Mathematical theater of the absurd
david petry
"The actual infinite is not required for the mathematics of the
physical world" (Soloman Feferman)
When we seek to understand the real world, we are forced to do
mathematics. It would be a very reasonable thing to do to define
mathematics in terms of what it is we are forced to do when we seek to
understand the real world. That is, we can and probably should takes
steps to ensure that mathematics stays in touch with reality. The
scientists use the notion of falsifiability as a criterion to
guarantee that science stays in touch with reality--to distinguish
science from philosophy, theology and pseudoscience--and that notion
could be incorporated into the foundations of mathematics where it
could serve to distinguish the mathematics of the physical world from
the mathematics of the metaphysical world; the mathematics of the
physical world must meet the criterion of being falsifiable. It would
be very reasonable to divide mathematics into (at least) two separate
subjects--the mathematics of the physical world, and the mathematics
of the metaphysical world--for exactly the same reasons that the
scientists keep science separate from philosophy, theology and
pseudoscience; the confusion caused by the failure to distinguish the
two subjects is an obstacle to progress in the sciences and in
technology. In particular, the general failure of people to recognize
that the sacred cows of metaphysical mathematics (e.g. Cantor's theory
of the actual infinite, and Godel's theorem) have nothing to tell us
about the physical world is a serious obstacle to progress in the
field of artificial intelligence.
Experience shows that it is virtually impossible to discuss the ideas
in the foregoing paragraph in these newsgroups; here, the inmates have
taken over the asylum. However, the interested reader may want to read
articles I've posted on this topic over the years.
http://groups.google.com/group/sci.math/msg/8245894cf9c14ac6?hl=en
http://groups.google.com/group/sci.math/msg/0845a6308e5d4633?hl=en
http://groups.google.com/group/sci.math/msg/40cc4610018d67de?hl=en
http://groups.google.com/group/sci.math/msg/859d0f3750a0e9dc?hl=en
http://groups.google.com/group/sci.math/msg/0025be708362cb7e?hl=en
Frederick Williams
>
> "The actual infinite is not required for the mathematics of the
> physical world" (Soloman Feferman)
How does he know?
--
... when we came back, late, from the hyacinth garden,
Your arms full, and your hair wet, I could not
Speak, and my eyes failed...
Neil W Rickert
>"The actual infinite is not required for the mathematics of the
>physical world" (Soloman Feferman)
That's his mistake.
>When we seek to understand the real world, we are forced to do
>mathematics.
Nonsense. We _choose_ to use mathematics. We are not _forced_ to
use mathematics.
> It would be a very reasonable thing to do to define
>mathematics in terms of what it is we are forced to do when we seek to
>understand the real world.
That's quite silly. Biology comes closer to "what we are forced
to do when we seek to understand the real world", though even that
is a stretch.
> That is, we can and probably should takes
>steps to ensure that mathematics stays in touch with reality.
Mathematics was never in touch with reality.
> The
>scientists use the notion of falsifiability as a criterion to
>guarantee that science stays in touch with reality--to distinguish
>science from philosophy, theology and pseudoscience-- ...
That's very doubtful. The philosophers (or some of them) claim
that falsifiability is important. But what the philosophers say
that scientists do has very little connection with what scientists
actually do. Falsificationism turns out to be unfalsifiable,
so by its own criteria it ought to be rejected.
LudovicoVan
> david petry <david_lawrence_pe...@yahoo.com> writes:
> >"The actual infinite is not required for the mathematics of the
> >physical world" (Soloman Feferman)
>
> That's his mistake.
This is interesting. Could you maybe elaborate a little bit?
[snip]
> > That is, we can and probably should takes
> > steps to ensure that mathematics stays in touch with reality.
>
> Mathematics was never in touch with reality.
Not more than Kantism.
> >scientists use the notion of falsifiability as a criterion to
> >guarantee that science stays in touch with reality--to distinguish
> >science from philosophy, theology and pseudoscience-- ...
>
> That's very doubtful. The philosophers (or some of them) claim
> that falsifiability is important. But what the philosophers say
> that scientists do has very little connection with what scientists
> actually do. Falsificationism turns out to be unfalsifiable,
> so by its own criteria it ought to be rejected.
Falsifiability is basic in philosophy, once an argument is running.
-LV
MoeBlee
wrote:
> When we seek to understand the real world, we are forced to do
> mathematics. It would be a very reasonable thing to do to define
> mathematics in terms of what it is we are forced to do when we seek to
> understand the real world.
And, aside from questions of what is meant by "the real world", it
would be reasonable also to think that mathematics is not defined as
you say above.
Surely, you don't mean your paragraph above as an ARGUMENT. That is,
surely you do not propose the following non sequitur:
We need mathematics for the physical sciences, THEREFORE, mathematics
is the study of what we need for the physical sciences.
> It would
> be very reasonable to divide mathematics into (at least) two separate
> subjects--the mathematics of the physical world,
Abstractions are not "metaphysical" merely for being abstractions
irrespective of empirical testing.
Sure, one may desire a mathematics for the physical sciences that
excludes what one considers to be extraneous abstractions, but then to
the extent that we may also desire that such a mathematics be formally
axiomatized, one is led to ask of you: Okay, so just what are your
axioms? Then, if you EVER (you never do) gave an answer, we could
evaluate your proposal on a number of criteria: including scope of
coverage of the physical sciences, intuitiveness, ease of use, and
your own criterion: exclusion of extraneous abstractions.
> the confusion caused by the failure to distinguish the
> two subjects is an obstacle to progress in the sciences and in
> technology.
You have an example where abstract mathematics hindered a scientist or
technician from whatever progress he or she would have made otherwise?
> In particular, the general failure of people to recognize
> that the sacred cows of metaphysical mathematics (e.g. Cantor's theory
> of the actual infinite, and Godel's theorem) have nothing to tell us
> about the physical world is a serious obstacle to progress in the
> field of artificial intelligence.
You have an example where abstract mathematics hindered a scientist or
technician from whatever progress he or she would have made otherwise
in artificial intelligence?
By the way, the incompleteness theorem does tell me something
substantive: It tells me not to waste my PHYSICAL actions trying to
find an algorithm that will settle any given problem in arithmetic.
> Experience shows that it is virtually impossible to discuss the ideas
> in the foregoing paragraph in these newsgroups;
There's no such virtual impossiblity. Rather, you tend to AVOID sharp
challenges to your arguments. Mostly you prefer to post a bunch of
quotes as mere polemical fodder, then followed by a post such as
above, then, often (though not always) to elide confronting some of
the best challenges to your posts.
> here, the inmates have
> taken over the asylum.
'inmates' and 'asylum' are interesting metaphors coming from you, as
you are a virtually certifiable anti-Semitic nutjob who advocates mass
executions of those "humanists" (and certain others?) who refuse to
renounce their their ways.
MoeBlee
Neil W Rickert
>On 5 May, 15:17, Neil W Rickert <rickert...@cs.niu.edu> wrote:
>> david petry <david_lawrence_pe...@yahoo.com> writes:
>> >"The actual infinite is not required for the mathematics of the
>> >physical world" (Soloman Feferman)
>> That's his mistake.
>This is interesting. Could you maybe elaborate a little bit?
I am taking "the mathematics of the physical world" to mean the
mathematics used by physicists. And some of the mathematics used
by physicists requires the axiom of choice.
I should add that Petry did not provide a citation. He might have
taken that statement completely out of context, and Feferman could
have intended something quite different from what I took it to say.
>> >scientists use the notion of falsifiability as a criterion to
>> >guarantee that science stays in touch with reality--to distinguish
>> >science from philosophy, theology and pseudoscience-- ...
>> That's very doubtful. =A0The philosophers (or some of them) claim
>> that falsifiability is important. =A0But what the philosophers say
>> that scientists do has very little connection with what scientists
>> actually do. =A0Falsificationism turns out to be unfalsifiable,
>> so by its own criteria it ought to be rejected.
>Falsifiability is basic in philosophy, once an argument is running.
I'm not sure what you are intending there. The idea of
falsificationism comes from Popper, and philosophy existed prior
to Popper. I can't say that I can think of where it is used, except
in connection with philosophy of science. Kuhn and Feyerabend both
disagreed. David Stove considered Popper to be an irrationalist
(because he denied induction, and his falsificationism was his way of
denying induction). According to C.I. Lewis, the fundamental laws
of any science are a priori (which would make them unfalsifiable).
That's from Lewis's 1923 paper "A Pragmatic Conception of the
A Priori".
MoeBlee
> LudovicoVan <ju...@diegidio.name> writes:
> >On 5 May, 15:17, Neil W Rickert <rickert...@cs.niu.edu> wrote:
> >> david petry <david_lawrence_pe...@yahoo.com> writes:
> >> >"The actual infinite is not required for the mathematics of the
> >> >physical world" (Soloman Feferman)
> >> That's his mistake.
> >This is interesting. Could you maybe elaborate a little bit?
>
> I am taking "the mathematics of the physical world" to mean the
> mathematics used by physicists. And some of the mathematics used
> by physicists requires the axiom of choice.
>
> I should add that Petry did not provide a citation.
It is in Feferman's book 'In The Light Of Logic' in which Feferman
offers a predicativist criticism of set theory.
MoeBlee
LudovicoVan
> LudovicoVan <ju...@diegidio.name> writes:
> >On 5 May, 15:17, Neil W Rickert <rickert...@cs.niu.edu> wrote:
> >> david petry <david_lawrence_pe...@yahoo.com> writes:
> >> >"The actual infinite is not required for the mathematics of the
> >> >physical world" (Soloman Feferman)
> >> That's his mistake.
> >This is interesting. Could you maybe elaborate a little bit?
>
> I am taking "the mathematics of the physical world" to mean the
> mathematics used by physicists. And some of the mathematics used
> by physicists requires the axiom of choice.
Nice, but... sorry to insist: could you maybe give any specific
example of an area of physics leveraging such mathematics? I mean, one
for all is enough, just to get an idea: I now near to nothing about
physics anyway (though not absolute zero).
> I should add that Petry did not provide a citation. He might have
> taken that statement completely out of context, and Feferman could
> have intended something quite different from what I took it to say.
>
> >> >scientists use the notion of falsifiability as a criterion to
> >> >guarantee that science stays in touch with reality--to distinguish
> >> >science from philosophy, theology and pseudoscience-- ...
> >> That's very doubtful. =A0The philosophers (or some of them) claim
> >> that falsifiability is important. =A0But what the philosophers say
> >> that scientists do has very little connection with what scientists
> >> actually do. =A0Falsificationism turns out to be unfalsifiable,
> >> so by its own criteria it ought to be rejected.
