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At 8:54 PM +0200 2/21/08, Avril Styrman wrote:
Pat and Chris, here's some answers and
qestions.
Quoting Pat Hayes <phayes@xxxxxxx>:
> >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.
If you are not satisfied with strict finitism, then take
Aristotle's potential infinity: you can take always more
and more, as much as is needed. However, because we are
only weak humans, we can only ever take a finite number
of anything, even though the finite border is vague.
This is why potential infinity can be reduced into
finitism: no matter how much we take, we always take
finitely many.
Of course we do, being (as you say) finite beings ourselves. But
our finiteness does not prevent us thinking about things larger
than ourselves, so please continue your thought, above. We always take
finitely many... of what? Or if you prefer the active form, we always
take finitely many from what? What is this bucket, or supply,
that we take these integers from? Is this not an infinite set? (Or do
you also not believe in sets?)
> 3. Continuous mathematics is certainly useful, I would claim
needed.
No need for transfinitism to be continuous. Transfinitism is
needed
only for being point-continuous. Trying to make the idea of point-
continuum intelligible does require transfinitism.
But even if one denies that the continuum is constructed
from points - a topic which gave rise to heated debate among
mathematicians as little as 150 years ago - it is hard to see how one
can rationally deny that it is possible to make indefinitely fine
discriminations of location on the continuum. So if a continuum exists
at all, then a point continuum exists alongside it, as it were.
I do not see why,
because mapping the point-continuum into nature only makes
movement
impossible. This was Zeno's reasoning
Zeno's reasoning was simply faulty, and Aristotle's response to
it was inadequate.
> 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.)
As you might know, many people think that the diagonal
argument proves nothing.
I have never seen a remotely convincing counter-argument to it,
however. I'd be delighted to see yours, if you have one.
> >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.)
I can in no way deny that it exists, once you have
typed it down.
But you DID deny it exists, in an earlier post. What then is your
actual position? If there is an upper bound, please tell us roughly
where it is, and I will undertake to immediately reply by naming a
number which is much larger than it.
It is a good criterion for
the
existence of numbers. But, there are
other criteria
too, such as the use of
numbers.
But we are here only discussing existence.
Numbers such as
Ack(Ack(5,5),5) might as well be considered solely
as numerals, or as character strings. Numbers like
1,2,3,4, are useful. For what do you use numbers
such as Ack(Ack(5,5),5)?
I enjoy contemplating very large numbers. What I do in
private is none of your business. The point at issue is, do they
exist? I take it that you have been denying that they do.
All the numbers that have a potential
applicability,
are within the borders of potential infinity, that can
in one sense be reduced into finitism. Why is potential
infinity not enough for you?
I have no idea what it means. What distinguishes a
potential infinity from infinity?
Does not a never ending totality bother
you?
No. It seems kind of obvious. Imagine a perspective view of
parallel lines. Its even in the Lord's prayer: "...for ever and
ever, amen."
How can it be a totality, if it never
ends?
How can it be completed, if it never ends?
What exactly do you mean by 'completed'? I am of course a finite
being, but I can think of an infinity as a whole.
> >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.
I interpret the three dots not as "ad infimum", but as
"ad enough".
You referred to "the series" - your words, above. Now,
what was IT that you were referring to, that thing called "the
series"? Your answer must be ONE thing.
The interpretation of the three dots
is
also one theme of Wittgenstein's critique of
transfinitism, and I recommend that you'd reconsider
your attitude towards it. Did you know that e.g.
Alan Turing participated into Wittgensten's lectures
on the foundations of mathematics? His attitute in
the lectures has some resemblance with your attitude.
Independently from this conversation, I have a very low opinion
of most of Wittgenstein's philosophy. I hesitate even to call it
philosophy: it smacks more of a collection of jottings. The
much-vaunted 'picture theory of meaning' in the Tractatus is clearly
absurd. I rank Wittgenstein somewhere near the writings of Lawrence of
Arabia. As for mathematics, there isn't the slightest evidence that
Wittgenstein understood anything about it at all.
> >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.
I can very well maintain that the limit is 0 in the
domain where the needed accuracy is 100
digits.
