must therefore
> be _wrong_? Is it so hard to accept: "this is beyond
> my grasp, so I leave it to my betters to handle"?
Why is it that I have no problem with hearing someone playing the piano
much better than I can? Why is it that I have no problem with seeing an
athlete skating on ice much better than I can? Why is it that I _have_
a problem with those who are "better" in mathematics? Good questions !
Han de Bruijn
Ross A. Finlayson
Hi Jesse,
I can, I think he'd like to explain to everybody exactly what he means,
and convince us all here of some new great idea.
I don't know him very well, I'm speaking with very limited, non-zero,
knowledge of his personality when it comes to talking about Harvey
Friedman the man you can assume I am not well qualified to comment
compared to himself, and vice versa.
That goes to his words too, Harvey is probably the most qualified
person to explain his own words, and Pi_0^1 is from Kolmogorov.
As he does, define his terms and so forth, and connect the related
meanings in agreeably sound ways, he's a voluble logician.
There is no universe in ZF. Positing the existence of one in a model
then leads to a problem because there is not one.
In terms of _the_ words, he's qualified to comment. Also, I think we
can agree that he's aware of this discussion, whether he is or not
being moot.
Ross
david petry
Gerry Myerson wrote:
> > > Does the biologist limit herself to what she can see
> > > through her microscope?
> >
> > Yes.
>
> So she never talks about oxygen and methane and uracil
> and other molecules that are too small to be seen in her microscope?
> She never talks about lions and tigers and bears (oh, my!)
> because they are too big to fit under her microscope?
> Biologists must be a lot more limited than I thought they were.
Actually, I think it is your own limited notion of what a microscope is
that is the problem here.
david petry
My point exactly! :-)
david petry
Herman Jurjus wrote:
> The way i understood Friedman (not this particular quote, but some of
> the other things he said), he claims that large cardinal axioms are
> (mathematically) relevant, -because- they influence what Pi01 sentences
> one can prove.
>
> So, he does claim that Pi01 sentences are the concrete things by which
> one can/should judge theories for their real, mathematical relevance.
> (And this is in line with what David Petry keeps saying).
>
> However, Friedman claims to have proofs that large cardinal axioms -do-
> make a difference about which Pi01 sentences can be proved. And
> precisely for that reason, he strongly -defends- large cardinal axioms
> as very meaningful and relevant also for the practicing mathematician.
>
> At least that's how i understand it.
I think you're right about Friedman's position, but let me make a
comment.
Feferman has asserted that the Pi01 implications of the large cardinal
axioms actually come from the assertion that the axioms are consistent
with ZFC, and not from the axioms themselves (at least that's how I
understand the situation). I haven't seen Friedman's response to
Feferman's criticism.
James Dolan
david petry <david_lawr...@yahoo.com> wrote:
given that you see the utility of not committing yourself in advance
to what constitutes a "microscope" it's peculiar that you don't see
the utility of not committing yourself in advance to what constitutes
a "mathematical microscope".
--
[e-mail address jdo...@math.ucr.edu]
Virgil
"david petry" <david_lawr...@yahoo.com> wrote:
Perhaps it is more by way of DP's inability to see what an anaalogy is.
Gerry Myerson
"david petry" <david_lawr...@yahoo.com> wrote:
Then perhaps you'll enlighten me.
Ross A. Finlayson
Hi Herman.
It's nice to hear from you.
People here have probably heard the parable of the elephant and the
blind men. Five blind guys stumble into an elephant. One feels a leg
and says "a rhinoceros". The other the trunk and "no, a snake." The
other, the tail, "a zebra". The others also fail in their
misunderstandings of what it is, I forget.
Burali-Forti is basically an elephant in ZF's parlor. Logicians hit
Russell and say "oh, no set of all sets." They notice something's
definitely there and say "oh, it must be a non-set class in the set
theory." Others similarly fail in their misunderstandings of what it
is. There must be a universe or there wouldn't be a quantifier, over
sets. It's elephants all the way up, the nearsighted don't crane well
enough, scaling elephants to just stare at the next one.
Last month, Petry noted that Zeilberger was having disillusionment
about transfinite cardinals, and wanted to go back to finitism. This
month, Petry notes Friedman observing a maximal element where none
exists in ZF and other theories supporting the powerset result, wanting
to get away from finitism. Finitist or infinitist, modern thinking
professional logicians, feeling and non, are moving past regular set
theories.
While that is so, it's still totally unpopular, don't fret. There is
time for you, too, to toe the line. Please save derisive laughter for
the end.
Of course, there can be only one theory.
Ross
Daryl McCullough
> "david petry" <david_lawr...@yahoo.com> wrote:
>
>> Gerry Myerson wrote:
>>
>> > > > Does the biologist limit herself to what she can see
>> > > > through her microscope?
>> > >
>> > > Yes.
>> >
>> > So she never talks about oxygen and methane and uracil
>> > and other molecules that are too small to be seen in her microscope?
>> > She never talks about lions and tigers and bears (oh, my!)
>> > because they are too big to fit under her microscope?
>> > Biologists must be a lot more limited than I thought they were.
>>
>> Actually, I think it is your own limited notion of what a microscope is
>> that is the problem here.
>
>Then perhaps you'll enlighten me.
When we are talking about biology (and by "we", I mean David Petry),
then "microscope" is used broadly to mean whatever it is the
biologists use to study whatever it is that they study. In contrast,
when we are talking about mathematics, then "microscope" is another
word for a computer.
--
Daryl McCullough
Ithaca, NY
--
NewsGuy.Com 30Gb $9.95 Carry Forward and On Demand Bandwidth
Herman Jurjus
Yes, that's right. But so what?
The ontological claim of the axioms ('there really is a universe of
sets, and it satisfies ZFC + whatever') is not the main point, anyway.
If it has mathematical advantages to 'imagine' such a universe, or
'reason as if' we have such a universe, then isn't that just as good?
(For the Pi01, concrete, mathematical consequences, anyway.)
I mean: couldn't we view 'ZFC+strong axioms' as a theory analogous to,
say, the physical theory that quarks exist?
In your terminology: couldn't we say that ZFC+.. can be seen as a theory
that does make testable claims about the computable, concrete world? And
that then Friedman's results mean that large cardinal axioms
do have testable consequences of the sort that you want?
--
Cheers,
Herman Jurjus
david petry
Rusin has already given us an example of a microscope that can be used
to observe lions and tigers and bears. Electron microscopes can be used
to observe small molecules. Particle accelerators can be used to
observe sub-atomic particles. These can all be thought of as
"microscopes".
david petry
Herman Jurjus wrote:
>> the Pi01 implications of the large cardinal
> > axioms actually come from the assertion that the axioms are consistent
> > with ZFC, and not from the axioms themselves
> Yes, that's right. But so what?
> The ontological claim of the axioms ('there really is a universe of
> sets, and it satisfies ZFC + whatever') is not the main point, anyway.
>
> If it has mathematical advantages to 'imagine' such a universe, or
> 'reason as if' we have such a universe, then isn't that just as good?
The mathematical advantages of imagining such a universe are imaginary,
and for applied mathematics, obfuscatory.
The whole point of my argument is that there are advantages to
accepting a reality check (something in the spirit of Ockham's Razor)
into the foundations of mathematics. In particular, if we want to
explain mathematics to a computer (artificial intelligence), then we
need to carefully examine the connections between mathematical concepts
and computation, and it would help to get rid of the imaginary universe
of Cantorian set theory.
ste...@nomail.com
> Herman Jurjus wrote:
>> If it has mathematical advantages to 'imagine' such a universe, or
>> 'reason as if' we have such a universe, then isn't that just as good?
> The mathematical advantages of imagining such a universe are imaginary,
> and for applied mathematics, obfuscatory.
> The whole point of my argument is that there are advantages to
> accepting a reality check (something in the spirit of Ockham's Razor)
> into the foundations of mathematics. In particular, if we want to
> explain mathematics to a computer (artificial intelligence), then we
> need to carefully examine the connections between mathematical concepts
> and computation, and it would help to get rid of the imaginary universe
> of Cantorian set theory.
Why would you assume that an AI would have any trouble with Cantor?
An AI capable of following a human readable mathematical proof
should have no more problem than a human. Perhaps you may
have a problem with Cantor, but it seems odd to insist
that AI's have to have the same problem.
Human intelligence does not think about the world exclusively in terms
of computation, and I see no reason to think an artificial intelligence
would have to be limited to seeing everything only in terms of
computation.
Stephen
david petry
> > The whole point of my argument is that there are advantages to
> > accepting a reality check (something in the spirit of Ockham's Razor)
> > into the foundations of mathematics. In particular, if we want to
> > explain mathematics to a computer (artificial intelligence), then we
> > need to carefully examine the connections between mathematical concepts
> > and computation, and it would help to get rid of the imaginary universe
> > of Cantorian set theory.
>
> Why would you assume that an AI would have any trouble with Cantor?
> An AI capable of following a human readable mathematical proof
> should have no more problem than a human.
If mathematics is nothing more than formal operations on meaningless
symbols, then Cantorian set theory is no different than any other part
of mathematics. But in fact, mathematical statements have *meaning*,
and this meaning is not equivalent to proof. The meaning of statements
which can be interpreted as predictions of the results of computational
experiments is clear (i.e. can easily be explained to an A.I.), while
the meaning of statements about the infinities of Cantorian set theory
is not clear.
Ross A. Finlayson
Well, you can look at Megill's Metamath, theorem prover software, and
how it interprets the nested intervals result as I last saw it. If it
weren't for a specific and unrelated result about the countability of
the rationals, it would show them uncountable, using its definitions.