> >Falsifiability is basic in philosophy, once an argument is running.
>
> I'm not sure what you are intending there. The idea of
> falsificationism comes from Popper, and philosophy existed prior
> to Popper. I can't say that I can think of where it is used, except
> in connection with philosophy of science. Kuhn and Feyerabend both
> disagreed. David Stove considered Popper to be an irrationalist
> (because he denied induction, and his falsificationism was his way of
> denying induction). According to C.I. Lewis, the fundamental laws
> of any science are a priori (which would make them unfalsifiable).
> That's from Lewis's 1923 paper "A Pragmatic Conception of the
> A Priori".
Surely the notion of falsifiability was not born with Popper, and it's
something different than falsificationism. I'd put it this way, but my
English will be tentative: falsifiability, among the related notions,
was born with post-socratic philosophy, when the connections to an
original spirituality got finally lost. That is where all arguments
started, and that is where clarity of thinking became a value
constantly in danger. Moreover, that is where the inductive problem
was born, for the man who had lost his spirit. The whole subsequent
history of the western world, through the middle ages (christianity),
then the modern ages (imperialism), and finally the post-modern ages
(post-imperialism), is the history of a lost spirit, then a lost
culture, finally a lost society.
My reply was stimulated by these words of yours: "Falsificationism
turns out to be unfalsifiable". I'd say: of course! Because the laws
of any science indeed cannot be scientific. I would think that is what
Lewis means too: philosophy of science is an a-priori to science, but
this is a guess.
-LV
fishfry
<799fc943-6e7a-4abe...@k19g2000prh.googlegroups.com>,
david petry <david_lawr...@yahoo.com> wrote:
> "The actual infinite is not required for the mathematics of the
> physical world" (Soloman Feferman)
>
>
> When we seek to understand the real world, we are forced to do
> mathematics. It would be a very reasonable thing to do to define
> mathematics in terms of what it is we are forced to do when we seek to
> understand the real world.
Silly. When I seek to go to the store for a bag of Fritos, I'm forced to
drive my car. Would it then be "very reasonable" to define driving as
going to the store for a bag of chips?
Math is a tool for understanding the real world. It's also a whole lot
more. It's that "whole lot more" that separates out the mathematicians
from the physical and social scientists, who use math to understand and
model the real world.
Neil W Rickert
>On May 5, 3:22=A0pm, Neil W Rickert <rickert...@cs.niu.edu> wrote:
>> I should add that Petry did not provide a citation. =A0
>It is in Feferman's book 'In The Light Of Logic' in which Feferman
>offers a predicativist criticism of set theory.
Thanks.
Neil W Rickert
>> I am taking "the mathematics of the physical world" to mean the
>> mathematics used by physicists. =A0And some of the mathematics used
>> by physicists requires the axiom of choice.
>Nice, but... sorry to insist: could you maybe give any specific
>example of an area of physics leveraging such mathematics? I mean, one
>for all is enough, just to get an idea: I now near to nothing about
>physics anyway (though not absolute zero).
Physicists use linear operatory theory in QM theories.
At a more basic level, the real number system as a continuum was a
development that was made to meet the needs of theoretical physics.
And the continuum is infinite (uncountably infinite).
>Surely the notion of falsifiability was not born with Popper, and it's
>something different than falsificationism. I'd put it this way, but my
>English will be tentative: falsifiability, among the related notions,
>was born with post-socratic philosophy, when the connections to an
>original spirituality got finally lost. That is where all arguments
>started, and that is where clarity of thinking became a value
>constantly in danger. Moreover, that is where the inductive problem
>was born, for the man who had lost his spirit. The whole subsequent
>history of the western world, through the middle ages (christianity),
>then the modern ages (imperialism), and finally the post-modern ages
>(post-imperialism), is the history of a lost spirit, then a lost
>culture, finally a lost society.
Okay. I have not much studied the history of philosophy.
Perhaps you are referring to the distinction between analytic and
synthetic propositions, where the synthetic ones would be considered
falsifiable.
LudovicoVan
> LudovicoVan <ju...@diegidio.name> writes:
> >> I am taking "the mathematics of the physical world" to mean the
> >> mathematics used by physicists. =A0And some of the mathematics used
> >> by physicists requires the axiom of choice.
> >Nice, but... sorry to insist: could you maybe give any specific
> >example of an area of physics leveraging such mathematics? I mean, one
> >for all is enough, just to get an idea: I now near to nothing about
> >physics anyway (though not absolute zero).
>
> Physicists use linear operatory theory in QM theories.
>
> At a more basic level, the real number system as a continuum was a
> development that was made to meet the needs of theoretical physics.
> And the continuum is infinite (uncountably infinite).
Thanks. I thought the continuum was an idea of the past.
> >Surely the notion of falsifiability was not born with Popper, and it's
> >something different than falsificationism. I'd put it this way, but my
> >English will be tentative: falsifiability, among the related notions,
> >was born with post-socratic philosophy, when the connections to an
> >original spirituality got finally lost. That is where all arguments
> >started, and that is where clarity of thinking became a value
> >constantly in danger. Moreover, that is where the inductive problem
> >was born, for the man who had lost his spirit. The whole subsequent
> >history of the western world, through the middle ages (christianity),
> >then the modern ages (imperialism), and finally the post-modern ages
> >(post-imperialism), is the history of a lost spirit, then a lost
> >culture, finally a lost society.
>
> Okay. I have not much studied the history of philosophy.
> Perhaps you are referring to the distinction between analytic and
> synthetic propositions, where the synthetic ones would be considered
> falsifiable.
Perhaps that is what *you* were referring to. :)
-LV
Jack Campin - bogus address
>>> of choice.
>> leveraging such mathematics?
Which in the case of Gleason's theorem uses not just the axiom of
choice but also the continuum hypothesis.
==== j a c k at c a m p i n . m e . u k === <http://www.campin.me.uk> ====
Jack Campin, 11 Third St, Newtongrange EH22 4PU, Scotland == mob 07800 739 557
CD-ROMs and free stuff: Scottish music, food intolerance, and Mac logic fonts
****** I killfile Google posts - email me if you want to be whitelisted ******
Daryl McCullough
>> Physicists use linear operator theory in QM theories.
>
>Which in the case of Gleason's theorem uses not just the axiom of
>choice but also the continuum hypothesis.
Are you sure about that? I've never seen a reference that suggested
that Gleason's theorem requires the continuum hypothesis. The continuum
hypothesis was, however, used by Pitowsky in coming up with a local
hidden variable theory (using nonmeasurable sets) for explaining the
EPR experiment for spin 1/2 particles.
--
Daryl McCullough
Ithaca, NY
Tim BandTech.com
> "The actual infinite is not required for the mathematics of the
> physical world" (Soloman Feferman)
I appreciate your context.
The numerous means of breaking down mathematics coupled with our
gradeschool education leaves little room to challenge mathematics
since we've got nearly hardwired principles driving our processor's
core routines. When we utter the word 'physical' does it inherently
mean measurable? I think this is nearly true and the word 'finite' is
nearby, so to what degree is physical inherently finite?
I plug the polysign numbers
http://www.BandTechnology.com/PolySigned
which expose support for spacetime including unidirectional time.
When pure math exposes congruence to physical reality will the
behavior be overlooked? It has been by existing mathematics. I am left
drawing the conclusion that these problems as you've dissected them
remain open. This is unconditionally true. The idea that we've gotten
something wrong is not something to be dreaded. That there is
something fundamental which remains to be constructed is reassuring.
This enlivens the modern gray age and eases the burden of learning the
details of past work.
- Tim
LudovicoVan
Hm, I'd say no, that is a misconception. A philosophy that has
discovered analysis (namely, ours) is a philosophy that primarily
falsifies the *analytical* process. E.g. philosophy (and only
philosophy) can falsify the unfalsifiability of falsificationism (a
philosophy of science).
-LV
herbzet
david petry wrote:
>
> "The actual infinite is not required for the mathematics of the
> physical world" (Soloman Feferman)
>
> When we seek to understand the real world, we are forced to do
> mathematics.
I just want to remark on how you have smoothly inserted, here
and below, the phrase "real world" for "physical world".
--
hz
david
wrote:
> david petry wrote:
>
> > "The actual infinite is not required for the mathematics of the
> > physical world" (Soloman Feferman)
>
> How does he know?
He has devoted his career to studying proof theory, where they
carefully examine what assumptions are needed to prove various
theorems. He is widely regarded as a leader in the field. If anyone
"knows", it is him.
MoeBlee
Just to be clear on that point: Feferman is among the great
mathematicians in set theory, and his philosophical views must be
worthy of serious consideration. But his stature in the field does not
alone give him sole or ultimate authority on such philosophical issues
which are subject to a wide range of viewpoints including those of
many other eminent mathematicians and philosphers.
MoeBlee
Dan Christensen
> physical world" (Soloman Feferman)
>
Are you going to suggest that we can only do number theory, for
example, for a finite number of elements? We would then have to do
away with the notion of addition and multiplication being a functions.
For x and y in N, we could not simply write x+y. This expression would
be not be well defined. Imagine how intractable even basic algebra
would be!
Maybe it would help to think of infinity as a convenient bookkeeping
device that may or may not reflect any aspect of physical reality (or
of the system being modelled). Just a thought.
Dan
Download my DC Proof software at http://www.dcproof.com
david
wrote:
> On May 5, 2:30 am, david petry <david_lawrence_pe...@yahoo.com> wrote:
>
> > "The actual infinite is not required for the mathematics of the
> > physical world" (Soloman Feferman)
>
> Are you going to suggest that we can only do number theory, for
> example, for a finite number of elements?
No.
http://groups.google.com/group/sci.math/msg/600ad9693523ba44?hl=en
Dan Christensen
> On May 7, 10:59 am, Dan Christensen <Dan_Christen...@sympatico.ca>
> wrote:
>
> > On May 5, 2:30 am, david petry <david_lawrence_pe...@yahoo.com> wrote:
>
> > > "The actual infinite is not required for the mathematics of the
> > > physical world" (Soloman Feferman)
>
> > Are you going to suggest that we can only do number theory, for
> > example, for a finite number of elements?
>
> No.
>
Whew!
> http://groups.google.com/group/sci.math/msg/600ad9693523ba44?hl=en
Here, you said, "But what about the real numbers? The basic idea is
that we have to find a way to approximate the set of real numbers by
entities that actually exist."