You did not previously refer to limits in a domain, you simply
said the limit of the series. Is this notion well-defined, in your
view, or must it depend upon the 'domain'? If so, we need a
mathematical theory of these domains.
Think
of normal transactions in a shop. If 5 cents would be
the smallest coin, then an item with price 5,551$ would
be rounded up into 5,55$, if you pay in
cash. No need
for transfinity here, and no need in anywhere else
either. The more accuracy you need, the more you get.
That is in the area of potential infinity.
I understand all that, of course. But now show me how to actually
provide a foundation for all this talk of 'potential'. One gets
similar difficulties when trying to formalize vague predicates such as
'large' or 'fairly small'. These ideas have never been satisfactorily,
rigorously, accounted for. If you feel you know how to do it, by all
means explain it to us, I will be all ears.
> >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.
The analysis of real numbers is similarly the story that tells
what they are. Taking a finitist view, we only need finitely
long rationals. Irrationals may be handled as symbols
that
denote a process that is executed with a
needed accuracy
How then do you distinguish between rationals and irrationals? Or
would you prefer that we had no such distinction?
> >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...
To exists as "being a completed totality" is very different
to
existing as "Generating".
No, to exist is to exist. There aren't different 'kinds' of
existence. And this has nothing to do with finite or infinite: its
simply what one means by 'exists'. I have no idea what 'to exist as
generating' can mean (except maybe that a generating process
exists, and can always be run to a next stage: but even to say that
last bit requires me to consider infinite models of my
axioms...)
Generation is a process, while being
is
not a process. For example, the Cantorist definition of
rationals
is the set that contains all a/b,
where a and b are naturals.
There is no generation there. Everything in transfinitism exists
solely as 'being'.
Everything described by any extant formalism exists solely as
'being'. To be IS to exist.
You just cannot overlook the Burali-Forti
paradox and say it is
nonsense.
I don't say that; but I do say that it is not a paradox.
It is one of a number of illustrations of the fact, now
well-understood, that mathematical formalisms can only capture
mathematical intuitions expressed within the system, rather than
outside the system. Thus the central question for a foundational
theory (such as ZFC) is, can all of mathematics be conducted within
this system? Which is a pragmatic question as much as a philosophical
one.
It is about the most fundamental feature
of classical
analysis, ZFC incorporated, that the law of the excluded middle
holds. Every theorem, that has any sense in it, is either true
or false. Every set in ZFC either exists or it does not exists.
The set of the natural numbers exists as being a completed
totality. So does its power set, and so on.
OK so far.
And this is the very
paradox. Everything in ZFC should exists as a completed totality,
but the axioms say that there are always a greater and greater
sets.
OK...
This is the paradox
?? ...what paradox? Why did you think there would be a
largest ordinal? You of all people should be content with this result,
I would have thought. ZFC proves that there is no largest ordinal.
This might be thought surprising, but it is not paradoxical. It might
be viewed as a critique of ZFC, indeed it was so viewed at one time;
but it is still not paradoxical.
, and it does not help to call it a
theorem that there is no greatest set/ordinal.
Its not a question of what you call it: it IS a theorem of ZFC.
That is simply a fact.
Chris argued about the same thing here:
> > What about the Burali-Forti paradox. All
Cantorist
> > sets exist as 'being', not as generating.
>
> This is not mathematics. There is no such thing in
mathematics as
> "Cantorist sets". Here's what we've got:
For any purported
> collection of things, ZFC either (i)
proves that it exists (as a set),
> (ii) proves that it doesn't exist, or (iii) does not decide the
matter
> either way. Those are the only facts pertinent to set
existence.
> Your talk of "existence as being" and "existence
as generating" is
> muddle-headed philosophical codswallop.
The idea of generation is from Brouwer's intuitionism, which
must be seen as a middle option between finitism and
transfinitism.
There, e.g. the real numbers are freely generating. This of course
faces the problem of the speed of the generation
Im afraid I agree with Chris that 'existence as generating' is
codswallop, Brouwer notwithstanding (and Brouwer's ideas have
never been made precise and rigorous: Heyting's formalization of
intuitionist reasoning was never accepted by Brouwer, although it is
now considered definitive.) But this talk of 'speed of generation' is
certainly not from Brouwer, and to me only emphasizes the coddish
nature of the wallop. Even if it made sense, what does it matter how
'fast' the integers 'grow'?