Thus in removing basically that lemma, as it is an axiom in that simple
expression of that result in vacuo, it generates incorrect results.
Garbage in, garbage out.
A variety of modern methods of mathematical logic have, for example,
infinite elements in the natural integers, even as necessary for
completeness of the theory of the natural integers, implicit
compactification of Suarez' continuum that is the set of natural
integers. Then again, people think that the Presburger formulation of
the natural integers is complete while the Peano formulation of the
natural integers is not, where a finite proof of a statement in one is
a finite proof of that statement in the other.
Ross
Gerry Myerson
"david petry" <david_lawr...@yahoo.com> wrote:
> Gerry Myerson wrote:
> > In article <1143053638.2...@i39g2000cwa.googlegroups.com>,
> > "david petry" <david_lawr...@yahoo.com> wrote:
> >
> > > Gerry Myerson wrote:
> > >
> > > > > > Does the biologist limit herself to what she can see
> > > > > > through her microscope?
> > > > >
> > > > > Yes.
> > > >
> > > > So she never talks about oxygen and methane and uracil
> > > > and other molecules that are too small to be seen in her microscope?
> > > > She never talks about lions and tigers and bears (oh, my!)
> > > > because they are too big to fit under her microscope?
> > > > Biologists must be a lot more limited than I thought they were.
> > >
> > > Actually, I think it is your own limited notion of what a microscope is
> > > that is the problem here.
> >
> > Then perhaps you'll enlighten me.
>
> Rusin has already given us an example of a microscope that can be used
> to observe lions and tigers and bears.
Dave Rusin has given us a joke (for which we are very grateful),
not an example of anything any biologist has ever used to observe
anything.
> Electron microscopes can be used to observe small molecules. Particle
> accelerators can be used to observe sub-atomic particles. These can
> all be thought of as "microscopes".
I suppose in the same way that large cardinal axioms and whatnot
can all be thought of as microscopes for mathematicians.
ste...@nomail.com
I see no evidence that your "meaningful" statements will be any more
easily explained to an A.I. than Cantorian set theory will be.
You seem to have decided in advance how an A.I. will work
with little or no justification. Human's can understand the
meaning of statements about infinities. Why do you assume
that an A.I. will automatically be deficient in this regard?
Stephen
david petry
Gerry Myerson wrote:
> > Electron microscopes can be used to observe small molecules. Particle
> > accelerators can be used to observe sub-atomic particles. These can
> > all be thought of as "microscopes".
>
> I suppose in the same way that large cardinal axioms and whatnot
> can all be thought of as microscopes for mathematicians.
We are far from communicating with each other.
david petry
>> The meaning of statements
> > which can be interpreted as predictions of the results of computational
> > experiments is clear (i.e. can easily be explained to an A.I.), while
> > the meaning of statements about the infinities of Cantorian set theory
> > is not clear.
>
> I see no evidence that your "meaningful" statements will be any more
> easily explained to an A.I. than Cantorian set theory will be.
An artificial intelligence can perform experiments in the world of
computation, but it cannot perform experiments in the world of the
infinite (nor can we). Note that performing experiments on the
formalism of set theory is performing experiments within the finite
world observable through computation.
Virgil
"david petry" <david_lawr...@yahoo.com> wrote:
Actually, an A.I. should have less trouble with an axiom system and its
logical consequences than with "meaning" and "reality".
Virgil
"david petry" <david_lawr...@yahoo.com> wrote:
An artificial intelligence that is incapable of logical deduction may be
artificial, but wouldn't be very intelligent.
ste...@nomail.com
So an artificial intelligence will have the same means
to understand set theory as a human does. Explaining
set theory to a human is not particularly hard, at least
for most of them. It is by no means the hardest part
of mathematics, and so there is no reason that it will
be particularly hard to explain to an AI.
Stephen
ste...@nomail.com
Yes. Currently computers are excellent at dealing with formal
operations on meaningless symbols. The challenge to achieving
AI is dealing with "meaning" and "reality". The idea that something
that is "more meaningful" is going to be easier to explain to
a computer seems quite backwards
Stephen
Gerry Myerson
"david petry" <david_lawr...@yahoo.com> wrote:
I'm sorry to learn that you are having difficulty comprehending
my messages. I'll try harder if you'll let me know how best I
can help.
cbr...@cbrownsystems.com
> ste...@nomail.com wrote:
> > In sci.math david petry <david_lawr...@yahoo.com> wrote:
>
> > > The whole point of my argument is that there are advantages to
> > > accepting a reality check (something in the spirit of Ockham's Razor)
> > > into the foundations of mathematics. In particular, if we want to
> > > explain mathematics to a computer (artificial intelligence), then we
> > > need to carefully examine the connections between mathematical concepts
> > > and computation, and it would help to get rid of the imaginary universe
> > > of Cantorian set theory.
> >
> > Why would you assume that an AI would have any trouble with Cantor?
> > An AI capable of following a human readable mathematical proof
> > should have no more problem than a human.
>
> If mathematics is nothing more than formal operations on meaningless
> symbols, then Cantorian set theory is no different than any other part
> of mathematics.
Not sure I agree with the above 100%. In particular, it seems to me
even given that the basic symbols may be "meaningless", there is a
meaningful pattern to the action of selecting certain axioms and rules
which compose those symbols into complex objects with more or less
"meaningful" structure.
At any rate, it doesn't address his question - why do you assume that,
given evidence that at least some humans think Cantor's proof is
meaningful, that an AI wouldn't also think it was meaningful, in
exactly the same way?
> But in fact, mathematical statements have *meaning*,
> and this meaning is not equivalent to proof.
>
To paraphrase Pres. Bill, "It all depends on what 'mean' means".
To me, whether or not a particular mathematical construct is meaningful
is not totally dependent on having an actual corresponding physical
structure which is modelled by that construct.
I would hazard that you would say that a lack of a physical analouge
implies that the construct is "mere" formalism, and not meaningful.
> The meaning of statements
> which can be interpreted as predictions of the results of computational
> experiments is clear (i.e. can easily be explained to an A.I.), while
> the meaning of statements about the infinities of Cantorian set theory
> is not clear.
This seems circular to me.
How will we know that you have really "explained" the mathematical
meaning of these statements to the AI, and that the AI isn't just
performing formal operations on (to the AI) meaningless symbols? If
Metamath proves a theorem, have we "explained" that theorem to the
program?
It seems likely that in order to pass a mathematical Turing test, an AI
must already be "contaminated" with the same types of mathematical
qualia that we ourselves prefer - otherwise, we'll probably dismiss it
as a clever, but ultimately reducible, construct.
I mean, you're a human intelligence, David; and look at how hard it is
to get other people to agree with your ideas. If I reject /your/
accusations of "mere formalism", how am I any less likely to do so for
an AI?
And who cares what an AI thinks, anyway? Why shouldn't the measure of
meaningful mathematics be that it is meaningful to /people/ (and, more
particularly, to myself)?
Cheers - Chas
abo
david petry wrote:
>
> If mathematics is nothing more than formal operations on meaningless
> symbols, then Cantorian set theory is no different than any other part
> of mathematics. But in fact, mathematical statements have *meaning*,
> and this meaning is not equivalent to proof. The meaning of statements
> which can be interpreted as predictions of the results of computational
> experiments is clear (i.e. can easily be explained to an A.I.), while
> the meaning of statements about the infinities of Cantorian set theory
> is not clear.
What if any axioms would you be willing to assert for mathematics?
Are you able to verify the Successor Axiom (every natural number has a
successor) by a computational experiment?
david petry
Explain how you can think of a large cardinal axiom as a microscope, in
the same (or similar) sense as a particle accelerator can be thought of
as a microscope.
david petry
cbr...@cbrownsystems.com wrote:
> At any rate, it doesn't address his question - why do you assume that,
> given evidence that at least some humans think Cantor's proof is
> meaningful, that an AI wouldn't also think it was meaningful, in
> exactly the same way?
Here's the idea: a statement is "meaningful" if you can give me a
hypothetical situation in which your hypothetical belief in the
statement would lead you make a different decision about a course of
action than if you didn't hold that belief. By that criterion,
mathematics in general is definitely meaningful, but Cantor's ideas
about infinite sets are not.
> > The meaning of statements
> > which can be interpreted as predictions of the results of computational
> > experiments is clear (i.e. can easily be explained to an A.I.), while
> > the meaning of statements about the infinities of Cantorian set theory
> > is not clear.
>
> This seems circular to me.
>
> How will we know that you have really "explained" the mathematical
> meaning of these statements to the AI, and that the AI isn't just
> performing formal operations on (to the AI) meaningless symbols? If
> Metamath proves a theorem, have we "explained" that theorem to the
> program?
As I have already explained, I'm taking the view that the world of
computation is a reality, and mathematics is the science that studies
the phenomena observed in that world. From this point of view, a
statement is meaningful if and only if it is a description of some
phenomenon observed in that world.
To prove that a statement is a formal theorem of some formal theory is
not the same as showing the the statement has a meaningful
interpretation.
> And who cares what an AI thinks, anyway? Why shouldn't the measure of
> meaningful mathematics be that it is meaningful to /people/ (and, more
> particularly, to myself)?