I understand that the set of real numbers (i.e. the set of Dedekind
cuts) can be constructed from the natural numbers using only the
axioms of set theory. Does a Dedekind cut not actually exist in the
sense you mean?
david petry
wrote:
> On May 8, 1:09 am, david <david_lawrence_pe...@yahoo.com> wrote:
>
> > On May 7, 10:59 am, Dan Christensen <Dan_Christen...@sympatico.ca>
> > wrote:
>
> > > On May 5, 2:30 am, david petry <david_lawrence_pe...@yahoo.com> wrote:
>
> > > > "The actual infinite is not required for the mathematics of the
> > > > physical world" (Soloman Feferman)
>
> > > Are you going to suggest that we can only do number theory, for
> > > example, for a finite number of elements?
>
> > No.
>
> Whew!
>
> >http://groups.google.com/group/sci.math/msg/600ad9693523ba44?hl=en
>
> Here, you said, "But what about the real numbers? The basic idea is
> that we have to find a way to approximate the set of real numbers by
> entities that actually exist."
>
> I understand that the set of real numbers (i.e. the set of Dedekind
> cuts) can be constructed from the natural numbers using only the
> axioms of set theory. Does a Dedekind cut not actually exist in the
> sense you mean?
A Dedekind cut is an infinite set. Infinite sets have only a
potential existence.
herbzet
Hence, the length of the diagonal of a unit square exists only potentially.
Did I say that right?
--
hz
david petry
Usually when we say something, we have a purpose for saying it. What
would be the purpose for saying that?
I can only repeat things I've already said. I don't know why you have
so much trouble understanding what I say.
The idea is that infinity is merely a useful fiction, or equivalently,
a figure a speech. The only properties that infinity has are useful
properties. That is, we're trying to understand the world we observe,
and any given observation contains only finite information. Infinity
is useful when it helps us (as a conceptual aid) to understand those
finite observations. When we think of infinity in that way, then we
say, just once, that infinity has only a "potential" existence, but
then we never ever have to mention the word "potential" again.
zzbu...@netscape.net
All of those reasons are also why self-assembling robots were built.
So it should be apparent to all, that digital computers are the
bastard child
of set theory, and have nothing to do with the real world.
Dan Christensen
wrote:
> On May 10, 7:33 pm, Dan Christensen <Dan_Christen...@sympatico.ca>
> wrote:
>
>
>
> > On May 8, 1:09 am, david <david_lawrence_pe...@yahoo.com> wrote:
>
> > > On May 7, 10:59 am, Dan Christensen <Dan_Christen...@sympatico.ca>
> > > wrote:
>
> > > > On May 5, 2:30 am, david petry <david_lawrence_pe...@yahoo.com> wrote:
>
> > > > > "The actual infinite is not required for the mathematics of the
> > > > > physical world" (Soloman Feferman)
>
> > > > Are you going to suggest that we can only do number theory, for
> > > > example, for a finite number of elements?
>
> > > No.
>
> > Whew!
>
> > >http://groups.google.com/group/sci.math/msg/600ad9693523ba44?hl=en
>
> > Here, you said, "But what about the real numbers? The basic idea is
> > that we have to find a way to approximate the set of real numbers by
> > entities that actually exist."
>
> > I understand that the set of real numbers (i.e. the set of Dedekind
> > cuts) can be constructed from the natural numbers using only the
> > axioms of set theory. Does a Dedekind cut not actually exist in the
> > sense you mean?
>
> A Dedekind cut is an infinite set. Infinite sets have only a
> potential existence.
I probably agree with you more than I don't. I'm no expert in this
area, but tend to agree with your paraphrasing of Gauss:
"The notion of a completed infinity doesn't belong in mathematics;
infinity is merely a figure of speech which helps us talk about
limits"
I don't know about your notion of "potential existence," though. Have
you been able to formalize this notion? Or is it merely a figure
speech as well. (Not being facetious. An honest question.)
Marshall
wrote:
>
> The idea is that infinity is merely a useful fiction, or equivalently,
> the world we observe,
> and any given observation contains only finite information. Infinity
> is useful when it helps us (as a conceptual aid) to understand those
> finite observations. When we think of infinity in that way, then we
> say, just once, that infinity has only a "potential" existence, but
> then we never ever have to mention the word "potential" again.
I agree with pretty much all of the above. The thing is, I don't
see that there is any least thing about it that is inconsistent
with current practice. In other words, yeah, that's what we're
already doing.
In fact, the above paragraph works just as well when you
substitute "seven" for "infinity."
Marshall
MoeBlee
wrote:
> That is, we're trying to understand the world we observe,
And other people also like finding out implications about pure
abstractions. I have no reason to accept your own limited notion of
what mathematics is.
MoeBlee
amy666
> The
> scientists use the notion of falsifiability as a
> criterion to
> guarantee that science stays in touch with
> reality--to distinguish
> science from philosophy, theology and
> pseudoscience--and that notion
> could be incorporated into the foundations of
> mathematics where it
> could serve to distinguish the mathematics of the
> physical world from
> the mathematics of the metaphysical world; the
> mathematics of the
> physical world must meet the criterion of being
> falsifiable. It would
> be very reasonable to divide mathematics into (at
> least) two separate
> subjects--the mathematics of the physical world, and
> the mathematics
> of the metaphysical world--for exactly the same
> reasons that the
> scientists keep science separate from philosophy,
> theology and
> pseudoscience; the confusion caused by the failure to
> distinguish the
> two subjects is an obstacle to progress in the
> sciences and in
> technology. In particular, the general failure of
> people to recognize
> that the sacred cows of metaphysical mathematics
> (e.g. Cantor's theory
> of the actual infinite, and Godel's theorem) have
> nothing to tell us
> about the physical world is a serious obstacle to
> progress in the
> field of artificial intelligence.
>
.. and science.
*** applause ***
regards
tommy1729
Bill Dubuque
>
> I probably agree with you more than I don't. I'm no expert in
> this area, but tend to agree with your paraphrasing of Gauss:
>
> "The notion of a completed infinity doesn't belong in mathematics;
> infinity is merely a figure of speech which helps us talk about limits"
That remark is a couple centuries old and is no longer correct.
Nowadays various notions of infinity are employed in many branches
of mathematics with great success. For an introduction accessible
to a bright layperson see Rudy Rucker's book "Infinity and the Mind".
--Bill Dubuque
Martin Michael Musatov
>
> "The actual infinite is not required for the
> mathematics of the
> physical world" (Soloman Feferman)
>
>
> When we seek to understand the real world, we are
> forced to do
> mathematics. It would be a very reasonable thing to
> do to define
> mathematics in terms of what it is we are forced to
> do when we seek to
> understand the real world. That is, we can and
> probably should takes
> steps to ensure that mathematics stays in touch with
> discuss the ideas
> in the foregoing paragraph in these newsgroups; here,
> the inmates have
> taken over the asylum. However, the interested reader
> may want to read
> articles I've posted on this topic over the years.
>
>
> http://groups.google.com/group/sci.math/msg/8245894cf9
> c14ac6?hl=en
>
> http://groups.google.com/group/sci.math/msg/0845a6308e
> 5d4633?hl=en
>
> http://groups.google.com/group/sci.math/msg/40cc461001
> 8d67de?hl=en
>
> http://groups.google.com/group/sci.math/msg/859d0f3750
> a0e9dc?hl=en
>
> http://groups.google.com/group/sci.math/msg/0025be7083
> 62cb7e?hl=en
>
lwa...@lausd.net
> In article
> <799fc943-6e7a-4abe-b11f-96a8203d5...@k19g2000prh.googlegroups.com>,
> david petry <david_lawrence_pe...@yahoo.com> wrote:
> > "The actual infinite is not required for the mathematics of the
> > physical world" (Soloman Feferman)
> > When we seek to understand the real world, we are forced to do
> > mathematics. It would be a very reasonable thing to do to define
> > mathematics in terms of what it is we are forced to do when we seek to
> > understand the real world.
> more. It's that "whole lot more" that separates out the mathematicians
> from the physical and social scientists, who use math to understand and
> model the real world.
But if a so-called "crank" proposed a alternate theory, that theory
isn't allowed to be part of that "whole lot more." Instead, the
standard set theorist would insist that the "crank" explain the
theory's applicability to science and the real world.
Unwritten rule #1: only standard theories are allowed to be part of
the "whole lot more." "Crank" theories must have an application to
the real world in order to be considered.
Also, the OP's quote of Feferman sounds very similar to Robinson's
comment about infinite sets not existing. Finitists "cranks" quote
Robinson all the time, yet are ridiculed for doing so.
Unwritten rule #2: only standard theorists are allowed to quote
Feferman's and Robinson's finitist comments. Also, Feferman and
Robinson are considered to be on the standard theorist side of the
debate despite their finitist quotes.
Jesse F. Hughes
> We can make excuses for however long we want still the facts
> stand:http://coding.derkeiler.com/pdf/Archive/General/comp.theory/2009-04/msg00122.pdf[P==NP][Martin
> Musatov]
Your act was cute for a little while, but this dull repetition is
tedious, not clever.
--
Jesse F. Hughes
"Run mathematicians, RUN!!! I'm coming for you. It may take a few
months, but I'll get [computer verification of my proof] and then your
lives will be ended as you previously knew it." -- JSH meets PVS
Dan Christensen
Thanks, Bill. For the record, I accept the existence of infinite sets
and an infinite hierarchy of orders of infinity. But I don't see that
David's notion of potential existence is all that useful. And while
you might be able to contrive quasi-numerical properties for oo,
doesn't Gauss (and David here) have a point that, strictly speaking,
it isn't necessary? Can you give an simple example in mathematical
theory where it is absolutely essential? It seems to me that even
statements like
Aleph0 + Aleph0 = Aleph0
can be restated in terms of mappings between sets.
herbzet
david petry wrote:
> herbzet wrote:
> > david petry wrote:
> > > A Dedekind cut is an infinite set. �Infinite sets have only a
> > > potential existence.
> >
> > Hence, the length of the diagonal of a unit square exists only potentially.
> >
> > Did I say that right?
>
> Usually when we say something, we have a purpose for saying it. What
> would be the purpose for saying that?
To clarify that the length (which is a number) of the diagonal does
not have an actual existence. Is that what you mean?
> I can only repeat things I've already said. I don't know why you have
> so much trouble understanding what I say.
I was dropped on my head as a child. Perhaps that has something to do
with it.
Or perhaps the problem I have understanding you stems from your inability
to express yourself clearly.
On the other hand, have you considered the possibility that the problem
may be that I *do* understand you?
Well, let's not bother any more with why I do or don't understand you,
'kay? It's off-topic really.
