Bill, (01)
potential infinity is Aristotle's idea, or I'm not aware of
any earlier theories that are similar. As is generally known,
Aristotle’s writings leave a lot of space for interpretation,
but the idea of potential infinity is quite clear. (02)
The problem is, that we cannot have a never ending totality,
but we don't want to be constrained into some fixed finite
limit either, such as 10 in the power of million.
Aristotle's solution is an arbiter between a fixed finite
limit and transfinity. (03)
But on the other hand to suppose that the infinite
does not exist in any way leads obviously to many
impossible consequences: there will be a beginning and
end of time, a magnitude will not be divisible into
magnitudes, number [e.g the number of years in the
past] will not be infinite. If, then, in view of the
above considerations, neither alternative seems
possible, an arbiter must be called in; and clearly
there is a sense in which the infinite exists and
another in which it does not. We must keep in mind
that the word ’is’ means either what potentially
potentially is or what fully is. Further, a thing is
infinite either by addition or by division. Now, as we
have seen, magnitude is not actually infinite. But by
division it is infinite. (There is no difficulty in
refuting the theory of indivisible lines) The
alternative then remains that the infinite has
potential existence. But the phrase ’potential
existence’ is ambiguous. When we speak of the
potential existence of a statue we mean that there
will be an actual statue. It is not so with infinite.
There will not be an actual infinite. (04)
... (05)
For generally the [potential] infinite has this mode
of existence: one thing is always being taken after
another, and each thing that is taken is always finite,
but always different. Physics, book 3, chapter 6. (06)
But the infinite does not exist potentially in the
sense that it will ever actually have separate existence;
it exists potentially only for knowledge. For the fact
that the process of dividing never comes to and end
ensures that this activity exists potentially, but not
that the infinite exists separately.
Metaphysics, book 9, chapter 6. (07)
Everything that is potential will one day be actual for Aristotle,
except for potential infinity. He seemed to have the picture that
matter is potentially infinitely divisibe: a rock could be divided
without a limit. Many of his ideas are out of date today, such as
the conception that the Universe circulates Earth, but nobody's
perfect: we have to apply those of his ideas that are still valid
and applicable. This is anyhow is the case in science in general. (08)
Simply, we can always take more and more. After 1 we can take 2,
after 2 we can take 3, and so forth, but the act of taking
limits the series to be always finite. In other words, if a
series exists, we have to construct it in a finite number of
steps. We can have as big numbers as we can type, but they are
always finite. This is why potential infinity can in one sense
be reduced into finitism. (09)
When transfinities are axiomatically postulated, a game can be
played with them, similarly as chess can be played, even when
it is postulated that the Queen is transfinite. (010)
Our account does not rob mathematicians of their
science, by disproving the actual existence of the
infinite in the direction of increase, in the sense
of the untraversable. In point of fact they do not
need the infinite and do not use it.
Physics, book 3, chapter 7. (011)
To rely only on mere thinking is absurd, for then
the excess or defect is not in the thing but in the
thought. One might think that one of us is bigger than
he is and magnify him ad infimum. But it does not
follow that he is bigger than the we are, just because
some one thinks he is, but only because he is the size
he is. The thought is an accident.
Physics [2] book 3, chapter 8. (012)
cheers, (013)
Avril (014)
Lainaus Bill Andersen <andersen@xxxxxxxxxxxxxxxxx>: (015)
> Hi Avril
>
> You used the phrase "potentially infinite" several times in your
> discussion below. I'm not sure I understand. Could you say more about
> what you mean by "potentially" so I can make sense of what a finitist
> could mean by "potentially infinite"?
>
> Cheers!
>
> Bill Andersen
> Ontology Works, Inc.
> 3600 O'Donnell Street, Suite 600
> Baltimore, MD 21224
> +1.410.675.1204 (w)
> +1.410.675.1201 (f)
> +1.443.858.6444 (m)
>
>
> On Mar 6, 2008, at 4:34 AM, Avril Styrman <Avril.Styrman@xxxxxxxxxxx>
> wrote:
>
> > Lainaus Christopher Menzel <cmenzel@xxxxxxxx>:
> >
> >>> "Having any set such as {1,2,3,...} that starts with number 1, and
> >>> has only successors of 1 as members, one after another, then, if the
> >>> set has a cardinality x, then x is also a member of the set"
> >>>
> >>> This axiom totally as objective as the interpretation of complete
> >>> induction in ZFC.
> >>
> >> Your "axiom" is incoherent twaddle and will remain so until you
> >> provide rigorous axioms or definitions for "number", "successor",
> >> "set", "member", and, especially, "cardinality" that are as rigorous
> >> as those found in ZFC. The only reason you are able to keep talking
> >> is that you refuse actually to cash your claims as mathematics. Your
> >> stock and trade is vagueness and ambiguity. The minute you try to
> >> turn it into real mathematics your "axiom" will vanish like a puff of
> >> smoke -- not that I expect you to try.
> >
> > I do not need to define things such as 'number' and 'successor',
> > because they are the most self-evident things in the world.
> > Instead, I can use them to define other things. For 'set' and
> > 'member', I can use the same axiom of extensionality that is in
> > ZFC -there is no transfinitism in extensionality. For cardinality
> > and rank, I can use constructive definitions, such as those in
> > Finitist set theory: www.cs.helsinki.fi/u/astyrman/FST.pdf
> >
> >
> >>> All the rest of the transf. hierarchy is built on omega-0. Having
> >>> omega-0 includes the very controversy of having a neverending as a
> >>> totality.
