|
At 10:15 AM +0200 2/13/08, Avril Styrman wrote:
John, Pat, Chris,
sorry for the delayed response. The roles in this discussion are
clear. I'm claiming that finitism is better than transfinitsim
because it is simpler, uncontroversial, and nothing more is
needed.
1. Its not simpler. It has to explain a host of new difficulties,
not the least of which is how it can account for the hundreds of
apparently obvious refutations of it (such as the elementary proof
that cannot be a largest integer.)
2. It is most certainly not uncontroversial. As this is an
elementary empirical fact, I will not provide arguments for it.
3. Continuous mathematics is certainly useful, I would claim
needed.
Pat and Chris are claiming that finitism
is somehow
not enough, even though they cannot show how, and hold on
to transfinitsim because they have been taught into it.
No, because it seems obvious. In fact I have been interested in
strict finitism for many years and have read everything about it I can
find. I am not impressed by the arguments I have seen so far, all of
which seem very weak and/or obviously faulty.
(By the way, your claim, repeated below, that mathematicians
believe in infinity only because we were taught to as babies and are
too uncritical to think for ourselves, is rather insulting. You might
want to think about other ways to try to convince people.)
John
is a by-stander who does not see the debate very important.
I have responded to all your questions, so please do continue
if you still disagree.
JS:
> But I would claim that the view held by nearly all
mathematicians
> until the latter part of the 19th century is coherent:
Infinity
> is a limit, not something that can be attained as a completed
> mathematical entity.
>
> In other words, one can accept the point that there is no
upper
> bound on the size of any integer, but the only sets that are
> legitimate objects of mathematical investigation are finite.
> That view is quite coherent, and nearly every mathematician
> accepted it as dogma in the first half of the 19th century.
>
> The dominant view about points in those days was the approach
> advocated by Aristotle and Euclid: a point is a
designated
> locus on a line, plane, or volume, not a "part" of the
line,
> plane, or volume. There is no upper bound on the number
of
> points that a mathematician might designate on a line, plane,
> or volume, but it is not permissible to talk about the
totality
> of all the points that one could designate -- because that is
> infinite, and not admissible as an object of mathematical
> investigation.
Yes, and the conception that line consists of points can be seen
as the very source of all transfinitism: transfinitism in needed
in order to make the point-continuum intelligible.
That is not the only reason, although it was historically
important. But note, Cantor's demonstration of the uncountability of
the reals is all about *numbers*, not about points. It nowhere refers
to geometry: it uses only the idea of a decimal expansion. One could
reject the set-theoretic account of the continuum, and still Cantor's
diagonalization is crying out for a refutation from any finitist. I'd
be interested to see a careful refutation of it (or, for that matter,
of the ancient and elementary proof that there is no largest number,
which I gave in an earlier posting.)
...
PH:
> Its reasonably coherent but it breaks down at the edges.
Up till this point, I haven't seen it breaking down in any
way.
I have.
One way to look at the
finitism-transfinitism debate
Wait. This phrase is like 'the creationist/evolutionist debate'.
There is no "finitism-transfinitism debate". Nobody in the
philosophy of mathematics takes finitism seriously as a coherent
foundational philosophy.
PH:
> Ack(Ack(5,5),5) can be
> constructed in a finite number of steps. Numbers
> so large that to write out their decimal
> expansion would take more than the
information
> capacity of the known universe can still be
> constructed in a finite number of steps. This is
> a real problem, even for 19th century
> mathematics. In fact, it was from wrestling with
> problems like this that 20th century mathematics
> emerged.
So, why not take this as the finitist limit: what you can
type down.
Using what notation? Decimal numerals are only one way to refer
to numbers: mathematics provides all kinds of ways. If I can use
"Ack( ..)" then I can refer to much larger numbers than if I
am restricted to decimal numerals.
It will alway be
finite.
If I am allowed to write omega-n, then I can refer to infinite
ordinals using a finite vocabulary. Are you saying that infinity
cannot be referred to? That is obviously false.
Of course, there is no
sense of typing Ack(Ack(5,5),5) because
you cannot
understand its magnity, and you have no you use for it.
What can I not understand about its magnitude? I know it is
larger than Ack(5,5), for example. But in any case, what I can
understand about it and whether it is any use (and how do you know
what uses I might have for very large numbers?) are irrelevant to the
proposition that it EXISTS. It seems obvious to me that it exists, as
does its square; indeed, as does the result of raising it to its own
power (a much bigger number than any that have been discussed
on this thread so far, but still one that has a successor.)