> The idea is that infinity is merely a useful fiction, or equivalently,
> a figure a speech.
A figure of speech? Sure, why not? Some people think that integers
are just figures of speech, useful fictions.
> The only properties that infinity has are useful properties.
I would say that if infinity is merely a figure of speech, then
the only properties it would have would be those we ascribe to it,
whether they be useful or not.
> That is, we're trying to understand the world we observe,
> and any given observation contains only finite information. Infinity
> is useful when it helps us (as a conceptual aid) to understand those
> finite observations. When we think of infinity in that way, then we
> say, just once, that infinity has only a "potential" existence, but
> then we never ever have to mention the word "potential" again.
Fine by me. I think it might be less contentious to say here that
infinity has a "hypothetical" existence, rather than a "potential"
existence. Hypothesizing entities for the purposes of explaining
phenomena is a hallowed tradition in the most hard-headed of the
sciences. E.g., quarks.
Personally, the word "potential" doesn't mean much to me in this context;
the phrases "potential set", "potential existence", "potential infinity",
etc., just don't seem meaningful english to me in this application.
--
hz
herbzet
Bill Dubuque wrote:
A lovely book!
--
hz
Aatu Koskensilta
> Bill Dubuque wrote:
>
>> For an introduction accessible to a bright layperson see Rudy
>> Rucker's book "Infinity and the Mind".
>
> A lovely book!
In several regards, yes. It does contain, though, many a fine example
of pointless philosophical posturing and gratuitous formal logic
chopping, with a few rather perplexing reflections on the
incompleteness theorems.
On an unrelated note, your sig-delimiter is broken. There should be a
space after the dashes.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Marshall
>
> > The idea is that infinity is merely a useful fiction, or equivalently,
> > a figure a speech.
>
> A figure of speech? Sure, why not? Some people think that integers
> are just figures of speech, useful fictions.
Indeed; I proposed that his thesis works as well with "seven"
in place of "infinity."
> Fine by me. I think it might be less contentious to say here that
> infinity has a "hypothetical" existence, rather than a "potential"
> existence. Hypothesizing entities for the purposes of explaining
> phenomena is a hallowed tradition in the most hard-headed of the
> sciences. E.g., quarks.
>
> Personally, the word "potential" doesn't mean much to me in this context;
> the phrases "potential set", "potential existence", "potential infinity",
> etc., just don't seem meaningful english to me in this application.
I might propose that these are meaningful English but not meaningful
mathematics.
Perhaps a better word choice would be "abstract."
Marshall
Aatu Koskensilta
> david petry <david_lawr...@yahoo.com> writes:
>
>>"The actual infinite is not required for the mathematics of the
>>physical world" (Soloman Feferman)
>
> That's his mistake.
Why do you think Solomon Feferman is mistaken? Before jumping into any
rash conclusion it's a good idea to take into account the context of
such statements as quoted above. Here you should recall Feferman has
been heavily involved in the project to determine what mathematical
principles are essential to these and those mathematical results and
techniques, in particular in the study of predicativism. It is
certainly plausible that most of the mathematical machinery necessary
in physics can be formalised so as to be conservative over PA or even
PRA. As the saying goes, the need for powerful mathematical principles
and abstractions is probably not a logical need, but a more subtle
matter.
> But what the philosophers say that scientists do has very little
> connection with what scientists actually do.
We may further observe that often what scientists say that scientists
do has very little connection with what scientists actually do.
> Falsificationism turns out to be unfalsifiable, so by its own
> criteria it ought to be rejected.
Why? No one has claimed the falsifiability criterion is a scientific
claim or theory, certainly not Popper.
Bill Dubuque
>herbzet <her...@gmail.com> writes:
>> Bill Dubuque wrote:
>>
>>> For an introduction accessible to a bright layperson see Rudy
>>> Rucker's book "Infinity and the Mind".
>>
>> A lovely book!
>
> In several regards, yes. It does contain, though, many a fine example
> of pointless philosophical posturing and gratuitous formal logic
> chopping, with a few rather perplexing reflections on the
> incompleteness theorems.
"gratuitous formal logic chopping" What does than mean?
If you think you could do better then, by all means, please do so.
I'm not aware of anything else worth recommending for such an audience.
Aatu Koskensilta
> "gratuitous formal logic chopping" What does than mean?
Logic chopping that is both gratuitous and formal. On reflection,
"gratuitously formal logic chopping" is perhaps a more apt
description, certainly of such sections as /A Technical Note on
Man-Machine Equivalence/.
> If you think you could do better then, by all means, please do so.
> I'm not aware of anything else worth recommending for such an
> audience.
My abilities or lack thereof as a popularizer are irrelevant. As
already conceded, it is in many regards a lovely book, and my carping
on some of its flaws is of course but an example of the well-known
phenomenon that experts will invariably find something to complain
about in any given popular exposition, however swell.
Bill Dubuque
You can find some examples of "necessary" uses of infinite ordinals
in some of my old posts on Goodstein's theorem, Kruskal's tree theorem,
etc. For example, see the links in my post [1] of 12 Nov 2002. Such
examples are not contrived - they occur frequently when one needs
to recursively define functions on complex tree-like structures.
In fact I first encountered such ideas while I was discovering the
first effective algorithm for computing limits and asymptotics
circa 1980 as an undergrad in the MIT Mathlab (Macsyma) group.
--Bill Dubuque
[1] http://google.com/groups?selm=y8zsmy6fiqc.fsf%40nestle.ai.mit.edu
Bill Dubuque
>Bill Dubuque <w...@nestle.csail.mit.edu> writes:
>>
>> "gratuitous formal logic chopping" What does than mean?
>
> Logic chopping that is both gratuitous and formal. On reflection,
> "gratuitously formal logic chopping" is perhaps a more apt
> description, certainly of such sections as /A Technical Note on
> Man-Machine Equivalence/.
Precisely what do you mean by "logic chopping"?
MoeBlee
> On May 5, 8:34 pm, fishfry <BLOCKSPAMfish...@your-mailbox.com> wrote:
>
> > In article
> > <799fc943-6e7a-4abe-b11f-96a8203d5...@k19g2000prh.googlegroups.com>,
> > david petry <david_lawrence_pe...@yahoo.com> wrote:
> > > "The actual infinite is not required for the mathematics of the
> > > physical world" (Soloman Feferman)
> > > When we seek to understand the real world, we are forced to do
> > > mathematics. It would be a very reasonable thing to do to define
> > > mathematics in terms of what it is we are forced to do when we seek to
> > > understand the real world.
> > Math is a tool for understanding the real world. It's also a whole lot
> > more. It's that "whole lot more" that separates out the mathematicians
> > from the physical and social scientists, who use math to understand and
> > model the real world.
>
> But if a so-called "crank" proposed a alternate theory, that theory
> isn't allowed to be part of that "whole lot more." Instead, the
> standard set theorist would insist that the "crank" explain the
> theory's applicability to science and the real world.
On what basis do you make that claim? Of course, to the extent that
one is interested in a foundational theory that provides for the
mathematics for the sciences, any theory proposed will be evaluated by
that criterion. But that does not preclude that one may also
appreciate a theory even if it is not a foundational theory that
provides for the mathematics for the sciences. First order group
theory does not in itself provide for all the mathematics for the
sciences, yet, of course, we may still be interested in first order
group theory.
> Unwritten rule #1: only standard theories are allowed to be part of
> the "whole lot more." "Crank" theories must have an application to
> the real world in order to be considered.
What is your basis for saying that?
> Also, the OP's quote of Feferman sounds very similar to Robinson's
> comment about infinite sets not existing. Finitists "cranks" quote
> Robinson all the time, yet are ridiculed for doing so.
They're ridiculed when they say that Robinson exemplifies the kind of
alternative mathematics they admire while not facing the fact that
Robinson uses ZFC to obtain his results.
> Unwritten rule #2: only standard theorists are allowed to quote
> Feferman's and Robinson's finitist comments. Also, Feferman and
> Robinson are considered to be on the standard theorist side of the
> debate despite their finitist quotes.
Well, Robinson worked in ZFC. Feferman worked in ZFC too, though
Feferman does also propose predicativist alternatives. As to "being on
a side", I don't know who even mentioned such "sides".
I don't know why you persist to drag yourself down with the whole
CARGO BIN of incorrect and irrelevant baggage you continue to take on.
MoeBlee
MoeBlee
wrote:
> It seems to me that even
> statements like
>
> Aleph0 + Aleph0 = Aleph0
>
> can be restated in terms of mappings between sets.
Of course it can be. So what?
MoeBlee
MoeBlee
> Bill Dubuque wrote:
> For an introduction accessible
> > to a bright layperson see Rudy Rucker's book "Infinity and the Mind".
>
> A lovely book!
It's a fun book. But it has at least one substantive error, as pointed
out by Torkel Frazen.
MoeBlee
Aatu Koskensilta
One might ask, What point is there to introducing cardinals and
ordinals as objects on their own right, instead of treating statements
about cardinalities and order-types as facon de parler for statements
about mappings between sets? One possible answer would be to note that
this allows e.g. for constructions in stages indexed by ordinals etc.
MoeBlee
Aatu Koskensilta
> Aatu Koskensilta <aatu.kos...@uta.fi> wrote:
>
>> On reflection, "gratuitously formal logic chopping" is perhaps a
>> more apt description, certainly of such sections as /A Technical
>> Note on Man-Machine Equivalence/.
>
> Precisely what do you mean by "logic chopping"?
Just the usual: scholastic, spurious argumentation, devious logical
machinations of little substance and so on.
Rucker's claim, that his bandying about essentially arbitrary
formalities demonstrates it's possible to formalise, and reason
rigorously about, such notions as "human mathematical intuition",
"mathematical universe" and what not, is laughable. Such notions are
not made a whit more definite by juggling formal predicates stipulated
to denote them, nor is the reasoning any more rigorous. What have
instead is a seductive illusion of exactness and rigour -- indeed,
Rucker's formalism itself embodies, in a way that is not immediately
apparent, many extremely dubious philosophical assumptions.
Bill Dubuque
> Bill Dubuque <w...@nestle.csail.mit.edu> writes:
>> Aatu Koskensilta <aatu.kos...@uta.fi> wrote:
>>
>>> On reflection, "gratuitously formal logic chopping" is perhaps a
>>> more apt description, certainly of such sections as /A Technical
>>> Note on Man-Machine Equivalence/.
>>
>> Precisely what do you mean by "logic chopping"?
>
> Just the usual: scholastic, spurious argumentation, devious logical
> machinations of little substance and so on.