So you're saying that there is no point in trying to distinguish
science from pseudoscience. Well, I disagree, and have previously
argued the point at length.
david petry
abo wrote:
> david petry wrote:
> >
> > If mathematics is nothing more than formal operations on meaningless
> > symbols, then Cantorian set theory is no different than any other part
> > of mathematics. But in fact, mathematical statements have *meaning*,
> > and this meaning is not equivalent to proof. The meaning of statements
> > which can be interpreted as predictions of the results of computational
> > experiments is clear (i.e. can easily be explained to an A.I.), while
> > the meaning of statements about the infinities of Cantorian set theory
> > is not clear.
>
> What if any axioms would you be willing to assert for mathematics?
The use of formalisms is undeniably a useful tool in mathematics, and
in fact, that's what mathematics is all about. But I would not define
mathematics in terms of axioms.
> Are you able to verify the Successor Axiom (every natural number has a
> successor) by a computational experiment?
We haven't been talking about "verification" here, an you haven't taken
the trouble to define what you mean by that.
What do you mean, and why can't you answer the question for yourself?
abo
david petry wrote:
> abo wrote:
> > david petry wrote:
> > >
> > > If mathematics is nothing more than formal operations on meaningless
> > > symbols, then Cantorian set theory is no different than any other part
> > > of mathematics. But in fact, mathematical statements have *meaning*,
> > > and this meaning is not equivalent to proof. The meaning of statements
> > > which can be interpreted as predictions of the results of computational
> > > experiments is clear (i.e. can easily be explained to an A.I.), while
> > > the meaning of statements about the infinities of Cantorian set theory
> > > is not clear.
> >
> > What if any axioms would you be willing to assert for mathematics?
>
> The use of formalisms is undeniably a useful tool in mathematics, and
> in fact, that's what mathematics is all about. But I would not define
> mathematics in terms of axioms.
I am not asking for you to define mathematics in terms of axioms. I am
asking what if any axioms you yourself would assert. If you are not
willing to assert any, then it is not just Cantor's set theory which is
causing problems for you...
>
>
> > Are you able to verify the Successor Axiom (every natural number has a
> > successor) by a computational experiment?
>
> We haven't been talking about "verification" here, an you haven't taken
> the trouble to define what you mean by that.
>
> What do you mean, and why can't you answer the question for yourself?
Sorry, I shouldn't have used "verified". I should have asked (using
the language you have set forth): is the Successor Axiom meaningful?
In case you are wondering, I do have an answer to the question, but
that is not what interests me. I am trying to understand your POV.
Either you are just waffling on, or you should be able to say, when
faced with simple concrete examples, whether a mathematical sentence
has meaning or not (and why).
Ross A. Finlayson
Have you heard of axiomless systems of natural deduction?
There's only one theory with no axioms.
All the true sentences are describe in terms of each other. It's
similar to the dichotomy or dialectic of points and lines or geometric
hyperspaces, numbers and number spaces, points and the physical
universe, er, or sets and elements or other natural constructions, that
basically exist.
Then, in large part it's a communicative logic, about not just you and
the universe or you and the void but you and another and the void, or
universe.
For thousands of years or more humanity has been trying to understand
reality, or numbers for example or other meaningful mathematical
constructs in their most primitive forms and what that means in terms
of everything else, the context, which in computer science terms in an
irregular universe with ubiquitous natural irregular elements has each
containing a reference to that most primitive excised element in its
complement int the universe or conveniently set of all sets.
That gets into what I call Janus' introspection and currency,
dually-self-intraconsistency, as precursor to even the paraconsistent
dialetheism that is so germane to modern technical philosophy today, in
the consideration of the ur-element of a comprehensive theory with no
non-logical or proper axioms that via unrestricted comprehension only
expresses true statements at a very fundamental level, because the
expression of what would be a false statement makes it true and changes
every other statement to agree because a paradox would be paradoxical.
A non-objectivist viewpoint, and that doesn't mean Rayndian which is
selfish, thus allows one to consider a consistent and complete theory.
So, then, an axiomless or null axiom theory, which is dually
universally axiomatized and either way enables reinterpretation of
Goedelian incompleteness, does enable natural deduction and induction
over the contents of then the universe in quantification over them all,
universal quantification and existential quantification which are much
the same thing, mechanistically, with very primitive ordinals, and thus
is perhaps richer than you might expect from a theory with no axioms in
terms of expressive content.
Ciao,
Ross F.
david petry
abo wrote:
> david petry wrote:
> > > What if any axioms would you be willing to assert for mathematics?
> >
> > The use of formalisms is undeniably a useful tool in mathematics, and
> > in fact, that's what mathematics is all about. But I would not define
> > mathematics in terms of axioms.
>
> I am not asking for you to define mathematics in terms of axioms. I am
> asking what if any axioms you yourself would assert. If you are not
> willing to assert any, then it is not just Cantor's set theory which is
> causing problems for you...
I am suggesting a different "mindset" for mathematics. We think of the
world of computation as something that simply exists. We think of
ourselves as being somehow thrown into the middle of it, and now we're
trying to understand it, and we proceed by creating models of it. So
"axioms" are rules by which we create the models. PA with restrictions
on the use of the existential quantifier is a good starting point to
create a model of the world of computation, and the successor axiom is
part of that.
cbr...@cbrownsystems.com
> cbr...@cbrownsystems.com wrote:
>
> > At any rate, it doesn't address his question - why do you assume that,
> > given evidence that at least some humans think Cantor's proof is
> > meaningful, that an AI wouldn't also think it was meaningful, in
> > exactly the same way?
>
> Here's the idea: a statement is "meaningful" if you can give me a
> hypothetical situation in which your hypothetical belief in the
> statement would lead you make a different decision about a course of
> action than if you didn't hold that belief. By that criterion,
> mathematics in general is definitely meaningful, but Cantor's ideas
> about infinite sets are not.
I don't see how you reach that conclusion; since your criterion
includes hypotheticals that are excessively broad in scope. "I will
kill your cat based on whether or not you can show that |P(X)| > |X|
in ZFC".
At any rate - you chose to justify your stand by claiming that an AI
would reject Cantor's infinities; and that therefore so should we. It's
clear that /you/ reject them - the question I asked was why do you
think an AI would neccessarily reject them?
>
>
>
> > > The meaning of statements
> > > which can be interpreted as predictions of the results of computational
> > > experiments is clear (i.e. can easily be explained to an A.I.), while
> > > the meaning of statements about the infinities of Cantorian set theory
> > > is not clear.
> >
> > This seems circular to me.
> >
> > How will we know that you have really "explained" the mathematical
> > meaning of these statements to the AI, and that the AI isn't just
> > performing formal operations on (to the AI) meaningless symbols? If
> > Metamath proves a theorem, have we "explained" that theorem to the
> > program?
>
> As I have already explained, I'm taking the view that the world of
> computation is a reality, and mathematics is the science that studies
> the phenomena observed in that world.
Yes, but why would an AI neccessarily share your views? Clearly, many
HIs (Human Intelligences) do not.
BTW, half of the flack you receive here comes from using the word
"meaningful" when you might better be served using the word "concrete"
or "scientific".
The other half of the flack comes from asserting that mathematics is a
science; a stand which is controversial, to say the least.
For example, if I substitute in your statements below:
> From this point of view, a
> statement is meaningful if and only if it is a description of some
> phenomenon observed in that world.
>
> To prove that a statement is a formal theorem of some formal theory is
> not the same as showing the the statement has a meaningful
> interpretation.
>
"From this point of view, a statement is scientific if and only if it
is a description of some phenomenon observed in that world.
"To prove that a statement is a formal theorem of some formal theory is
not the same as showing the the statement has a scientific
interpretation."
I would agree with both of these statements. I would disagree, however,
with the stand that mathematics is only concerned with generating
statements which have scientific interpretations.
I would also disagree that the only mathematical statements which have
scientific interpretations are those which are strictly models of a
finite boolean algebra.
> > And who cares what an AI thinks, anyway? Why shouldn't the measure of
> > meaningful mathematics be that it is meaningful to /people/ (and, more
> > particularly, to myself)?
>
> So you're saying that there is no point in trying to distinguish
> science from pseudoscience. Well, I disagree, and have previously
> argued the point at length.
That's a strawman argument. I'm not claiming mathematics is a science,
you are. I'm as anti-pseudoscience as the next intellectual! And I
think the claim that if mathematics is a not a science, then it is a
religion, is a false dichotomy.
And I still see no particular reason why an AI, which claims to be able
to ascertain meaning in mathematical statements, should not agree with
my understanding of "meaning".
Cheers - Chas
david petry
cbr...@cbrownsystems.com wrote:
> david petry wrote:
> > Here's the idea: a statement is "meaningful" if you can give me a
> > hypothetical situation in which your hypothetical belief in the
> > statement would lead you make a different decision about a course of
> > action than if you didn't hold that belief. By that criterion,
> > mathematics in general is definitely meaningful, but Cantor's ideas
> > about infinite sets are not.
>
> I don't see how you reach that conclusion; since your criterion
> includes hypotheticals that are excessively broad in scope. "I will
> kill your cat based on whether or not you can show that |P(X)| > |X|
> in ZFC".
>From my point of view, what you are saying is quite flaky. I don't
know how to respond.
Virgil
"david petry" <david_lawr...@yahoo.com> wrote:
> cbr...@cbrownsystems.com wrote:
>
> > At any rate, it doesn't address his question - why do you assume that,
> > given evidence that at least some humans think Cantor's proof is
> > meaningful, that an AI wouldn't also think it was meaningful, in
> > exactly the same way?
>
> Here's the idea: a statement is "meaningful" if you can give me a
> hypothetical situation in which your hypothetical belief in the
> statement would lead you make a different decision about a course of
> action than if you didn't hold that belief. By that criterion,
> mathematics in general is definitely meaningful, but Cantor's ideas
> about infinite sets are not.