> >>
> >> There is no controversy among real mathematicians. The actual
> >> infinite is at the heart of nearly all contemporary mathematics,
> >> including in particular the real analysis that underlies physics.
> >> The
> >> existence of the transfinite is a simple consequence of the axioms of
> >> ZF set theory.
> >
> > The term "the natural numbers" can be used very well without
> > having to commit to anything transfinite, by maintaining that
> > the series is potentially infinite. It is in the heart of
> > mathematics of course. I'm sure that also you understand the
> > twist in having a never ending as a totality.
> >
> > * * * * * * * * * * * * * * * * * * * *
> > If you can show there are numbers bigger than the infinite, your head
> > whirls. [124] p.16.
> >
> > I have always said you can't speak of all numbers, because
> > there's no such
> > thing as 'all numbers'. But that's only the expression of a
> > feeling.
> > Strictly, one should say, . . . "In arithmetic we never are talking
> > about
> > all numbers, and if someone nevertheless does speak in that way,
> > then he so
> > to speak invents something - nonsensical - to supplement the
> > arithmetical
> > facts." (Anything invented as a supplement to logic must of course
> > be
> > nonsense). [122] XII.129, [121] XII.448.
> >
> > . . . A searchlight sends out light into infinite space and so
> > illuminates
> > everything in its direction, but you can't say it illuminates infini
> > ty.
> > [122] XII.142, [121] XII.490.
> >
> > The infinite number series is only the infinite possibility of
> > finite series
> > of numbers. It is senseless to speak of the whole infinite number
> > series, as
> > if it, too, were an extension. [122] XII.144, [121] XII.504.
> >
> > If I were to say "If we were acquainted with an infinite extension,
> > then it
> > would be all right to talk of an actual infinite", that would really
> > be like
> > saying, "If there were a sense of abracadabra then it would be all r
> > ight to
> > talk about abracadabraic sense perception". [122] XII.144, [121] XII
> > .511.
> >
> > But why is it easier to imagine life without end than an endless
> > series in
> > space? Somehow, it's because we simply take the endless life as nev
> > er
> > complete, whereas the infinite series in space ought, we feel,
> > already to
> > exist as a whole. [122] XII.145, [121] XII.515.
> >
> > Let's imagine a man whose life goes back for an infinite time and wh
> > o says
> > to us: 'I'm just writing down the last digit of p, and it's a
> > 2. Every day
> > of his life he has written down a digit, without ever having begun;
> > he has
> > just finished. This seems utter nonsense, and a reductio ad absurdum
> > of the
> > concept of an infinite totality. [122] XII.145, [121] XII.516.
> >
> > . . . what is infinite about endlessness is only the endlessness
> > itself.
> > [122] XII.145, [121] XII.519.
> > * * * * * * * * * * * * * * * * * * * *
> >
> >> Although it cannot be proved mathematically, due to
> >> Gödel's theorem, over a century of rigorous testing and use suggests
> >> there is every reason to believe these axioms are consistent and only
> >> cranks like you argue there are "problems" without the least
> >> mathematical evidence or competence. If you think there are problems
> >> with ZF -- despite the fact that you cannot prove its inconsistency
> >> --
> >> then the only rational mathematical response is to provide a
> >> theoretical alternative so there is actually something to discuss.
> >> But you obviously lack the ability to do this.
> >
> > The problems are evident, but you just deny them, similarly as a
> > priest in the year 1600 would deny alternative gods. This is very
> > natural for a human being: "My language is good, my country
> > is good, my theory is good".
> >
> > Why do you need an alternative for something that is useless? I'm not
> > aiming to give an alternative, but I'm only writing a thesis about
> > the problems of transfinity. The problems are not in the coherence
> > of the axioms, but the problems are in what the axioms say. Similarly,
> > Alice in the Wonderland is totally coherent, but it is only a story.
> >
> > Again, tell me one thing where transfinitism is really used, other
> > than
> > in turning contradiction into contraction, proving that Cantor's set
> > exists, and so forth. The proofs that require transfinitism are not
> > really required in practice. They are not required in space flights,
> > cosmology, physics, chemistry, computer science, you name it.
> >
> >
> >>> I have clearly argued that potential infinity is totally enough for
> >>> the needs of the man kind.
> >>
> >> You have done no such thing. Your talk of "potential infinity" is
> >> useless to science and a distraction to this forum until you actually
> >> provide a mathematical theory that realizes the idea and demonstrate
> >> its adequacy for natural science and the theory of computation.
> >
> > For the theory of computation, Turing machine is an implementation
> > of potential infinity. The memory tape of a TM is potentially
> > infinite. It is not actually infinite/transfinite. The potential
> > infinity is totally enough and very useful. Show me one actual
> > problem or application where it is not enough, and I'll turn into
> > a transfinitist.
> >
> >
> > Avril
> >
> >
> >
> >
> >
> > [124] Cora Diamond (editor): Wittgenstein's Lectures on the Foundati
> > ons of
> > Mathematics, Cambridge, 1939.
> >
> > [122] Ludwig Wittgenstein: Philosophical Remarks. Edited by Rhus
> > Rhees and
> > translated into English by Raymond Hargreaves and Roger White. Basil
> > Blackwell, 1975. First German edition 1964.
> >
> > [121] LudwigWittgenstein: Filosofisia Huomautuksia (Philosophical
> > Remarks).
> > Translated by Heikki Nyman. Werner Söderström, 1983.
> >
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