I suspect that the term has some fuzzy meaning to philosophers
but I doubt many mathematicians have any clue what you mean.
Do you have any examples in mind from Rucker's book, esp.
those having to do with the topic at hand? (infinity)
> Rucker's claim, that his bandying about essentially arbitrary
> formalities demonstrates it's possible to formalise, and reason
> rigorously about, such notions as "human mathematical intuition",
> "mathematical universe" and what not, is laughable. Such notions are
> not made a whit more definite by juggling formal predicates stipulated
> to denote them, nor is the reasoning any more rigorous. What have
> instead is a seductive illusion of exactness and rigour -- indeed,
> Rucker's formalism itself embodies, in a way that is not immediately
> apparent, many extremely dubious philosophical assumptions.
I metioned Rucker as a popular introduction to infinity in mathematics -
not philosophy.
--Bill Dubuque
Aatu Koskensilta
> Aatu Koskensilta <aatu.kos...@uta.fi> wrote:
>> Bill Dubuque <w...@nestle.csail.mit.edu> writes:
>>
>>> Precisely what do you mean by "logic chopping"?
>>
>> Just the usual: scholastic, spurious argumentation, devious logical
>> machinations of little substance and so on.
>
> I suspect that the term has some fuzzy meaning to philosophers but I
> doubt many mathematicians have any clue what you mean. Do you have
> any examples in mind from Rucker's book, esp. those having to do
> with the topic at hand? (infinity)
I already mentioned the section /A Technical Note on Man-Machine
Equivalence/. I don't off-hand recall any equally atrocious examples
of formal mumbo-jumbo having to do with infinity.
It is apparent even in passages dealing with infinity that Rucker has
a philosophical axe to grind -- he explicitly peddles with gay abandon
a pet solution to many a classical philosophical conundrum. As noted,
the text is in many ways a splendid book, but I would suggest the
innocent reader takes Rucker's more philosophical (and rather
idiosyncratic) reflections as mere rhetorical flourish, ignoring his
musings on incompleteness, robots and so on altogether.
MoeBlee
> MoeBlee <jazzm...@hotmail.com> writes:
> > On May 11, 9:44 pm, Dan Christensen <Dan_Christen...@sympatico.ca>
> > wrote:
>
> >> It seems to me that even statements like
>
> >> Aleph0 + Aleph0 = Aleph0
>
> >> can be restated in terms of mappings between sets.
>
> > Of course it can be. So what?
>
> One might ask, What point is there to introducing cardinals and
> ordinals as objects on their own right, instead of treating statements
> about cardinalities and order-types as facon de parler for statements
> about mappings between sets? One possible answer would be to note that
> this allows e.g. for constructions in stages indexed by ordinals etc.
I quite agree as to the convenience of that. But still it could be
done equivalently in the primitive language, if that were demanded.
(I don't think we're disagreeing about anything here, are we?)
MoeBlee
lwa...@lausd.net
> Bill Dubuque <w...@nestle.csail.mit.edu> writes:
> > I suspect that the term has some fuzzy meaning to philosophers but I
> > doubt many mathematicians have any clue what you mean. Do you have
> > any examples in mind from Rucker's book, esp. those having to do
> > with the topic at hand? (infinity)
> a philosophical axe to grind -- he explicitly peddles with gay abandon
> a pet solution to many a classical philosophical conundrum. As noted,
> the text is in many ways a splendid book, but I would suggest the
> innocent reader takes Rucker's more philosophical (and rather
> idiosyncratic) reflections as mere rhetorical flourish, ignoring his
> musings on incompleteness, robots and so on altogether.
Rucker's book sounds a lot like the book I've read a while back,
"Kingdom of Infinite Number" (don't remember the author). Both
books attempt to introduce set theory to the layperson, but
both make mistakes. I remember how "Kingdom" not only makes
the common error that aleph_1 is _defined_ to equal 2^aleph_0,
but even claims that CH is equivalent to the statement that omega
(not omega_1, but _omega_) is equal to the continuum!
Another book, "1, 2, 3, Infinity," (Gamow? is the author?) defines
aleph_1 as 2^aleph_0 and then declares ~CH to be equivalent to
be equivalent to the statement that the set of irrationals has a
cardinality strictly between aleph_0 and aleph_1 (which is wrong
for several reasons)!
I believe I might have read Rucker many years ago. I forget what
errors he makes (but I know that they're more sophisticated than
those involving aleph_1 or CH), but I believe that it's one of the
first
places where I've heard of the operation of tetration (where Rucker
defines epsilon_0 to be the ordinal omega^^omega).
Of course, this begs the question, why do I read Rucker or Gamow
but not real textbooks that don't make these errors? That's an
easy one: Rucker and Gamow are less expensive than most
textbooks, and their books are easier to find. Major bookstores are
much more likely to carry Rucker or Gamow than textbooks, while
you're lucky if Amazon carries the right textbook. Also, I might
have read Rucker or Gamow at a library -- a _public_ library, not a
university library that carries textbooks but requires that one be
affiliated with the university to use.
I'm aware that many so-called "cranks" read Rucker or Gamow,
then post errors regarding aleph_1 or CH on Usenet, then are
confused when the standard theorists tell them that they're wrong
since the "crank" merely quotes Rucker/Gamow. I do wish that
Rucker, Gamow, and others wouldn't make such errors, since the
"crank" is more likely to read them than a full-priced textbook.
MoeBlee
> why do I read Rucker or Gamow
> but not real textbooks that don't make these errors? That's an
> easy one: Rucker and Gamow are less expensive than most
> textbooks
Wow, if ever there were a time to say "penny wise, pound foolish"!
Reminds of the song lyric about people who know the price of
everything and the value of nothing at all.
> while
> you're lucky if Amazon carries the right textbook
I don't know what textbook you might want but that Amazon doesn't have
it used?
You can get a lot of used math textbooks (especially Dover editions)
fairly cheap at Amazon.
Though, there are plenty of errors (usually typos) even in some of the
best textbooks. I find that a part (albeit small) of the learning
process is reading textbooks carefully and critically so that I can
spot typos and re-interpret in that context.
MoeBlee
Marshall
> On May 12, 3:00 pm, lwal...@lausd.net wrote:
>
> > why do I read Rucker or Gamow
> > but not real textbooks that don't make these errors? That's an
> > easy one: Rucker and Gamow are less expensive than most
> > textbooks
>
> Wow, if ever there were a time to say "penny wise, pound foolish"!
>
> Reminds of the song lyric about people who know the price of
> everything and the value of nothing at all.
Suppes' "Axiomatic Set Theory" on Amazon:
http://www.amazon.com/Axiomatic-Set-Theory-Patrick-Suppes/dp/0486616304
is $10.36 new. Used copies are as low as $5.79.
Approximately the same price as a single ticket to see
"Ghosts of Girlfriends Past" starring Matthew McConaughey
and Jennifer Garner.
Celebrity photographer Connor Mead (MATTHEW McCONAUGHEY)
loves freedom, fun and women...in that order. A committed
bachelor with a no-strings policy, he thinks nothing of breaking
up with multiple women on a conference call while prepping
his next date.
Rudy Rucker's "Infinity and the Mind" $16.47 on Amazon.
Whoops, gotta run; no link; non-sequitur not found.
Marshall
Aatu Koskensilta
> I'm aware that many so-called "cranks" read Rucker or Gamow, then
> post errors regarding aleph_1 or CH on Usenet, then are confused
> when the standard theorists tell them that they're wrong since the
> "crank" merely quotes Rucker/Gamow. I do wish that Rucker, Gamow,
> and others wouldn't make such errors, since the "crank" is more
> likely to read them than a full-priced textbook.
The problem with Rucker's book is not that it contains technical
errors, but rather that it presents, without being very explicit about
it, Rucker's personal reflections on various philosophical and
metaphysical matters, reflections he apparently personally finds
immensely satisfying but which are by no stretch of the imagination an
essential part of the mathematical understanding of the infinite, the
incompleteness theorems, non-standard analysis, Richard's paradox,
what have you. (I just re-read Rucker's book and can happily report it
is much worse than I recalled, literally full of ill-founded
metaphysical waffle and rather baffling philosophical claims offered
essentially without any justification whatever.)
Of course, if one wishes to mount an out-and-out attack on standard
mathematics, a deeper understanding is required than can be obtained
from popular expositions. Equally of course, if one wishes merely to
present some alternate ideas, developing them in some mathematically
and conceptually fruitful way, it is a good idea not to tie such ideas
and developments to silly claims about standard mathematics, at least
publicly. Quoting Kreisel commenting on Brouwer's missionary work --
in which Brouwer placed considerable emphasis on the "negative aspect"
to intuitionism -- from the Biographical Memoirs of Royal Society
article on Brouwer (available on-line for free):
Brouwer's own doctrinaire presentation was not only philosophically
dubious but practically unsuccessful because it didn't really convey
his views. It is highly likely that, at an early stage, his own work
benefitted greatly from two very usual consequences of any
doctrinaire position. He was able to develop his ideas vigorously,
first because he had put out of his mind all but the matter in hand;
and second, because weaknesses of a position are less 'disturbing' if
(one thinks) there are no alternatives.
No doubt this quote partially reflects Kreisel's happy-go-lucky
eclectic milk-sop attitude to foundations[1], but there is an important
if trivial truth to be appreciated, namely that it is futile to expect
people to pay attention to one's ideas merely as a result of endless
bickering about how rotten the accepted ideas are -- rather, what is
needed are striking demonstrations of fruitfulness, of conceptual,
mathematical, philosophical insights to be gleaned from one's pet
ideas.
Footnotes:
[1] Astute readers will no doubt have already surmised this is an
attitude I find very congenial.
Aatu Koskensilta
Certainly, but to show that this is so we must show e.g. that we can
assign to well-orderings set representatives in some canonical way
(von Neumann ordinals, sets of order-isomorphic sets of least possible
rank, ...), effectively establishing that we can indeed talk of such
things as order-types, cardinal numbers, and so on, as objects on
their own right in set theory, i.e. giving a set theoretic
reduction. Without such a reduction, and without introducing ordinals,
cardinals, etc. as primitive, it wouldn't be possible to carry out
constructions in stages indexed by ordinals etc. In this sense the set
theoretic reduction does more mathematical work than could be squeezed
out of treating cardinalities, order types, etc. as merely facon de
parler for talk about mappings and what not.
Marshall
> but there is an important
> if trivial truth to be appreciated, namely that it is futile to expect
> people to pay attention to one's ideas merely as a result of endless
> bickering about how rotten the accepted ideas are -- rather, what is
> needed are striking demonstrations of fruitfulness, of conceptual,
> mathematical, philosophical insights to be gleaned from one's pet
> ideas.