Then the fact that I, as a believer that Cantor's proof is meaningful
have chosen to respond to you, whereas as a non-believer, I would not
have done, verifies by your own criteria that Cantor's ideas are
meaningful!
> As I have already explained, I'm taking the view that the world of
> computation is a reality, and mathematics is the science that studies
> the phenomena observed in that world. From this point of view, a
> statement is meaningful if and only if it is a description of some
> phenomenon observed in that world.
Then how is your description meaningful? Attitudes of mind are not of
themselves observable.
>
>
> > And who cares what an AI thinks, anyway? Why shouldn't the measure of
> > meaningful mathematics be that it is meaningful to /people/ (and, more
> > particularly, to myself)?
>
> So you're saying that there is no point in trying to distinguish
> science from pseudoscience.
Then David accepts the latter label himself? Mathematicians do not say
that mathematics is science at all, they reject that fallacy entirely,
so it is not pseudo-anything. So that if anything is to be pseudo, it
must be David.
Virgil
"david petry" <david_lawr...@yahoo.com> wrote:
> The use of formalisms is undeniably a useful tool in mathematics, and
> in fact, that's what mathematics is all about. But I would not define
> mathematics in terms of axioms.
Non-mathematicians are hardly in a position to "define" mathematics at
all.
Virgil
"david petry" <david_lawr...@yahoo.com> wrote:
> abo wrote:
> > david petry wrote:
> > > abo wrote:
>
> > > > What if any axioms would you be willing to assert for mathematics?
> > >
> > > The use of formalisms is undeniably a useful tool in mathematics, and
> > > in fact, that's what mathematics is all about. But I would not define
> > > mathematics in terms of axioms.
> >
> > I am not asking for you to define mathematics in terms of axioms. I am
> > asking what if any axioms you yourself would assert. If you are not
> > willing to assert any, then it is not just Cantor's set theory which is
> > causing problems for you...
>
> I am suggesting a different "mindset" for mathematics.
The mindset that mathematicians currently have is producing more new
mathematics than physicists can handle.
Ross A. Finlayson
What's your criterion?
Hancher, everyone here has a sincere interest in mathematics. Your
quip there is just a stupid ad hominem. You're just being a flaming
troll, and it's evident that you're that way with a wide variety of
different people, none of whom is going to hoodwink everybody else
without your input, as they engage in serious and lighthearted
discussions with others. Obviously this is a gentle personal jibe.
You don't speak for, us. Perhaps on the other newsgroups you care to
troll your plainly non-mathematical snipings would be topical, here
they are not. You have been unweighting your opinion. I don't feel
that I have to say that, not something nice.
And now for something different: In appreciable senses, the
transfinite cardinals are mathematics, whether or not they are unsound,
that physics does not need. The Fraenkel likened them to an illness.
Less than absolute zero is hotter than any finite positive temperature.
Mathematics is defined in terms of definitions.
Harvey is aware of this discussion. I invited him to comment. He
might.
Ciao,
Ross F.
abo
Ross A. Finlayson wrote:
>
> Have you heard of axiomless systems of natural deduction?
>
written as inferences, then yes I have heard of them. Sorry, I was
using axioms as shorthand for "assumption"; and in the case when the
assumption is made as an inference, I would like to know the inference.
>
> All the true sentences are describe in terms of each other. It's
> similar to the dichotomy or dialectic of points and lines or geometric
> hyperspaces, numbers and number spaces, points and the physical
> universe, er, or sets and elements or other natural constructions, that
> basically exist.
And here and the rest, I'm afraid you lost me.
<snip>
abo
david petry wrote:
.
>
> I am suggesting a different "mindset" for mathematics.
Is it fair to say then that your point is not what mathematics *is*,
but what mathematics *should* be... ?
> We think of the
> world of computation as something that simply exists. We think of
> ourselves as being somehow thrown into the middle of it, and now we're
> trying to understand it, and we proceed by creating models of it.
"We" think of ...? Who is we? I don't think of it this way at all.
Is this just the royal "we"? And I'm sorry I don't understand what you
mean be "we proceed by creating models of it". Is this some variant on
"mathematical theorems are not discovered; they are created." ?
> So
> "axioms" are rules by which we create the models. PA with restrictions
> on the use of the existential quantifier is a good starting point to
> create a model of the world of computation, and the successor axiom is
> part of that.
It sounds here that you are saying the Successor Axiom has meaning. If
that is so, my follow-up is: what prediction of *computation* is the
Successor Axiom making? It's making a prediction - if you have a
natural number, then you can find a natural number that is one greater
- but this prediction does not seem to be a computation at all.
Kent Paul Dolan
> Hancher, everyone here has a sincere interest in
> mathematics.
Nonsense.
James Harris on his record stands as more than
sufficient counterexample, and he is hardly alone.
Many are here merely to glorify their egos by
quarreling with accepted mathematics they happen to
be unable to comprehend or accept. That's a sincere
interest in ego gratification, not in mathematics.
If they had "a sincere interest in mathematics",
they'd be busy studying textbooks and technical
reports and journal articles to repair their flaws
in comprehension, not shepherding infinitely
meaningless discussions such as the current OP is
doing in this current thread, shifting ground
constantly, avoiding even answering each cogent
rebuttal, arguing strawmen and flawed analogies such
as what does or does not constitute a "microscope",
rather than addressing the subject matter:
mathematics.
So many complete innumeracies are being added to
this thread, by many posters who should, but don't,
know better, it's obvious that those "sincerely
interested in mathematics" are in a minority, but
addressing the points thus uttered one by one would
overflow most spool directories.
IMEO
xanthian.
cbr...@cbrownsystems.com
> cbr...@cbrownsystems.com wrote:
> > david petry wrote:
>
> > > Here's the idea: a statement is "meaningful" if you can give me a
> > > hypothetical situation in which your hypothetical belief in the
> > > statement would lead you make a different decision about a course of
> > > action than if you didn't hold that belief. By that criterion,
> > > mathematics in general is definitely meaningful, but Cantor's ideas
> > > about infinite sets are not.
> >
> > I don't see how you reach that conclusion; since your criterion
> > includes hypotheticals that are excessively broad in scope. "I will
> > kill your cat based on whether or not you can show that |P(X)| > |X|
> > in ZFC".
>
> .From my point of view, what you are saying is quite flaky. I don't
> know how to respond.
Well, you could start by responding to the main body of my post, which
reiterates the question you seem intent on evading: How does anything
you're saying relate to whether or not an AI would accept or reject
"Cantorian infinities"?
I already know that /you/ don't accept them, and that you further
propose to delineate acceptable statements via application of a
property you name "meaningful", for which you provide various
definitions to help us "get it".
But given that many /humans/ don't agree with your restriction to only
those statements which have the property you name "meaningful", why
should an /AI/ neccessarily agree with you?
That agreement is a seminal justifcation for your program, is it not?
Cheers - Chas
PS: I sincerely hope you didn't take my remarks above as the slightest
threat to you or your cat - I was imagining ME being threatened with
the life of MY cat by another; and the words in quotes were intended to
be those of the villian. I truly apologize if you took any offense!
Virgil made my point much more simply - the fact that we are typing
here is evidence that actions can follow from beliefs; and that's why I
thought your criterion was "excessively broad".
Ross A. Finlayson
That's agreeable. It's a question of interpretation, though. I don't
read Harris' posts, he descended into ad hominem nonsense in basically
his deep-seated desire to be mathematically productive, driven there by
others' ad hominem nonsense where besides that he is generally affable
if a one-track mind of sorts about his unfactoring or whatever he's
calling multiplication or multiplicative partitions today. The point
there is being that he's happy to talk about mathematics, and does if
left undisturbed.
This is an open community and suffers the pathologies and enjoys the
niceties of other open communities of people with interest and not
necessarily agreement where there is enough of an assembled brainshare
to cogently argue divisive points at hand. This is in the context of
extended discussion about, well, basically from my perspective
infinity, which has occupied a lion's share of mathematical
conceptualization for all of recorded history. There are not many
contentious issues on this newsgroups among all the sharing of
knowledge. There just aren't many contentious issues in mathematics,
because there are unequivocably correct answers to well-defined
questions.
Usenet, thirty years old or so, is in ways a new forum in the context
of these kinds of discussions, in comparison to parcel post of personal
and group communications among groups and individuals of the previous
millenia. You can be quite assured that posts are read by relatively
many, if not many in an absolute scale. You read some of them. As
well, where this is thankfully a discussion of interested parties who
are skilled and knowledgeable in the subject area in absolute terms,
you'll notice that in general discussions remain topical, there's
basically politeness in moderate effusion, and the posters who are
overwhelmingly male (hi Barb) generally feel macho about their
contributions, deriving satisfaction, because math is hard, and to
impress with an elegant formulation in mathematics is sublime, for
those who appreciate those kinds of things.
There are people who read and contribute to sci.math and sci.logic who
can compete in terms of ability with anyone in the world in many of a
myriad styles of specialization in mathematical practice and theory,
not everyone, not necessarily myself, exceptional individuals. Again,
only a small proportion of those practicants are aware of this kind of
forum, many have interest but not the time, points being that seriously
interested mathematics affcianados moreso than cross-posting usenet
rabble-rousers, pundits, the nut gallery, etcetera represent people who
come here for mathematics, and yes we do know what we're talking about.
I use it as a soap box to promote my own personal theories, selfishly,
theories of mathematical logic, because I feel reasonable that is the
correct thing to do, because I see benefit in that notion for more than
myself, altruism _and_ self-promotion, and I do believe what I say.