PH:
> Finitism seems much more intuitive when dealing
> with very large numbers than when dealing with
> very small ones, ie their reciprocals. If all our
> series have to be finite and we cannot talk of
> limits, it becomes impossible to give an adequate
> foundation for calculus, for example. On the
> whole, I think that the mathematicians have done
> a fairly good job and we would all be better off
> leaving it to them, and focusing on matters of
> more direct importance to our engineering.
We can very well talk about limits without having to
use anything infinite. Take the series 1/2, 1/4, 1/8, ...
the limit of the series is obviously 0.
You just did talk about something infinite, by your casual use of
the three-dots notation, and saying 'the series'. That series is an
infinite object.
You must
give a certain level of accuracy that is needed
for the problem at hand, like building a house or
assigning the accuracy of a float variable on a
computer program. If your accuracy is 100 digits,
then you only need 100 digits, and the limit of
the series is 0 because within the accuracy of 100
digits there are only zeros 0.000....0
That does not establish that the limit is zero. To establish
that, you have to also show that if your accuracy is 10|3 digits,
then.. and if your accuracy is 10|4 digits, then, ... and so on. And
that 'and so on' has no finite stopping place.
The good part of finitism here is, that
even if one
is a Cantorist, his programs use finite approximations
in any case. This example also shows that finitism
actually drives the thinker into thinking, and not
just burying the thinking into 'limits of infinite
serieses'.
To turn your ad hominism on its head for a moment, I wonder if
you really understand the notion of a limit. Since it is easy to show
that the notion cannot be defined in a strict finitist framework (or,
to be more exact, that any definition of it reduces to triviality
there - just use induction to get to the finite end of the finite
series, and you will attain any approximable limit), this may not be
surprising.
> > I am not interested in the ideology in itself, but only how
the
> > ideology should affect mathematics and all science for that
matter.
CM:
> The "ideology" in question would makes natural science
impossible.
> Notably, strict finitism, even coherently argued, does not
permit
> enough real analysis (which involves not only the infinite, but
the
> uncountably infinite) to do physics.
What does it make impossible?
Real analysis; differential geometry, for example.
Give one single example of something
infinite that the human kind needs in some way?
You can make the distinction to countable and uncountable infinity
only after you have entered into the transfitist realm. If you
have
done that, you implicitly accept transfinitism. If you have not
entered,
it makes no sense to talk about any innumerable infinities.
Transfinitism,
like Islam and Christianity, is the sort of a thing that requires
belief
into it. Without the belief, it has no meaning.
Of course, Löwenheim-Skolem theorem states that every thing that
one
can 'do' with an innumberable model can be done with a numerable
model
too.
> > It is currently a minor ideology, but will be (hopefully
soon) also
> > the general mathematical ideology.
CM:
> It will never be, largely because it is viewed almost
universally
> among actual mathematicians as, at worst, a delusion, and at best
a
> novelty, a curiousity, like developing an internally
consistent
> defense of a flat earth.
I can't see any sense in that analogy. And I believe that in the
end,
every scientist is a finitist, simply because simpler is better
than
complex and useless.
Funny, those are the very reasons that most scientists and
virtually all mathematicians are NOT finitists.
> > It is not just an ideology like some religion. It is evident
that
> > everything paradoxical should be pruned off from logic.
CM:
> Indeed. And if you could show one single paradox in
contemporary
> mathematics you might be on to something. Note this does
not mean
> producing an informal argument involving undefined terms with
unstated
> premises. It means taking the
*axioms* of any branch of mathematics
> -- let's say ZF set theory plus Choice, since pre-ZF set theory
was a
> fairly rich source of genuine paradox -- and *demonstrating*
a
> contradiction, that is, producing from those axioms a deduction
of A
> and not-A, for some proposition A. (Note if you can do
this, you've
> probably got a Fields Medal coming.) Until you can do that,
any
> claims to the effect that classical, infinitary mathematics
is
> paradoxical is on the same logical footing as, say,
Scientology.
Fair enough. What about the Burali-Forti
paradox. All Cantorist sets exist
as 'being', not as generating. So,
For the record, I utterly fail to follow this piece of your argument.
BUt never mind...
there should also exist the greatest
ordinal
OMEGA, the universal closure of ZFC. But the axioms say there are
always
greater and greater ordinals. This is a
case of A and not A.