Hear hear!
Marshall
MoeBlee
> The problem with Rucker's book is not that it contains technical
> errors,
I would call his overlooking (as noted by Franzen) that the Godel
machine (or whatever it's called in the book) doesn't have to tell the
truth a technical/conceptual error to the extent that the notion of
the machine is an analogy with a technical concern, so that Rucker is
overlooking that a theory might not be arithmetically sound.
MoeBlee
Aatu Koskensilta
> I would call his overlooking (as noted by Franzen) that the Godel
> machine (or whatever it's called in the book) doesn't have to tell
> the truth a technical/conceptual error to the extent that the notion
> of the machine is an analogy with a technical concern, so that
> Rucker is overlooking that a theory might not be arithmetically
> sound.
Rucker explicitly stipulates the Universal Truth Machine with which
G�del is having a conversation only utters arithmetical truths. The
problem is not technical but conceptual. It is simply unclear what we
are to make of Rucker's stipulations concerning the fable he's
offering, and if we try to take them seriously, as a description of a
hypothetical situation, the whole setup simply breaks up, unable carry
the metaphysical and philosophical weight put on it by Rucker. This
point is driven home in a delightful manner by Torkel's alternative
ending to the fable.
MoeBlee
> MoeBlee <jazzm...@hotmail.com> writes:
> > I would call his overlooking (as noted by Franzen) that the Godel
> > machine (or whatever it's called in the book) doesn't have to tell
> > the truth a technical/conceptual error to the extent that the notion
> > of the machine is an analogy with a technical concern, so that
> > Rucker is overlooking that a theory might not be arithmetically
> > sound.
>
> Rucker explicitly stipulates the Universal Truth Machine with which
> problem is not technical but conceptual. It is simply unclear what we
> are to make of Rucker's stipulations concerning the fable he's
> offering, and if we try to take them seriously, as a description of a
> hypothetical situation, the whole setup simply breaks up, unable carry
> the metaphysical and philosophical weight put on it by Rucker. This
> point is driven home in a delightful manner by Torkel's alternative
> ending to the fable.
Okay, I should not have said that he overlooks but rather that he
stipulates for no good reason. Because Franzen shows (in CORRECT
alignment with the interesting mathematics) what happens when the
machine speaks falsely.
Though the mistake is in the informal allegory, I think it is a
technical mistake in the sense that it is is borne froma technical
mistake; it's an incorrect analogy with the actual pertinent
mathematics. But I don't think we need to debate over the matter. It's
fine with me that one may regard the error as merely conceptual or as
conceptual/technical.
MoeBlee
MoeBlee
> On May 12, 4:46 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
>
>
>
> > MoeBlee <jazzm...@hotmail.com> writes:
> > > I would call his overlooking (as noted by Franzen) that the Godel
> > > machine (or whatever it's called in the book) doesn't have to tell
> > > the truth a technical/conceptual error to the extent that the notion
> > > of the machine is an analogy with a technical concern, so that
> > > Rucker is overlooking that a theory might not be arithmetically
> > > sound.
>
> > Rucker explicitly stipulates the Universal Truth Machine with which
> > Gödel is having a conversation only utters arithmetical truths. The
> > problem is not technical but conceptual. It is simply unclear what we
> > are to make of Rucker's stipulations concerning the fable he's
> > offering, and if we try to take them seriously, as a description of a
> > hypothetical situation, the whole setup simply breaks up, unable carry
> > the metaphysical and philosophical weight put on it by Rucker. This
> > point is driven home in a delightful manner by Torkel's alternative
> > ending to the fable.
>
> Okay, I should not have said that he overlooks but rather that he
> stipulates for no good reason.
But wait a minute, before I concede the point, I should look at the
book tonight. For I don't recall that Rucker does say that the machine
only tells truth (as opposed to the machine claiming only to tell
truth).
MoeBlee
Aatu Koskensilta
> But wait a minute, before I concede the point, I should look at the
> book tonight. For I don't recall that Rucker does say that the
> machine only tells truth (as opposed to the machine claiming only to
> tell truth).
Examining the relevant passages more carefully, I now find Rucker's
fable starts with the stipulation that G�del is introduced, by
someone, to a machine /supposed to be/ a Universal Truth Machine,
capable of answering any question correctly.
Covering my face in shame, I now readily admit Rucker's colourful
account of this historic meeting is not only misleading and
conceptually unfounded but also simply mistaken on a technical
point. Rather feeble I offer in my defense the claim that my earlier
description of Rucker's G�del waffle is still quite accurate,
understood more generally, even if in this instance we can discern a
specific technical error.
Aatu Koskensilta
> But wait a minute, before I concede the point, I should look at the
> book tonight. For I don't recall that Rucker does say that the
> machine only tells truth (as opposed to the machine claiming only to
> tell truth).
Examining the relevant passages more carefully, I now find Rucker's
fable starts with the stipulation that G�del is introduced, by
someone, to a machine /supposed to be/ a Universal Truth Machine,
capable of answering any question correctly.[1]
Covering my face in shame, I now readily admit Rucker's colourful
account of this historic meeting is not only misleading and
conceptually unfounded but also simply mistaken on a technical
point. Rather feebly I offer in my defense the claim that my earlier
description of Rucker's G�del waffle is still quite accurate,
understood more generally, even if in this instance we can discern a
specific technical error.
Footnotes:
[1] I am basing my comments on the Finnish translation of the book,
which alas is not particularly good. Caveat lector. (For some
unfathomable reason the translator has chosen to inform Finnish
readers of Cantor's having published several papers with English
titles, always a sign of shoddy work.)
david petry
> david petry wrote:
> > herbzet wrote:
> > > david petry wrote:
> > > > A Dedekind cut is an infinite set. Infinite sets have only a
> > > > potential existence.
>
> > > Hence, the length of the diagonal of a unit square exists only potentially.
>
> > > Did I say that right?
>
> > Usually when we say something, we have a purpose for saying it. What
> > would be the purpose for saying that?
>
> To clarify that the length (which is a number) of the diagonal does
> not have an actual existence. Is that what you mean?
If, in your own mind, you equate the length with an infinite set, then
I guess you might say that the length has a "potential" existence. I
guess. But if you recognize that we have an algorithm to compute the
length to any desired accuracy, and you equate the length with that
algorithm, then you would probably want to say that the length has an
actual existence, although, again, I don't know why you would want to
say that.
Dan Christensen
Thanks again, Bill. I looked up Goodstein's theorem at Wiki.
I realize this may not be possible, but can you (or anyone else) give
a brief, simple example -- something, errr... for the bright
layperson, as you put it?
Dan
herbzet
Aatu Koskensilta wrote:
> herbzet writes:
> > Bill Dubuque wrote:
> >
> >> For an introduction accessible to a bright layperson see Rudy
> >> Rucker's book "Infinity and the Mind".
> >
> > A lovely book!
>
> In several regards, yes. It does contain, though, many a fine example
> of pointless philosophical posturing and gratuitous formal logic
> chopping, with a few rather perplexing reflections on the
> incompleteness theorems.
Sounds a lot like sci.logic! :-0
Still, it's engaging and relatively accessible to a bright layperson,
as BD says.
If you get a chance to address the "perplexing reflections" on the
incompleteness theorems, I'd love to see your thoughts.
> On an unrelated note, your sig-delimiter is broken. There should be a
> space after the dashes.
I leave out the space on purpose -- I hope it's not too annoying to
edit out my sig.
--
hz
david petry
> herbzet <herb...@gmail.com> writes:
> > Bill Dubuque wrote:
>
> >> For an introduction accessible to a bright layperson see Rudy
> >> Rucker's book "Infinity and the Mind".
>
> > A lovely book!
>
> In several regards, yes. It does contain, though, many a fine example
> of pointless philosophical posturing and gratuitous formal logic
> chopping, with a few rather perplexing reflections on the
> incompleteness theorems.
Interesting. FWIW, I read at least significant parts of that book
many years ago, and that reading confirmed for me my suspicions that
modern mathematics has taken a strange detour away from truth and
reality, and has entered the realm of mysticism and fantasy.
Aatu Koskensilta
> FWIW, I read at least significant parts of that book many years ago,
> and that reading confirmed for me my suspicions that modern
> mathematics has taken a strange detour away from truth and reality,
> and has entered the realm of mysticism and fantasy.
It is understandable reading Rucker's book should leave one with such
an impression -- on re-reading I find it much worse than I
recalled. Similarly, one might obtain a rather befuddling picture of
the theory of evolution from reading Teilhard's musings on the
subject. It is best to leave such metaphysical fantasies aside when
trying to wrap one's head around the actual substance of the subjects
involved, satisfying as the speculations might be to author, and those
sharing their penchant for philosophical mumbo-jumbo the only apparent
use of which is to make people's heads swim in pleasant confusion.
herbzet
Marshall wrote:
> herbzet wrote:
> >
> > > The idea is that infinity is merely a useful fiction, or equivalently,
> > > a figure a speech.
> >
> > A figure of speech? �Sure, why not? �Some people think that integers
> > are just figures of speech, useful fictions.
>
> Indeed; I proposed that his thesis works as well with "seven"
> in place of "infinity."
Yeah, I saw your post after I composed mine. So of course I thought
"Drat, he beat me to it." Then I thought, "Maybe I should allude to
Marshall's post in my reply?". Then I thought, "No, that would
suggest I got the idea from him." Then I thought, "So what? Are
you that desperate for intellectual credit?" Then I thought,
"Oh, fuck it" and posted it as written. Then I lay down for
awhile.
> > Fine by me. �I think it might be less contentious to say here that
> > infinity has a "hypothetical" existence, rather than a "potential"
> > existence. �Hypothesizing entities for the purposes of explaining
> > phenomena is a hallowed tradition in the most hard-headed of the
> > sciences. �E.g., quarks.
Oh, yeah, I wanted to add "or phlogiston", but I couldn't remember
the word at the time.
> > Personally, the word "potential" doesn't mean much to me in this context;
> > the phrases "potential set", "potential existence", "potential infinity",
> > etc., just don't seem meaningful english to me in this application.
>
> I might propose that these are meaningful English but not meaningful
> mathematics.
>
> Perhaps a better word choice would be "abstract."
Perhaps -- "hypothetical" seemed good in that context. Besides,
what *isn't* an abstract object in logic/math? (Please don't
answer "seven"!)
--
hz
herbzet
herbzet wrote:
> If you get a chance to address the "perplexing reflections" on the
> incompleteness theorems, I'd love to see your thoughts.