I felt honored that Harvey replied to me and said he was surprised
because he'd never really heard of sci.logic before, and was
intellectually stimulated by its sophistication. He asks if there are
other fora similar in nature, basically in the caliber and volume of
postings along the lines of sci.logic or FOM the mailing list, those
having a large shared readership, in terms of the day's methods of
informal public communications among serious practitioners and
theorists in academia, privacy, and the public about nascent and
cutting-edge issues of relevance in concern to mathematical logicians.
What else is there?
That's a good question.
This discussion group is _obviously_ not peer-reviewed,
brick-and-mortar, print bound and microfiched, stitched encasings on
dead trees of the corpus of collected knowledge that is the library of
collected human knowledge that is rapidly moving towards general
electronic dissemination. This basically _is_ a part of the "open
Internet discussion group."
http://www.arxiv.org/
http://citeseer.ist.psu.edu/
http://scholar.google.com/
Those are Internet access points to basically publicly available
research papers in high technology and so forth, for example with the
HEP and so forth. The local library has quite more specialized and
in-depth research tools available, Athena, Medline, Euler,
what-have-you.
It _is_ definitely peer-reviewed, in a very open manner that anyone can
see, and refer to later basically irrepudiably; although there are some
concerns about that, distribution is decentralized. There is some
protection for the authors, who hold their own copyrights of their
posts, for lack of a better word, barring eschewment. What it is is
cheap self-publishing, with distribution, and spam is roundly denied,
in the sense of the round file. Many of the smaller groups are more
dedicated with higher signal/noise, eg sci.stat.math,
sci.math.symbolic, actual contention in mathematics generally is right
on sci.math and sci.logic, basically as a casual discussion group with
some dozens or score of regulars, generally known by their names,
words, and opinions. If you want to hammer out ideas in mathematics,
you can find someone here to educatedly discuss them with you,
basically telling you all about it.
Pertti Lounesto once argued that Usenet could not be a vehicle for the
advancement of scientific discussion, I disagreed. Usenet is dying,
you heard it here first, film at 11. There are many specialist "blogs"
and many have at least three readers which is great, some many more,
sometimes those are closely held, not very... interactive, representing
discussion, not just erudition. Triality is quadratic.
http://www.google.com/search?q=infinity+Cantor+blog
http://groups.google.com/groups/search?q=infinity+Cantor
Cool, Rucker has a web-log. He is perhaps the most recognized
popularizer of standard late 70's theory of infinite ordinals and
cardinals. If you're interested in technical infinities and haven't
read it you should read "Infinity and the Mind", skipping the parts
about Rucker's mind.
There are dozens, if not hundreds, of coffee-jacked hackers with broad
mathematical knowledge waiting to read the next post, and capable of
writing it.
So, here's a question: what other open, if filtered, discussion groups
are there with large memberships about specifically mathematics and
mathematical logic, for serious, interested mathematicians to put forth
their views?
Ciao,
Ross F.
Ross A. Finlayson
Hi,
I'm not quite sure what to tell you. If you're interested in further
explanations of those words, there are some thousands of pages of
expository material here about it.
I think the axioms of ZFC, besides regularity, are theorems, or
inferences, of reduction to an axiomless system of natural deduction.
In calling a theory, or A theory, the null axiom theory, axiomless,
that doesn't mean without general inference with the predicate calculus
and truth tables and so forth. It just means that there are no
non-logical or proper axioms, instead in the practice of reason
inferring, or deducing, in the constructive or intuitionist sense, and
both from irrelevantly mutually falsifiable alternatives, acceptable
truths. Only truths are encoded, and that's rather standard.
M. abo, you could consider a theistic approach to this mathematical
logic. Humans have no capability to assert truth in axiomatization,
only to acknowledge it in agreement with each other, that which already
is. To axiomatize is a usurpation, and it's not necessary to derive
agreeble, and even fundamental, truths.
Assertion of axioms leads in very well-known manners generally
attributed to Goedel to incompleteness of axiomatized theories with
would-be meaning. In a sense, then, as there are truths in that theory
that can not be expressed, they are not, in that sense of
incompleteness there is inconsistency, as all true statements are true,
and thus reactive invalidation ensues.
Consider Zermelo-Fraenkel set theory, basically the gold standard.
There are only sets in the theory, quantify over sets, that's not a
set, error.
These things can be quite clear. All the true statements are described
in terms of each other.
Thank you for your reply.
Ciao,
Ross F.
Herman Jurjus
> Herman Jurjus wrote:
>
>>> the Pi01 implications of the large cardinal
>>> axioms actually come from the assertion that the axioms are consistent
>>> with ZFC, and not from the axioms themselves
>
>> Yes, that's right. But so what?
>> The ontological claim of the axioms ('there really is a universe of
>> sets, and it satisfies ZFC + whatever') is not the main point, anyway.
>>
>> If it has mathematical advantages to 'imagine' such a universe, or
>> 'reason as if' we have such a universe, then isn't that just as good?
>
> The mathematical advantages of imagining such a universe are imaginary,
Not if Friedman is right.
> and for applied mathematics, obfuscatory.
You may be right, but i doubt that applied mathematics doesn't also use
lots of 'leading intuitions in the background'.
> The whole point of my argument is that there are advantages to
> accepting a reality check (something in the spirit of Ockham's Razor)
> into the foundations of mathematics.
So you not only demand that mathematical theories have observable,
concrete -consequences- (and are judged by these), but that the
assumptions in themselves must all be so concrete and down-to-earth?
Not even physicists demand that of their theories. So why should
mathematicians try to be more catholic than the pope?
In particular, if we want to
> explain mathematics to a computer (artificial intelligence), then we
> need to carefully examine the connections between mathematical concepts
> and computation, and it would help to get rid of the imaginary universe
> of Cantorian set theory.
Your ideas may very well lead to very interesting alternative
mathematics, one day. Why don't you simply start working on that, and
simply ignore the Boetians? Anyway, why couldn't both forms of
mathematics be allowed existence in this world, living next to each
other? At least for the first few decades or so.
(Remember non-euclidean geometry? Look what it did to the once glorious
and exclusive position of Euclidean Space. It's all possible, but
everyone's got to start small and modest.)
--
Cheers,
Herman Jurjus
Herman Jurjus
> Herman Jurjus wrote:
BTW, do you happen to know Nik Weaver's ideas as well?
What's your opinion on that?
--
Cheers,
Herman Jurjus
david petry
cbr...@cbrownsystems.com wrote:
> How does anything
> you're saying relate to whether or not an AI would accept or reject
> "Cantorian infinities"?
I don't understand why I have to keep repeating this.
An A.I. lives in the world of computation. It can perform experiments
in the world of computation. And thus, a statement which makes
predictions about the results of computational experiments is something
the A.I. could put to the test, and hence the A.I. could understand its
meaning. The A.I. has no more access to the world of the infinite than
we do.
david petry
abo wrote:
> david petry wrote:
> .
> >
> > I am suggesting a different "mindset" for mathematics.
>
> Is it fair to say then that your point is not what mathematics *is*,
> but what mathematics *should* be... ?
Not really.
> > We think of the
> > world of computation as something that simply exists. We think of
> > ourselves as being somehow thrown into the middle of it, and now we're
> > trying to understand it, and we proceed by creating models of it.
>
> "We" think of ...? Who is we?
Those who accept the new mindset.
> And I'm sorry I don't understand what you
> mean be "we proceed by creating models of it". Is this some variant on
> "mathematical theorems are not discovered; they are created." ?
I'm suggesting that mathematics can be done exactly the same way that
science in general is done.
> > So
> > "axioms" are rules by which we create the models. PA with restrictions
> > on the use of the existential quantifier is a good starting point to
> > create a model of the world of computation, and the successor axiom is
> > part of that.
>
> It sounds here that you are saying the Successor Axiom has meaning. If
> that is so, my follow-up is: what prediction of *computation* is the
> Successor Axiom making? It's making a prediction - if you have a
> natural number, then you can find a natural number that is one greater
> - but this prediction does not seem to be a computation at all.
S(3) = 3+1 = 4
If we're going to explain the "meaning" of the successor axiom to an
A.I., we must explain to the A.I. what every human knows but does not
say: when we test an assertion, we test it with small values of the
variables, and leave the large values for another day.,
david petry
Herman Jurjus wrote:
> BTW, do you happen to know Nik Weaver's ideas as well?
> What's your opinion on that?
Thanks for mentioning him. It looks interesting. I'm going to do a
little reading ....
abo
david petry wrote:
> abo wrote:
> > david petry wrote:
> > .
> > >
> > > I am suggesting a different "mindset" for mathematics.
> >
> > Is it fair to say then that your point is not what mathematics *is*,
> > but what mathematics *should* be... ?
>
> Not really.
I'll toss my hands in the air on this one !
>
> > > We think of the
> > > world of computation as something that simply exists. We think of
> > > ourselves as being somehow thrown into the middle of it, and now we're
> > > trying to understand it, and we proceed by creating models of it.
> >
> > "We" think of ...? Who is we?
>
> Those who accept the new mindset.
So it's pretty close to the royal "we"....
>
> > And I'm sorry I don't understand what you
> > mean be "we proceed by creating models of it". Is this some variant on
> > "mathematical theorems are not discovered; they are created." ?
>
> I'm suggesting that mathematics can be done exactly the same way that
> science in general is done.
>
Of course it *can*. But it's not done like that way now and I doubt
that it will be done that way anywhere in the near future.