No, its a case where one has to admit that naive intuitions break
down. That is why this is no longer classified as a paradox. There is
no greatest ordinal (in fact, I don't find this surprising at all: why
would one think there was??) The set of all sets is not itself a set
(or perhaps, does not exist); and many other examples like this are
all handled in the same way. So this is not a paradox in the sense
that Chris was asking for. Again, you might not like this way
of thinking, but it is the way that has been used, and it makes sense
and appears to be internally consistent.
In order to justify his cause, Cantor
turned the paradox into a
definition: it is in the essence of the 'truly' infinite to be such
that
it is the greatest and simultaneously not the greatest. A
Cantorist
can of course invent more and more supplements to logic, such
as the 'class' of all ordinals, but this is only an escape
route
from an evident paradox.
But one an say that about almost any new mathematical insight.
Such rhetoric isn't constructive or useful (and it certainly isnt
doing mathematics.)
I just wait to see in what exact way you
disagree, because it seems as
though you have already decided that you will disagree before even
reading
this. The reason for this is that you believe that transfinitism is
true,
and you want to defend it.
We think it is true because there are so many convincing
arguments for it, and it seems obviously, intuitively, true.
I believe that finitism is a better
view, but
I have also a better justification: there are no abracadabraic things
in
finitism, and it is all that a scientist needs. Your turn to show
something
where finitism is not
enough.
We already have.
> >Cantorists only try to escape
the evident implications of
> >complete induction. Sure, they don't want to have
infitely
> >big naturals, so they just deny them. This is the
doctrine
> >of the Cantorists: just deny the
> >implications of your own theory.
>
> ? What makes you feel that this is an
> implication? It seems simply obvious to me that
> the set of natural numbers contains only finite
> numbers but is itself infinite. I think this has
> been obvious to me since I was a fairly young
> child. There is nothing paradoxical about it. It
> has nothing to do with naming conventions.
This is exactly the case: the Cantorist interpretation
of complete induction has been taught to you as a child,
and that's why it is hard to get rid of it. It is the
same as every child, and most of the adults, holds that
"my country is good and my language is good".
This is starting to get beyond merely insulting, and verging on
the insane. I knew that there were infinitely many numbers (though I
might not have expressed it that way) a long time before I had heard
of Cantor. Im not sure exactly, but I know it was before I was seven
years old, because I had an argument with a friend about it at that
age. His view was that there was a largest number but that only God
knew what it was. I said God could add one to it, and he said that
God could, but wouldn't ever actually do it, so it would always
be the largest.
> > Having a set {1,2,3, ..., n}, its order type is n
CM:
> That is an unordered set. It has no order type.
Of course a set such as {1,2,3} has an order type: 3,
and is also ordered!
No, it isn't. If you want to refer to the ordered set, use
different brackets:
<1,2,3, ... , n>
This means that you don't even
need the axiom of choice to well-order it : )
CM:
> Good grief. It's worse than I thought. We have here
the locus
> classicus of all anti-infinitary crackpottery. It is
actually pretty
> challenging to try to figure out exactly what mental quirk it is
that
> leads some people to deny there are infinite sets of natural
numbers
> on the grounds that, if there were, there would have to be an
infinite
> natural number. It is, truly, just the strangest thing.
I mean, what
> can one say in response? Imagine if someone argued that,
because the
> cries of small children in pain gave him pleasure, it was
morally
> permissible for him to poke small children with a sharp stick.
What
> could you say in response? Anyone proffering the argument
would
> obviously be confused (to say the least) about the concept of
moral
> permissibility. But if you were to try to point out, e.g.,
that
> finding something pleasurable does not entail that it is
morally
> permissible, the reply would simply be that it obviously does.
Trying
> to clarify the relevant moral concepts to him would simply, in
his
> eyes, confirm his belief that it is you who is confused.
I'm afraid
> that those who find the inference from "S is infinite"
to "S has an
> infinite member" compelling are just as confused about
relatively
> simply mathematical concepts, and that both arguing with them
and
> trying to unconfuse them is just as fruitless.
You have learned the convention: omega-0 is the order type of an
inductive set, such as the set of the natural numbers. Along this
convention, you have learned that there are infinitely many
naturals, but these are all finite. There is no objective way
to decide that the convention that you have learned is somehow
better than the below convention:
1. Having a set {1,2,3, ..., n}, the cardinality of the set is n.
2. The cardinality of the set grows as n grows.
3. If the cardinality is infinite, there must be an infinite n
If there is a neverending amount of members, all different, and
the next always greater than the first, it implies that there
are also infinitely big/long members.