I see now that you probably meant "perplexing philosophical reflections"
rather than "mathematical errors or ambiguities".
If there's actual mistakes in the incompleteness mathematics, that's
what I'd be interested in. The philosophical consequences he draws
are clearly something different and debatable.
--
hz
herbzet
Aatu Koskensilta wrote:
> Rucker's claim, that his bandying about essentially arbitrary
> formalities demonstrates it's possible to formalise, and reason
> rigorously about, such notions as "human mathematical intuition",
> "mathematical universe" and what not, is laughable. Such notions are
> not made a whit more definite by juggling formal predicates stipulated
> to denote them, nor is the reasoning any more rigorous. What have
> instead is a seductive illusion of exactness and rigour -- indeed,
> Rucker's formalism itself embodies, in a way that is not immediately
> apparent, many extremely dubious philosophical assumptions.
Still, it's fun stuff, with some good semi-rigorous stuff mixed in.
Provocative for the general reader -- hopefully, the general reader
would be intrigued enough to read other stuff.
--
hz
herbzet
I'm just trying to figure out what are the consequences of your
assertion that infinite sets have only a potential existence.
E.g., the real numbers may be construed as Dedekind cuts of
rational numbers, which cuts are, as you say, infinite, and
therefore, according to your edict, have only a potential existence.
This would suggest that you consider real numbers as having
only a potential existence.
But considering what you say above, and your previous posts,
I'm guessing that you accept *some* real numbers as having
"actual" existence -- e.g., sqrt(2), pi, the Louisville
constant -- because the algorithm for their construction
can be finitely given.
So, I'm guessing that the breakdown for you is:
Computable number = actually existent number
Non-computable number = potentially existent number
whereas my general impression was that people who subscribe
to the potential/actual number distinction usually have it that
rational number = actually existent number
irrational number = potentially existent number
or maybe
algebraic number = actual
non-algebraic number = potential.
I must say that I don't really grasp what it means to
describe a set or a number as having a potential existence,
although I might understand something like "There potentially
exists an even number greater than 2 that is not the sum of
two primes" as an epistemological assertion.
--
hz
Daryl McCullough
>I'm just trying to figure out what are the consequences of your
>assertion that infinite sets have only a potential existence.
David Petry's posts are pure expressions of emotion (in particular,
hate). They have no propositional content.
--
Daryl McCullough
Ithaca, NY
herbzet
Daryl McCullough wrote:
> herbzet says...
>
> >I'm just trying to figure out what are the consequences of your
> >assertion that infinite sets have only a potential existence.
>
> David Petry's posts are pure expressions of emotion (in particular,
> hate). They have no propositional content.
As long as he confines himself to the maths, I'm not averse
to discussing with him.
I think he's trying to be mathematical.
This potential/actual debate seems to have been infused with
a lot of religious-type hysteria since Kronecker, at least.
Like civilization is at stake.
Not sure why that is.
--
hz
david petry
> I'm just trying to figure out what are the consequences of your
> assertion that infinite sets have only a potential existence.
Infinity may be thought of as a figure of speech, and figures of
speech have no consequences at all.
As Herman Jurjus used to like to say, we should think about infinite
sets in a way analogous to the way we think about infinitesimals--they
don't really "exist"; they can't be used to prove anything that can't
be proved without them; but they do have a place within a certain
conceptual framework which helps us reason about mathematical ideas.
Bill Dubuque
>> examples are not contrived - they occur frequently when one needs
>> to recursively define functions on complex tree-like structures.
>> In fact I first encountered such ideas while I was discovering the
>> first effective algorithm for computing limits and asymptotics
>> circa 1980 as an undergrad in the MIT Mathlab (Macsyma) group.
>>
>
> Thanks again, Bill. I looked up Goodstein's theorem at Wiki.
>
> I realize this may not be possible, but can you (or anyone else)
> give a brief, simple example -- something, errr... for the bright
> layperson, as you put it?
Did you peruse the links in my prior post [0] cited above?
It's first link is to my sci.logic post [1] on 11 Dec 95
which contains an introduction to Goodstein's theorem.
--Bill Dubuque
[1] http://groups.google.com/groups?selm=y8zogx9m4i4.fsf%40nestle.ai.mit.edu
Chris Menzel
<lwa...@lausd.net> said:
> On May 12, 10:49ï¿œam, Aatu Koskensilta <aatu.koskensi...@uta.fi>
> wrote:
>> Bill Dubuque <w...@nestle.csail.mit.edu> writes:
>> > I suspect that the term has some fuzzy meaning to philosophers but
>> > I doubt many mathematicians have any clue what you mean. Do you
>> > to do with the topic at hand? (infinity)
>>
>> It is apparent even in passages dealing with infinity that Rucker has
>> a philosophical axe to grind -- he explicitly peddles with gay
>> abandon a pet solution to many a classical philosophical conundrum.
>> As noted, the text is in many ways a splendid book, but I would
>> suggest the innocent reader takes Rucker's more philosophical (and
>> rather idiosyncratic) reflections as mere rhetorical flourish,
>> ignoring his musings on incompleteness, robots and so on altogether.
>
> Rucker's book sounds a lot like the book I've read a while back,
> "Kingdom of Infinite Number" (don't remember the author). Both books
> attempt to introduce set theory to the layperson, but both make
> more sophisticated than those involving aleph_1 or CH), .
I'd be very surprised if Rucker's contains any mathematical errors. (As
Aatu notes, it is chockablock with dubious philosophical claims but is
nonetheless a very entertaining read.)
> ...Of course, this begs the question,
> ...why do I read Rucker or Gamow but not real textbooks that don't
> make these errors? That's an easy one: Rucker and Gamow are less
> expensive than most textbooks,
But not all, at least not appreciably. Levy's excellent text _Basic Set
Theory_, for example, is under $20 and can be found used for even less.
Suppes' _Axiomatic Set Theory_ is 10 bucks new.
> and their books are easier to find.
Easier than typing "http://amazon.com"?
Chris Menzel
<aatu.kos...@uta.fi> said:
> MoeBlee <jazz...@hotmail.com> writes:
>
>> I would call his overlooking (as noted by Franzen) that the Godel
>> machine (or whatever it's called in the book) doesn't have to tell
>> the truth a technical/conceptual error to the extent that the notion
>> of the machine is an analogy with a technical concern, so that
>> Rucker is overlooking that a theory might not be arithmetically
>> sound.
>
> Rucker explicitly stipulates the Universal Truth Machine with which
> problem is not technical but conceptual. It is simply unclear what we
> are to make of Rucker's stipulations concerning the fable he's
> offering, and if we try to take them seriously, as a description of a
> hypothetical situation, the whole setup simply breaks up, unable carry
> the metaphysical and philosophical weight put on it by Rucker. This
> point is driven home in a delightful manner by Torkel's alternative
> ending to the fable.
Rest his soul.
Chris Menzel
> MoeBlee <jazz...@hotmail.com> writes:
>
>> But wait a minute, before I concede the point, I should look at the
>> book tonight. For I don't recall that Rucker does say that the
>> machine only tells truth (as opposed to the machine claiming only to
>> tell truth).
>
> Examining the relevant passages more carefully, I now find Rucker's
> someone, to a machine /supposed to be/ a Universal Truth Machine,
> capable of answering any question correctly.[1]
>
> Covering my face in shame, I now readily admit Rucker's colourful
> account of this historic meeting is not only misleading and
> conceptually unfounded but also simply mistaken on a technical
> point. Rather feebly I offer in my defense the claim that my earlier
> understood more generally, even if in this instance we can discern a
> specific technical error.
I wish this were the worst gaffe I've ever made on sci.logic. ;-)
amy666
> In article
> <799fc943-6e7a-4abe...@k19g2000prh.goog
> legroups.com>,
> david petry <david_lawr...@yahoo.com> wrote:
>
> > "The actual infinite is not required for the
> mathematics of the
> > physical world" (Soloman Feferman)
> >
> >
> > When we seek to understand the real world, we are
> forced to do
> > mathematics. It would be a very reasonable thing
> to do to define
> > mathematics in terms of what it is we are forced to
> do when we seek to
> > understand the real world.
>
> Silly. When I seek to go to the store for a bag of
> Fritos, I'm forced to
> drive my car. Would it then be "very reasonable" to
> define driving as
> going to the store for a bag of chips?
>
> Math is a tool for understanding the real world. It's
> also a whole lot
> more. It's that "whole lot more" that separates out
> the mathematicians
> from the physical and social scientists, who use math
> to understand and
> model the real world.
relativity is a " whole lot more "
it has an extra dimension , space time curvature , equations and other abstractions.
however its serve as a model for reality.
as for good pure math , it has applications in the sense of use , meaning and tools in other branches of math.
there is no such thing going on with aleph_73.
although aleph_73 might exist , i only show the uselessness here , not even considering the absurdity of non-consistant and/or randomly picked assumptions*.
( randomly picked assumptions = axioms of set theory )
however i respect your opinion ...
regards
tommy1729
BURT
wrote:
> "The actual infinite is not required for the mathematics of the
> physical world" (Soloman Feferman)
>
> When we seek to understand the real world, we are forced to do
> mathematics. It would be a very reasonable thing to do to define
> mathematics in terms of what it is we are forced to do when we seek to
> steps to ensure that mathematics stays in touch with reality. The
> scientists use the notion of falsifiability as a criterion to
> guarantee that science stays in touch with reality--to distinguish
> science from philosophy, theology and pseudoscience--and that notion
> could be incorporated into the foundations of mathematics where it
> could serve to distinguish the mathematics of the physical world from
> the mathematics of the metaphysical world; the mathematics of the
> physical world must meet the criterion of being falsifiable. It would
> be very reasonable to divide mathematics into (at least) two separate
> subjects--the mathematics of the physical world, and the mathematics
> of the metaphysical world--for exactly the same reasons that the
> scientists keep science separate from philosophy, theology and
> pseudoscience; the confusion caused by the failure to distinguish the
> two subjects is an obstacle to progress in the sciences and in
> technology. In particular, the general failure of people to recognize
> that the sacred cows of metaphysical mathematics (e.g. Cantor's theory
> of the actual infinite, and Godel's theorem) have nothing to tell us
> about the physical world is a serious obstacle to progress in the
> field of artificial intelligence.
>
> Experience shows that it is virtually impossible to discuss the ideas
> in the foregoing paragraph in these newsgroups; here, the inmates have
> taken over the asylum. However, the interested reader may want to read
> articles I've posted on this topic over the years.