>
>
> > > So
> > > "axioms" are rules by which we create the models. PA with restrictions
> > > on the use of the existential quantifier is a good starting point to
> > > create a model of the world of computation, and the successor axiom is
> > > part of that.
> >
> > It sounds here that you are saying the Successor Axiom has meaning. If
> > that is so, my follow-up is: what prediction of *computation* is the
> > Successor Axiom making? It's making a prediction - if you have a
> > natural number, then you can find a natural number that is one greater
> > - but this prediction does not seem to be a computation at all.
>
> S(3) = 3+1 = 4
OK but the prediction is not computational in nature, but ontological.
Someone who believes in uncountable cardinals could just as easily come
up with computational equations involving them. So why don't you
accept that they exist? Somewhere you are making a distinction as to
the type of computation which is permitted, and I presume you are
restricting yourself to the type of computation which can be done by a
computer. But real computers don't obey the Successor Axiom - for
every computer there is a largest number which they can store.
>
> If we're going to explain the "meaning" of the successor axiom to an
> A.I., we must explain to the A.I. what every human knows but does not
> say: when we test an assertion, we test it with small values of the
> variables, and leave the large values for another day.,
That wouldn't explain the meaning of the successor axiom, but only how
to get the successor in the domain of small natural numbers.
david petry
abo wrote:
> david petry wrote:
> > I'm suggesting that mathematics can be done exactly the same way that
> > science in general is done.
> >
>
> Of course it *can*. But it's not done like that way now and I doubt
> that it will be done that way anywhere in the near future.
Nevertheless, I'm suggesting that it would be a really good idea to do
it that way.
> > > > So
> > > > "axioms" are rules by which we create the models. PA with restrictions
> > > > on the use of the existential quantifier is a good starting point to
> > > > create a model of the world of computation, and the successor axiom is
> > > > part of that.
> > >
> > > It sounds here that you are saying the Successor Axiom has meaning. If
> > > that is so, my follow-up is: what prediction of *computation* is the
> > > Successor Axiom making? It's making a prediction - if you have a
> > > natural number, then you can find a natural number that is one greater
> > > - but this prediction does not seem to be a computation at all.
> >
> > S(3) = 3+1 = 4
>
> OK but the prediction is not computational in nature, but ontological.
> Someone who believes in uncountable cardinals could just as easily come
> up with computational equations involving them. So why don't you
> accept that they exist?
The formalism -- the game of manipulating meaningless symbols -- most
certainly "exists".
> Somewhere you are making a distinction as to
> the type of computation which is permitted, and I presume you are
> restricting yourself to the type of computation which can be done by a
> computer.
Yes, and hence, computations that can actually be called computations.
> But real computers don't obey the Successor Axiom - for
> every computer there is a largest number which they can store.
Which I thought I answered in the next paragraph.
> > If we're going to explain the "meaning" of the successor axiom to an
> > A.I., we must explain to the A.I. what every human knows but does not
> > say: when we test an assertion, we test it with small values of the
> > variables, and leave the large values for another day.,
> That wouldn't explain the meaning of the successor axiom, but only how
> to get the successor in the domain of small natural numbers.
That's enough to give it meaning.
Jesse F. Hughes
And it has no less access to the world of the infinite than we do,
right?
And yet a whole slew of folks think that ZF is meaningful in some
sense. So an AI would have no less reason to find it meaningful than
human mathematicians do.
But, of course, this is really a week argument in any case. We don't
choose our mathematical foundations on the grounds that one day, we
might have to explain them to HAL 9000.
--
"Many argue that its programmers have turned out shoddy programs, but
[their] objective is to make profit, not superlative programs per
se. By the profit criterion, Microsoft has been one of the greatest
companies in the history of this country." -- ADTI defends Microsoft
abo
david petry wrote:
> abo wrote:
> > david petry wrote:
>
> > > I'm suggesting that mathematics can be done exactly the same way that
> > > science in general is done.
> > >
> >
> > Of course it *can*. But it's not done like that way now and I doubt
> > that it will be done that way anywhere in the near future.
>
> Nevertheless, I'm suggesting that it would be a really good idea to do
> it that way.
Suggest away. By the way, this would again seem to indicate that your
point is about what mathematics *should* be, not what mathematics *is*.
Well obviously I don't think you did !
>
> > > If we're going to explain the "meaning" of the successor axiom to an
> > > A.I., we must explain to the A.I. what every human knows but does not
> > > say: when we test an assertion, we test it with small values of the
> > > variables, and leave the large values for another day.,
>
> > That wouldn't explain the meaning of the successor axiom, but only how
> > to get the successor in the domain of small natural numbers.
>
> That's enough to give it meaning.
This is beginning to become an argument with diminishing returns...
Try it this way... The small values of the natural numbers are also the
small values of the ordinals. You want to permit an inference from
these small values to all the natural numbers but not to all the
ordinals. I don't see any non-circular reason why you think you can
make this distinction.
Virgil
"david petry" <david_lawr...@yahoo.com> wrote:
It lives in a world governed by rules (axioms), so if it cannot
understand that world, what sort of world *can* it "understand"?
Virgil
"david petry" <david_lawr...@yahoo.com> wrote:
> I'm suggesting that mathematics can be done exactly the same way that
> science in general is done.
Then let us counter-suggest that science be done exactly the way that
mathematics is done.
Makes as much sense. At least to mathematician.
Virgil
"david petry" <david_lawr...@yahoo.com> wrote:
> I'm suggesting that mathematics can be done exactly the same way that
> science in general is done.
And, as there are thousands of different theories as to how science is
done, will there also be thousands of theories as to how mathetmaics is
to be done?
Virgil
"david petry" <david_lawr...@yahoo.com> wrote:
> abo wrote:
> > david petry wrote:
>
> > > I'm suggesting that mathematics can be done exactly the same way that
> > > science in general is done.
> > >
> >
> > Of course it *can*. But it's not done like that way now and I doubt
> > that it will be done that way anywhere in the near future.
>
> Nevertheless, I'm suggesting that it would be a really good idea to do
> it that way.
Can David give a universally acceptable definition of "how science is
done" so that we can see more clearly what he is suggesting?
david petry
Jesse F. Hughes wrote:
> "david petry" <david_lawr...@yahoo.com> writes:
> > An A.I. lives in the world of computation. It can perform
> > experiments in the world of computation. And thus, a statement which
> > makes predictions about the results of computational experiments is
> > something the A.I. could put to the test, and hence the A.I. could
> > understand its meaning. The A.I. has no more access to the world of
> > the infinite than we do.
>
> And it has no less access to the world of the infinite than we do,
> right?
>
> And yet a whole slew of folks think that ZF is meaningful in some
> sense.
Yeah, in a sense they can't really explain except with circular
reasoning.
>So an AI would have no less reason to find it meaningful than
> human mathematicians do.
Yeah, you could always program a virus into the A.I.
cbr...@cbrownsystems.com
> cbr...@cbrownsystems.com wrote:
>
> > How does anything
> > you're saying relate to whether or not an AI would accept or reject
> > "Cantorian infinities"?
>
> I don't understand why I have to keep repeating this.
>
Allow me to amplify on my question.
> An A.I. lives in the world of computation.
More exactly, an AI "lives" in a world of boolean algebra: implemented
with transistors, forming a collection of NAND and NOR gates.
Certain patterns of NAND and NOR gates can be interpreted as various
mathematical structures; for example a certain collection of gates can
be interpreted as "adding two natural numbers". But that interpretation
is /external/ to the fundamental underlying boolean algebra.
You imply that the isomorphism between performing certain calculations
in a boolean algebra and "adding natural numbers" will be "meaningful"
to an AI; but that the interpretation of a boolean algebra as an
argument regarding infinities will be "meaningless".
But it may be that your AI rejects the concept of the rational numbers
or even the naturals as "meaningless formalisms" - only calculations in
finite boolean algebras have any "real" meaning.
Yes, such an AI might state, humans may interpret such-and-such a
calculation in a boolean algebra as a confirmation that "2+3=5", or
"for each prime p there is another prime q with p < q < p!+1"; but
these interpretations are "mere formalisms", in exactly the same sense
that Cantor's proof can be represented as a formal manipulation of
symbols. All these expressions can be calculated, but have no "real"
meaning, beyond their existence as certain confirmable calculations in
a boolean algebra.
Why do you believe that an AI would be open minded enough to accept the
natural numbers as an abstraction of a particular boolean algebra, but
not open minded enough to consider even more abstract notions?
> It can perform experiments
> in the world of computation. And thus, a statement which makes
> predictions about the results of computational experiments is something
> the A.I. could put to the test, and hence the A.I. could understand its
> meaning.
At root, these experiments are calculations in a boolean algebra. The
meaning of these calculations is expressed by some formal mapping
between that algebra and some other mathematical structure. The choice
of mappings is what I would call "understand its meaning".
Where does the choice of mapping come from, and why should we assume
that an AI would choose one set of mappings over another?
> The A.I. has no more access to the world of the infinite than
> we do.
Right. I would also argue it has no /less/ access to the world of the
infinite. So given our similar situations, how does this answer why
people might accept infinities, but an AI must reject them?
Cheers - Chas
PS: Suggested reading:
cbr...@cbrownsystems.com
Jesse F. Hughes wrote:
> But, of course, this is really a weak argument in any case. We don't
> choose our mathematical foundations on the grounds that one day, we
> might have to explain them to HAL 9000.
"I, for one, welcome our new alien masters!"