NO IT DOESN'T !!! Here, YOU explain to US why you believe this,
which seems so obviously false. Why can't see that the integers are an
obvious counter-example?
This interpretation of
complete induction is totally as
objective as that which you
are holding. Yet another interpretation is that induction
simply stops after a vague finite limit.
AS:
> >I am not interested in the ideology in itself, but only
how
> >the ideology should affect mathematics and all science
for
> >that matter. It is currently a minor ideology, but will
be
> >(hopefully soon) also the general mathematical ideology.
PH:
> This is vanishingly unlikely. It is just easier
> to be a Platonist when doing actual mathematics.
You can do it as easily by being a finitist, but without
paradoxes. You can talk of all natural numbers, meaning
those that make sense. Just forget the 'unnatural'
natural numbers.
You have to explain what you mean by 'make sense' and
'unnatural'. And you have to do this in mathematical
terms.
> >It is not just an ideology like some religion. It is
evident
> >that everything paradoxical should be pruned off from
logic.
PH:
> Of course. But one can work paradox-free without adopting
strict
> finitism.
And this is the point: being paradox-free. Transfinitism is the
very
swamp of contradictions, as I have shown above
It isn't, and you havn't.
, while finitism manages
to do all that needs to be done, without
paradoxes.
Its rife with paradoxes. There's the largest-number paradox. All
finite sets of numbers have a largest member. If the set of all
numbers is finite, then there is a largest number, call it N.
N+1 is larger than N: paradox. Show me how to get past that one, and
I'll give you another.
> >Do you disagree with
Wittgenstein?
>
> Yes. Almost everything he wrote, in fact.
Do you disagree with Aristotle too:
Yes.
> >If the set has infinitely many
members, there should be infinitely big
> >memebers too.
PH:
> No, that is simply a mistake. There are indeed
> arbitrarily large members, and in fact infinitely
> many of them, but there need not be infinitely
> large members.
The conception of arbitrary is severely rotten. It
is supposed to be a number randomly selected from within
the set of all natural
numbers.
No. Nothing about random. I can rephrase the point without using
the a-word: for any given size, there is a number in the set larger
than that size. That is all I meant by 'arbitrary'.
> >Another way to explain this is
that do you have any
> >idea of N? How big is N? If you have no idea about it,
> >and no use for it, why do you postulate N in the first
> >place?
>
> Um.. the point of the proof was to show that
> there cannot be such an N. I postulate it in
> order to show that such a postulation leads to a
> contradiction.
There cannot be the greatest N if N is exactly 100
of any other exact number. But we can talk about
a vague border for the intelligible
natural numbers.
You can talk about it, but I'd like to know what you mean. I have
no idea what a vague border could be. Give us some of this
much-vaunted finitist mathematics. What is a vague boundary? What
properties does it have, what theorems are true of it?
PH:
> No, its not nonsense. I can imagine any number. I
> claim this is true. Prove me wrong. You have to
> show that a number exists such that I cannot ever
> imagine it. Your move.
This only proves that the intepretation of induction is
subjective:
one says it is nonsense, and the other says that it is nonsense to
say that it is nonsense.
But ok, if you can imagine it, then explain somehow the meaning of
Ackerman(Ackerman(5 5) 5), and how you will use it. What is its
application and utility?
I said nothing about application or utility. I said only that I
can imagine it (you havn't refuted this claim) and that it exists
(which I understand you to be denying?)
Your move.
PH:
> It seems pretty easy to write down
>
> A(A(A(5,5),5),5)
>
> or for that matter
>
> 10|(10|(10|100))
>
And again, how will you intend to apply those numbers,
and do you understand them?
I understand them, yes. I don't claim that they are particularly
useful; only that they exist.
... the Cantorist conception is
contradictory
>
> No, its not contradictory. You may dislike it, but it is
consistent.
A neverendig as a totality, unnatural natural numbers, undefinable
selection of arbitrary numbers, Burali-Forti paradox, and you
call it consistent.
Yes, indeed I do.
>
> >Nobody needs it,
>
> Mathematics needs it.
That part of mathematics that the human kind actually
needs e.g. in space flights, hospitals, nuclear plants,
computers, experimental fission facilities, AI, does
certainly not need transfinitism.