>
> http://groups.google.com/group/sci.math/msg/8245894cf9c14ac6?hl=en
>
> http://groups.google.com/group/sci.math/msg/0845a6308e5d4633?hl=en
>
> http://groups.google.com/group/sci.math/msg/40cc4610018d67de?hl=en
>
> http://groups.google.com/group/sci.math/msg/859d0f3750a0e9dc?hl=en
>
> http://groups.google.com/group/sci.math/msg/0025be708362cb7e?hl=en
Sizes of infinity of the infinitely small are definitions of the
finites. This is the way to see finite sizes in the world.
Mitch Raemsch
lwa...@lausd.net
wrote:
> On Tue, 12 May 2009 15:00:48 -0700 (PDT), lwal...@lausd.net
> > ...Of course, this [raises] the question [changed since Menzel
> > insisted on providing the link to the explanation of the word]
> > ...why do I read Rucker or Gamow but not real textbooks that don't
> > make these errors? That's an easy one: Rucker and Gamow are less
> > expensive than most textbooks,
> But not all, at least not appreciably. Levy's excellent text _Basic Set
> Theory_, for example, is under $20 and can be found used for even less.
> Suppes' _Axiomatic Set Theory_ is 10 bucks new.
Interesting enough, I already own both of those books! I first
purchased Levy after being strongly encouraged by sci.math
posters to buy a book. Then I was told that Levy isn't a good
book for beginners, so I bought the Suppes book as well.
More often than not, when I complain about book prices, I'm
complaining about books about theories other than ZFC. Most
books about alternate theories really are expensive. But
Menzel's post is, I admit, a fair comparison since Levy,
Suppes, Rucker, and Gamow all discuss standard set theory.
> > and their books are easier to find.
> Easier than typing "http://amazon.com"?
Actually, I found Levy in a bookstore and Suppes via Amazon,
but as for Rucker and Gamow, I actually found those in a
library (a _public_ library, not a university library to which
I have no access but might actually carry Levy or Suppes).
herbzet
david petry wrote:
> herbzet wrote:
>
> > I'm just trying to figure out what are the consequences of your
> > assertion that infinite sets have only a potential existence.
>
> Infinity may be thought of as a figure of speech, and figures of
> speech have no consequences at all.
So, is the square root of 2 just a figure of speech?
Or does that depend on whether I think of it as a Dedekind cut rather
than as an unending algorithm?
> As Herman Jurjus used to like to say, we should think about infinite
> sets in a way analogous to the way we think about infinitesimals--they
> don't really "exist"; they can't be used to prove anything that can't
> be proved without them;
Not sure about that; Bill Dubuque is posting some good links which
suggest otherwise.
> but they do have a place within a certain
> conceptual framework which helps us reason about mathematical ideas.
--
hz
Neil W Rickert
>Neil W Rickert <ricke...@cs.niu.edu> writes:
>> david petry <david_lawr...@yahoo.com> writes:
>>>"The actual infinite is not required for the mathematics of the
>>>physical world" (Soloman Feferman)
>> That's his mistake.
>Why do you think Solomon Feferman is mistaken? Before jumping into any
>rash conclusion it's a good idea to take into account the context of
>such statements as quoted above.
I already stated in an earlier post, that Petry may have taken
Feferman's comment out of context, and that Feferman might not have
actually been mistaken when seen in context.
> As the saying goes, the need for powerful mathematical principles
>and abstractions is probably not a logical need, but a more subtle
>matter.
For sure. Physicists are studying physics, not logic. And they often
use their physical intuition to guide them. This is what tends to lead
to the kind of mathematical thinking used by mathematical platonists,
and is why platonism is a good mathematical fit for physics.
>> But what the philosophers say that scientists do has very little
>> connection with what scientists actually do.
>We may further observe that often what scientists say that scientists
>do has very little connection with what scientists actually do.
Agreed.
>> Falsificationism turns out to be unfalsifiable, so by its own
>> criteria it ought to be rejected.
>Why? No one has claimed the falsifiability criterion is a scientific
>claim or theory, certainly not Popper.
Popper was attempting to distinguish between what has an empirical
basis and what is religion. If falsificationism is religion,
we ought to discard it.
amy666
> On May 6, 11:04 pm, david
> <david_lawrence_pe...@yahoo.com> wrote:
> > On May 5, 4:13 am, Frederick Williams
> <frederick.willia...@tesco.net>
> > wrote:
> >
> > > david petry wrote:
> >
> > > > "The actual infinite is not required for the
> mathematics of the
> > > > physical world" (Soloman Feferman)
> >
> > > How does he know?
> >
> > He has devoted his career to studying proof theory,
> where they
> > carefully examine what assumptions are needed to
> prove various
> > theorems. He is widely regarded as a leader in the
> field. If anyone
> > "knows", it is him.
>
> Just to be clear on that point: Feferman is among the
> great
> mathematicians in set theory, and his philosophical
> views must be
> worthy of serious consideration. But his stature in
> the field does not
> alone give him sole or ultimate authority on such
> philosophical issues
> which are subject to a wide range of viewpoints
> including those of
> many other eminent mathematicians and philosphers.
>
> MoeBlee
lets turn the tables again !
i dare anyone to give 3 conjectures that can only be proved with the actual infinite and in no other way.
the 'neilists' will turn up again , but no math answers will.
regards
tommy1729
MoeBlee
> lets turn the tables again !
Only you know in what way you think you're turning any tables
> i dare anyone to give 3 conjectures that can only be proved with the actual infinite and in no other way.
If they're conjectures (in the sense of that they remain conjectures
today) then we don't know (in say, ZFC) whether there is a proof. But
to say "In ZFC, P can only be proven with Q" is to say:
If G |- P, then Q (or some Q* equivalent in ZFC to Q) is in G.
But I hardly think you've even thought the matter through that far.
If your challenge is simply that for any formula P, there are
consistent axioms that prove P but do not include any logical
equivalent of "there exist infinte sets" (as that is given some
particular formulation), then no one claims otherwise.
On the other hand, of course for any P, if P is proven from some set
of axioms S, then there is finite conjunction C of members of S such
that C->P is logically equivalent with P (some people seem illogically
to think that that is some kind of question begging).
What you imagine this has to do with Feferman, only you can say.
> the 'neilists' will turn up again , but no math answers will.
I have no idea what 'the neilists' is supposed to refer to.
MoeBlee
Michael Press
<1dc49767-a4ce-4bcd...@d38g2000prn.googlegroups.com>,
david petry <david_lawr...@yahoo.com> wrote:
> On May 10, 7:33�pm, Dan Christensen <Dan_Christen...@sympatico.ca>
> wrote:
> > On May 8, 1:09�am, david <david_lawrence_pe...@yahoo.com> wrote:
> >
> > > On May 7, 10:59�am, Dan Christensen <Dan_Christen...@sympatico.ca>
> > > wrote:
> >
> > > > On May 5, 2:30�am, david petry <david_lawrence_pe...@yahoo.com> wrote:
> >
> > > > > "The actual infinite is not required for the mathematics of the
> > > > > physical world" (Soloman Feferman)
> >
> > > > Are you going to suggest that we can only do number theory, for
> > > > example, for a finite number of elements?
> >
> > > No.
> >
> > Whew!
> >
> > >http://groups.google.com/group/sci.math/msg/600ad9693523ba44?hl=en
> >
> > Here, you said, "But what about the real numbers? The basic idea is
> > that we have to find a way to approximate the set of real numbers by
> > entities that actually exist."
> >
> > I understand that the set of real numbers (i.e. the set of Dedekind
> > cuts) can be constructed from the natural numbers using only the
> > axioms of set theory. Does a Dedekind cut not actually exist in the
> > sense you mean?
>
> A Dedekind cut is an infinite set. Infinite sets have only a
> potential existence.
No, a Dedekind cut consists of exactly two sets.
--
Michael Press
Aatu Koskensilta
> No, a Dedekind cut consists of exactly two sets.
Whether a Dedekind cut consists of two sets or not is largely a matter
of taste -- that is, one can define Dedekind cuts either way, without
it making any mathematical difference.
david petry
> >>> "The notion of a completed infinity doesn't belong in mathematics;
> >>> infinity is merely a figure of speech which helps us talk about limits"
>
> >> That remark is a couple centuries old and is no longer correct.
> You can find some examples of "necessary" uses of infinite ordinals
> in some of my old posts on Goodstein's theorem, Kruskal's tree theorem,
I wonder why you put "necessary" in scare-quotes?
Certainly Goodstein's sequence grows faster than any sequence that can
be defined within PA, but it does seems that there's a gap in your
argument that we can conclude from that the infinite ordinals are
necessary for the proof of Goodstein's theorem, and hence more than
the notion of potential infinity is needed for the proof. For one
thing, I believe that countable ordinals can be handled within a
conceptual framework in which infinity is held to have only a
potential existence. But also, can't Goodstein's theorem be proved
within second order PA, which would suggest that a notion of completed
infinity is not really needed.
Frederick Williams
> >
> > > "The actual infinite is not required for the mathematics of the
> > > physical world" (Soloman Feferman)
> >
>
> He has devoted his career to studying proof theory, where they
> carefully examine what assumptions are needed to prove various
> theorems. He is widely regarded as a leader in the field. If anyone
> "knows", it is him.
When the physical world is understood then (and no sooner) will anyone
know what its mathematics is. Nor should we overlook the possibility
that different mathematics may be applicable to (the same) physical
world.
--
... when we came back, late, from the hyacinth garden,
Your arms full, and your hair wet, I could not
Speak, and my eyes failed...
Martin Musatov
the assembly if the Lord." -Deuteronomy:1
f18<i>Hebrew of a dog</i>--MMM
Martin Musatov
Michael Press
Bill Dubuque <w...@nestle.csail.mit.edu> wrote:
> [1] http://groups.google.com/groups?selm=y8zogx9m4i4.fsf%40nestle.ai.mit.edu
Some of the links are dead.
Goodstein's theorem alone is novel and interesting and readily accessible.
What turns the trick is the rather more difficult theorem of
Kirby and Paris where they prove that Goodstein's theorem
is not provable in Peano Arithmetic. So when raising these
matters it is important to mention Kirby and Paris at the
same time, as does the Wikipedia article on Goodstein's theorem.
--
Michael Press
Aatu Koskensilta
> Aatu Koskensilta <aatu.kos...@uta.fi> writes:
>
>>No one has claimed the falsifiability criterion is a scientific claim
>>or theory, certainly not Popper.
>
> Popper was attempting to distinguish between what has an empirical
> basis and what is religion.
No he wasn't.