-- Kent Brockman
Cheers - Chas
Jesse F. Hughes
> Jesse F. Hughes wrote:
>> "david petry" <david_lawr...@yahoo.com> writes:
>
>> > An A.I. lives in the world of computation. It can perform
>> > experiments in the world of computation. And thus, a statement which
>> > makes predictions about the results of computational experiments is
>> > something the A.I. could put to the test, and hence the A.I. could
>> > understand its meaning. The A.I. has no more access to the world of
>> > the infinite than we do.
>>
>> And it has no less access to the world of the infinite than we do,
>> right?
>>
>> And yet a whole slew of folks think that ZF is meaningful in some
>> sense.
>
> Yeah, in a sense they can't really explain except with circular
> reasoning.
Well, I don't particularly see any difference between the
meaningfulness of set theory without the axiom of infinity and set
theory with it.
And the difference sure as heck isn't any clearer with your bad
metaphors of computer-as-microscope (especially since, as we all know,
computers can prove the same darn things as us).
>>So an AI would have no less reason to find it meaningful than
>> human mathematicians do.
>
> Yeah, you could always program a virus into the A.I.
Keep them metaphors coming! Very persuasive!
--
Jesse F. Hughes
"That's what's annoying about Usenet as some loser will state a case,
get their ass kicked, but STILL keep coming back as if nothing
happened." -- James Harris explains his strategy.
cbr...@cbrownsystems.com
> Jesse F. Hughes wrote:
<snip>
> >So an AI would have no less reason to find it meaningful than
> > human mathematicians do.
>
> Yeah, you could always program a virus into the A.I.
This is what I meant when I previously said:
"It seems likely that in order to pass a mathematical Turing test, an
AI must already be "contaminated" with the same types of mathematical
qualia that we ourselves prefer."
One person's "virus" may be another's "useful program".
Cheers - Chas
david petry
cbr...@cbrownsystems.com wrote:
> But it may be that your AI rejects the concept of the rational numbers
> or even the naturals as "meaningless formalisms" - only calculations in
> finite boolean algebras have any "real" meaning.
The A.I. will have a model of the underlying computer it is running on,
and integers are an important part of that model (e.g. they are used to
compute memory addresses)
> Why do you believe that an AI would be open minded enough to accept the
> natural numbers as an abstraction of a particular boolean algebra, but
> not open minded enough to consider even more abstract notions?
It will accept any abstraction that lives within the world of
computation.
> > The A.I. has no more access to the world of the infinite than
> > we do.
>
> Right. I would also argue it has no /less/ access to the world of the
> infinite. So given our similar situations, how does this answer why
> people might accept infinities, but an AI must reject them?
People accept religious and mystical notions, and an AI could be
programmed to accept them also, whatever that may mean. But those
religious and mystical notions are not essential for understanding the
world in which we live, and the A.I. will probably be aware of that.
david petry
Hmmm. If we want our A.I.'s to make strategic decisions in the real
world, then teaching them about set theory would at best be harmless,
but at worst could severely interfere with the functioning of the A.I.
However, if we want our A.I.'s to keep us company with good
conversation, then they better learn set theory, if that's a topic we
like to discuss.
Jesse F. Hughes
> People accept religious and mystical notions, and an AI could be
> programmed to accept them also, whatever that may mean. But those
> religious and mystical notions are not essential for understanding the
> world in which we live, and the A.I. will probably be aware of that.
Can you *not* understand how weak this argument is? Why not argue
about what angels can understand and how important it is to form a
reasonable mathematics for talking with *them*?
Tell you what: you create the artificial intelligence and then I
promise I'll change mathematics so that our new buddy can understand
it.
--
"You got more out of it
than I put into it last night.
Who were you thinking of when we were loving last night?"
-- Texas Tornadoes
david petry
Jesse F. Hughes wrote:
> "david petry" <david_lawr...@yahoo.com> writes:
>
> > People accept religious and mystical notions, and an AI could be
> > programmed to accept them also, whatever that may mean. But those
> > religious and mystical notions are not essential for understanding the
> > world in which we live, and the A.I. will probably be aware of that.
>
> Can you *not* understand how weak this argument is?
It wasn't an argument. It was a hint to help the reader find the answer
for himself.
Kent Paul Dolan
> I don't understand why I have to keep repeating
> this.
Probably because no one can believe you repeat such
obvious impractical nonsense, and out of mercy are
attempting to give you a chance to escape from the
hole you've dug yourself.
> An A.I. lives in the world of computation.
Well, no. An A.I. lives in the world of software,
which may involve computation at base, but covers
much richer concepts at its expressive levels.
> It can perform experiments in the world of
> computation.
So? Even sticking to the finite, all but a vanishing
fraction of those "experiments" would not have time
to run to completion before the heat death of the
known universe. Experiments aren't a very good way
to do mathematical research.
> And thus, a statement which makes predictions
> about the results of computational experiments is
> something the A.I. could put to the test,
Perhaps, but with almost no chance of seeing that
test come to a result. What good does that do?
> and hence the A.I. could understand its meaning.
Being "Artificially Intelligent" doesn't guarantee
being capable of "understanding" in any meaningful
sense.
> The A.I. has no more access to the world of the
> infinite than we do.
Certainly it does, certainly we do; neither of us
happens to _live_ there, but we have plenty of mere
access. Software to do symbolic reasoning has
existed for decades. Mathematical reasoning about
infinities has traditionally been done using symbol
sets including symbols for various infinities,
together with rulesets for transformations of those
symbols. There is nothing inherent in the subject
matter being infinities which makes those symbols
and rulesets any more difficult for an A.I. than any
other symbolic reasoning task.
Your way doesn't seem _to me_ to be a useful way to
do mathematics, that almost all experiments will
fail to reach a result in a time short enough to be
usable by the human species, much less the
initiating researcher, whether carbon based or
silicon based.
Contrast to explorations done using infinities,
which _are_ capable of reasoning about numbers
"bigger than huge" without automatically failing
simply from the magnitude of the subject matter, and
you'll soon realize why most practicing
mathematicians won't be interested in your model of
mathematics.
xanthian.
In about 1970, I took a class in partial
differential equations from a U of Hawaii professor
whose then "kitchen table" project was counting all
possible continuous paths from the center to the
edge of the unit disk. I suspect that was a _very_
large number, and not one you'd want to set an AI to
calculating without access to Cantor's
hierarchy/lattice of infinities. Nevertheless, it
was a mathematically interesting topic to someone
skilled in mathematics.
cbr...@cbrownsystems.com
> cbr...@cbrownsystems.com wrote:
>
> > But it may be that your AI rejects the concept of the rational numbers
> > or even the naturals as "meaningless formalisms" - only calculations in
> > finite boolean algebras have any "real" meaning.
>
> The A.I. will have a model of the underlying computer it is running on,
> and integers are an important part of that model (e.g. they are used to
> compute memory addresses)
That's how you choose to view it. It might be that an AI will view its
memory as a 64 dimensional hypercube, with 2 addresses per dimension.
The "usual" traversal of all existing such elements (what those crazy
humans call 'integer addressing') is only one arbitrary method of
accomplishing such a traversal, and not neccessarily the "best" one in
every situation. Given some ordering of the dimensions, perhaps a gray
code will seem more "natural" for this task. Or perhaps no particular
traversal will be prefered at all; and our obsession with a particular
class of incomplete traversals of the memory hypercube will seem
meaningless.
If you argue that conventional programming practices rely heavily on
our intuitions about the integers, and so by example an AI will imitate
them, then we begin the process of "contamination" - which brings the
question of "which interpretations are meaningful" back into the human
sphere.
"Garbage in - garbage out". My guess is that any mathematical meaning
that we find in an AI will be a meaning that was ultimately put there
by us.
<snip>
> > > The A.I. has no more access to the world of the infinite than
> > > we do.
> >
> > Right. I would also argue it has no /less/ access to the world of the
> > infinite. So given our similar situations, how does this answer why
> > people might accept infinities, but an AI must reject them?
>
> People accept religious and mystical notions, and an AI could be
> programmed to accept them also, whatever that may mean. But those
> religious and mystical notions are not essential for understanding the
> world in which we live, and the A.I. will probably be aware of that.
Who knows what an AI would consider "essential"?
In retrospect, perhaps AI's will have /more/ access to the world of the
infinite. After all, I know that I am finite temporally, and that
someday I will die - and I'm not talking about trillions of trillions
of years from now: there's no way I'll live to see, say, 200.
On the other hand, considering Moore's Law as a given to an algorithm
running on a computer, the future looks brighter all the time -
continually increasing complexity and no end in sight! And it's easy to
port an algorithm "perfectly" to new hardware. What evidence in the
"world of computation" is there that this process will ever stop?
Perhaps, after realizing that a computer which is finite yet unbounded
in capacity would have certain limitations, it might wonder if these
same limitations would apply to a non-finite computer. Would that be
mysticism, heresy, or simple curiousity?
All this is of course just blowing hot air: in your terms, it's
pseudo-science. This is why it seems crucial to me to have an
equivalent to the Turing Test - we're talking about an AI that
"understands" certain mathematical statements, without any means for
verifying that any particular AI actually "understands" anything.
Cheers - Chas
Gerry Myerson
"david petry" <david_lawr...@yahoo.com> wrote:
> Gerry Myerson wrote:
> > In article <1143167615.6...@i40g2000cwc.googlegroups.com>,
> > "david petry" <david_lawr...@yahoo.com> wrote:
> >
> > > Gerry Myerson wrote:
> > >
> > > > > Electron microscopes can be used to observe small molecules. Particle
> > > > > accelerators can be used to observe sub-atomic particles. These can
> > > > > all be thought of as "microscopes".