They most certainly need real analysis and differential geometry.
If you can show how to re-create these fields (and, say, topology,
catastrophe theory, etc.) within a strictly finitist philosophy of
mathematics, you will at the very least be able to publish many papers
in journals devoted to the philosophy of mathematics.
> the question is, just how big a
number can we understand?
>
> (1) What exactly do you mean by 'understand'? And
> (2) why does this matter? We can certainly refer
> to and reason about and prove properties of very,
> very large numbers: much larger than could
> possibly be physically represented as a numeral.
What use can there be for a number that cannot even
be typed down? If there is no use, why
postulate it?
I DID write it down. And the 'use' is that its a lot easier to
allow it to exist than to deny its existence. It doesn't need to be
'postulated': its existence follows from the ordinary assumptions of
arithmetic. Denying its existence is what would take a great
deal of explaining. See below.
We can define a vague border on how big a number we
can ever apply in practice. We can think that our proofs
are limited to cover only that range. By doing this, we can
give up the unnatural natural numbers, and simultaneously
we give up also the whole hierarchy of transfinities.
Consider N!, where N is an integer. Clearly N! is greater than N,
for any N. Moreover, as N increases, N! increases much faster, so that
the amount by which N! is greater than N increases as N increases.
This means that if there is an upper limit to the set of numbers, even
if it is 'vague', that N! crosses that upper boundary a lot sooner
than N does. So there must be a lot of numbers between the largest N
such that N! vaguely exists and the largest N, all of which exist but
their factorials don't. How can that happen? Now re-run this argument
for any other rapidly-increasing function of N, say 2|N or N|N, or
even Ack(N,N). What happens to these functions past the point where
their value would be too large to be a 'real' integer, yet their
argument is still quite small? Do they somehow "stop
working"? But they are all defined by finitely describable, even
quite compact, algorithms which perform ordinary arithmetic
operations, so what is there to go wrong? Or do they actually produce
results, but these very, very big results are not counted as
"numbers"? Why not? But more to the point, whatever you call
them, if they exist then classical infinitary mathematics seems to
apply to them. Or... what?? You have to say SOMETHING about arguments
like this, other than that you don't like their conclusions.
> >JS:
> >> Many 19th century mathematicians strongly objected
to that way
> >> of talking, and I sympathize with them. But
those mathematicians
> >> would *never* agree to a fixed upper bound on the
integers, such
> >> as 10**120, Ackermann(5 5), or any other finite
integer.
> >
> >The border does not have to be fixed, but vague:
something
> >that can be understood. There is no clear border of the
> >greatest natural number, because the border is vague.
>
> A vague border on a set of integers is a notion
> that cries out for more detailed explication.
Do you think that a it requires any less explanation than the
conception
of arbitrary? A good explanation of it is the following:
take as big/accurate numbers that can in principle find
application.
All these are below Cantorist arbitrary, with probability 1. The
probablility
1 because if you select, even though the selection is impossible,
randomly
a natural numebr between 0 and omega-0, the probability that the
number
is so great that it can be typed within an acceccible part of the
universe,
is 1.
That doesn't make finitist sense. Your arguments seem to
presuppose the denial of your own intended conclusions. By the way,
the mathematical theory you seem to be groping towards is measure
theory, which makes very good sense of how to talk of the
probabilities of selections from infinite sets. I think your point can
be stated as: the set of all 'accurate' numbers must have measure
zero. But that still doesn't account for how this (finite!) set can
have a 'vague' upper limit.
Pat
PS. There is a lot of fascinating work on rapidly-increasing
functions on integers. One can derive results which are close to
paradoxical-sounding, such as the existence of the busy beaver
function which, however, cannot be computed or even have a computable
upper bound. None of this requires infinitary mathematics, but it
poses quite a challenge for a strict finitist perspective. You can
follow this up in Wikipedia starting with 'large number'.
--
---------------------------------------------------------------------
IHMC
(850)434 8903 or (650)494 3973 home
40 South Alcaniz St.
(850)202 4416 office
Pensacola
(850)202 4440 fax
FL 32502
(850)291 0667 cell
http://www.ihmc.us/users/phayes
phayesAT-SIGNihmc.us
http://www.flickr.com/pathayes/collections
_________________________________________________________________
Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/
Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/
Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
Shared Files: http://ontolog.cim3.net/file/
Community Wiki: http://ontolog.cim3.net/wiki/
To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx (01)
|