> > > >
> > > > I suppose in the same way that large cardinal axioms and whatnot
> > > > can all be thought of as microscopes for mathematicians.
> > >
> > > We are far from communicating with each other.
> >
> > I'm sorry to learn that you are having difficulty comprehending
> > my messages. I'll try harder if you'll let me know how best I
> > can help.
>
> Explain how you can think of a large cardinal axiom as a microscope, in
> the same (or similar) sense as a particle accelerator can be thought of
> as a microscope.
I've never used either one in my life, so I can only go on hearsay,
but I gather that either one can be used to get results you can't
get by other means.
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
Daryl McCullough
>cbr...@cbrownsystems.com wrote:
>> Right. I would also argue it has no /less/ access to the world of the
>> infinite. So given our similar situations, how does this answer why
>> people might accept infinities, but an AI must reject them?
>
>People accept religious and mystical notions, and an AI could be
>programmed to accept them also, whatever that may mean. But those
>religious and mystical notions are not essential for understanding the
>world in which we live, and the A.I. will probably be aware of that.
How do you know what is essential? The only *examples* of intelligence
we have are human beings. And you, rather than *observing* humans, and
formulating testable *hypotheses* about how humans do what they do.
Anyway, you are confusing yourself. The mathematics that an AI might
*use* has nothing (necessarily) to do with the mathematics needed for
us to *design* an AI. Most humans have no grasp of mathematics beyond
simple arithmetic, but they can accomplish many tasks that seem beyond
current computer programs.
As usual, your talk about "religious and mystical notions" is just
ridiculous. Set theory is about *abstraction*, and that is certainly
a key tool of intelligence. Certainly I agree with you that
higher-power set theory probably has little in the way of practical
applications, but so does your idea of the computer as a "microscope".
Why do we need such a microscope? What is an example of a practical
application of it?
--
Daryl McCullough
Ithaca, NY
--
NewsGuy.Com 30Gb $9.95 Carry Forward and On Demand Bandwidth
Daryl McCullough
>Jesse F. Hughes wrote:
>> And yet a whole slew of folks think that ZF is meaningful in some
>> sense.
>
>Yeah, in a sense they can't really explain except with circular
>reasoning.
It's not circular to say that all known mathematical results
can be formulated in ZF. It's not circular to say that ZF
has an extremely high consistency strength (that is, ZF can
prove the consistency of many other theories such as PA, but
not vice-versa).
What you really mean is that ZF is meaningful in a way that
can't really be explained to a person holding his fingers in
his ears.
Ross A. Finlayson
Hi Daryl, hey how's it going,
I don't see "every mathematical truth" as being expressible in ZF. For
example, in a very close relation to that theory, ZFC, many results
that depend on the axiom of choice are generally considered true
statements.
It is not _true_ to say that all known mathematical results can be
formulated in ZF.
The reason a lot of people are looking for some "better" theory than ZF
is because of basically that they would prefer their foundations of
mathematics, and thus a wide variety of other thought and theoretical
and metatheoretical processing, to be as consistent and complete as
possible. That's basically occurred since the invention of ZF, a
theory in which _many_ mathematical truths can be expressed, for
example with the notions of Church and Quine and so forth, today with
the Aczel and Holmes and apparently all these other people, pretty much
everybody, including myself.
There is no universe in ZF, a universal set or domain of discourse or
universal quantification over sets with powerset and union. There's no
class of all classes in ZF with classes. No theory axiomatized regular
escapes having no universe and basically incompleteness, in the idea of
some inconsistency.
Zermelo saw well-ordering as self-evident and Fraenkel I think is quite
humorous, Fraenkel is really great. He said, perhaps out-of-context or
tongue-in-cheek, that transfinite cardinals were a disease of
mathematics, and for him is named the logical system in which those
results are generally expressed.
I'd agree that ZF has meaning. So does a system with no complex
numbers, or, say, no irrational numbers, as that was thought to be the
case of all numbers in a prior mathematical era. These days it's
generally recognized for those not to be the case.
Burali-Forti, the "paradox" named after Cesare Burali-Forti that a set
of ordinals is an ordinal, means that when you quantify over ordinals
in ZF you're lying to yourself. A reason ZF is regular is because of
Russell, and Cantor.
That's not necessarily wrong.
So, quantify over sets. While you're at it, well-order the reals. You
can't in ZF, except perhaps well-order the reals.
There are theories besides ZF that are, arguably, better, basically by
the pragmata of being more complete, and consistent, with "all known
mathematical results."
I wish I knew more of the "all known mathematical results." Luckily,
there can be only one consistent and complete theory, so, I know there
is at least one.
Ciao,
Ross F.
david petry
Daryl McCullough wrote:
> Certainly I agree with you that
> higher-power set theory probably has little in the way of practical
> applications, but so does your idea of the computer as a "microscope".
> Why do we need such a microscope? What is an example of a practical
> application of it?
Here's an example I have discussed at length before.
Mathematicians don't know how to prove the Reimann hypothesis. So what
do they do? They compute (i.e. use the mathematicians' microscope) to
compute billions of the zeros of the zeta function, both looking for a
possible counterexample, and also looking for patterns in those zeros,
hoping to find insights which will help them find a proof.
MoeBlee
I don't think anyone denies that mathematics should NOT use this
"microscope". But, even as you mention, mathematicians also seek
proofs. These are often proofs not just that a certain possible
counterexample is not a counterexample, but rather that there simply
are no counterexamples. That is core mathematics since the ancients,
and since the late 19th century infinite sets are posited to make the
proofs strictly from axioms and rules of inference. You just seem to
not like that that is what mathematics, or at least a vast chunk of it,
IS. You want mathematics to confine itself to methods of empirical
science, while mathematics is not empirical science, otherwise it would
have been, all this time, since the ancients, only studying empirical
phenomenon.
Your charge, that positing infinite sets is mysticism, is unfounded
since infinite sets are intended as abstractions but need not be (and
almost always are not) claimed to have any affect upon physical events.
In fact, it is the LACK of a claim of affect upon observable phenomenon
with the LACK of allegory with persons, things, and events that ensures
that the postiting of infinite sets is NOT mysticism. Now, whatever
mathematics can be developed without infinite sets is welcome, and may,
depending on the axioms, be preferred for not having to posit
infinitude. But as long as there are also useful (even if only to
mathematicians themselves) conclusions to be drawn consistently from
infinite sets, then theories with infinite sets will also be objects of
study, your stipulative definition of 'mathematics' notwithstanding.
By the way, in the Nik Weaver paper, doesn't he have infinite sets?
MoeBlee
MoeBlee
> david petry wrote:
> I don't think anyone denies that mathematics should NOT use this
> "microscope".
Should be 'claims' instead of 'denies'.
david petry
MoeBlee wrote:
> By the way, in the Nik Weaver paper, doesn't he have infinite sets?
Countable ones.
Dik T. Winter
And you claim that is a practical application? Unless a couterexample is
found, it is not practical at all. And I know, I have been busy with
programs doing quite a few of those calculations. They gave neither
insight not results. As such a complete waste of time (except that they
did give a counter-example to Merten's conjecture).
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
David Bernier
Computers aren't mathematical genies. Odlydzko computed many zeros
of the Riemann zeta function and was able to prove with the aid
of computer(s) that all the zeros found (in the critical strip)
had a real part of 1/2. Going up the half-line Re(s)=1/2,
Im(s)>=0, he and others computed the differences in
the imaginary part of a zero on the half-line and
the next one going in increasing imaginary part direction.
These differences get smaller and smaller on average.
By rescaling [stretching the half-line non-uniformly],
the distance between a point (zero) on the stretched
half-line and the next one stays constant (on average).
On the stretched half-line, the points successive points "repel".
This is a different behaviour than that of a time-line
with points marking clicks of a Geiger counter (Poisson process).
Scientists/mathematicians have studied something called
GUE (Gaussian unitary ensemble). Related to GUE
is the problem of how the n eigenvalues of
a random nxn unitary matrix are distributed
on the unit circle in the complex plane.
The idea of random unitary matrices in the
group U(n) of nxn unitary matrices can be formalized
by using the Haar measure on the compact group U(n);
U(n) is compact, so the measure is both left- and
right-invariant. The measure of the whole U(n)
is set to 1 to get a probability measure.
With random unitary nxn properly defined as above,
it makes sense to talk of the n
eigenvalues lamda_1, ... lambda_n obtained by going
counterclockwise on the unit circle starting at 1.
What is the probability density function for
the n values n/(2*pi) arg(lambda_{j+1}/lambda_{j}) [ 1<=j<n]
and n/(2*pi) arg(lambda_1 / lambda_n) for a random
nxn unitary matrix? (*)
For "large" n, the statistics of the scaled
arg(.) values above for eigenvalues
of random nxn unitary matrices are very close to
the scaled differences in the imaginary parts
of pairs of successive zeros of zeta; with the
right scaling, the data from Odlydzko's computations
when used to plot the scaled imaginary part differences
are very close to the known (but not obvious)
p.d.f.'s for the random variables in (*) above.
See for example Fig. 1 in the paper:
Katz, N.; Sarnak, P. : `` Zeroes of zeta functions and symmetry",
Bull. Amer. Math. Soc. 36 (1999), 1-26.
So many authors (mathematicians, physicists) have done
work related to the above and this article is already long.
The Katz and Sarnak paper mentioned above has a long references
section to the (original) research papers.
Katz and Sarnak paper available here:
http://www.ams.org/bull/1999-36-01/S0273-0979-99-00766-1/home.html
David Bernier