> Ignorant rubbish. It makes no
sense to talk about what *should* exist
> in ZFC, only what does or doesn't. It is in fact a *simple*
theorem
> of ZFC that there is no largest ordinal. Period. That
is, we have
> case (ii) above. The Burali-Forti paradox simply never gets
off the
> ground in ZFC. That was the *point* of *axiomatizing* set
theory in
> the first place. What possesses you even to talk about ZFC
when you
> clearly have never studied it and don't understand even its
most
> elementary theorems?
As I already answered Pat, it does not make it any better to say
"Hey, it is not a bug, it is a feature". The law of the
excluded
middle together with the ZFC axioms give birth to the BF
paradox.
No, they DONT. Im sorry, you are just flat wrong here. There is
no B-F paradox in ZFC.
There is nothing more special in it, than
what Cantor said: it
is in the nature of the truly infinite that it is not exhausted,
and thereby Cantor turned a paradox into a definition. Also this
is not logic, because all that your counter argument proves, is
that it is totally a subjective option whether the BF is or is
not a paradox. And this proves also that the whole (subjective)
framework should be abandoned.
It proves nothing at all.
> > But the axioms say there are always greater and greater
ordinals.
>
> Well, it's a theorem, but yes.
>
> > This is a case of A and not A.
>
> It is exactly not such a case. It is a case of not-A.
Period.
You give a too friendly interpretaion of ZFC. It is in the nature
of ZFC that all that exists, exists as a completed
totality.
You are trying to force a philosophical view onto a formal
theory, and complaining that it does not fit. Chris is saying that if
it doesn't fit, forget the philosophy and stick to the formalism. That
after all is what the formalism was designed for in the first
place.
Assume
that the hierarchy does not exists as a completed totality:
you must assume that because there is no greatest ordinal. There
is no greatest ordinal, but there simultaneously must always be
greater and greater ordinals. This immediately brings up two more
questions.
1) Can you see the analogy with hierarchy of transfinity and
the natural numbers?
Yes, of course. But it is a superficial analogy, and mathematical
foundations are not made of analogies.
2) if the hierarchy is not complete, and
similarly there is no
greatest set, then there must be some sort of a generation
going on.
Why? All I need say is that the elements all exist but their
totality does not (or if it does, is not of the same kind that they
are, eg is a 'proper class'). Things do not "go on" in
mathematics. There need be no 'generator' : indeed, the very notion
hardly seems meaningful in the very-large-ordinal domain.
How fast does the hierarchy generate?
This is a question you
must answer, if you maintain that the series is not
complete.
No, it is a question I can, and emphatically do, reject as
meaningless.
I could also say that it is as insulting from you to
say things that you have said about Wittgenstein, as
it is insulting to a muslim to say to say similar
things about Mohammed.
As I have long suspected, having attended several of their
meetings, it seems that Wittgensteinian scholarship is a form of
religious fundamentalism.
> >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?
It is not very important to me just how it is: just how the
complete
induction should be interpreted, because I argue the there is no
real need to have complete induction at all. My strategy is to
show that the interpretation of complete induction is
_subjective_.
Because it is subjective, it cannot have anything to do with
logic, that should not be subjective at all.
Anything invented as a
supplement to logic
must of course be
nonsense
-Wittgenstein,
Philosophical Remarks XII.129.
I explained another subjective interpretaion of complete
induction to Chris below.
You have not answered my challenge. Convice me that what you say
above is true: that "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."
It seems to me that the infinite set of numerals {1,2,3,...} is
an obvious counterexample. This is a never-ending collection of
members, each bigger than the one before it, and each of them is
obviously finite. What mistake have I made?
> >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.
By saying that arbitrary numbers are unnatural, I mean
that they are practically very close to transifinite
numbers.
You have to explain it mathematically. What does
'practically close' mean?
> >, 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.
Take the Aristotelian view,
OK, go ahead. Tell me the Aristotelian flaw in the above
reasoning.
> > > >Do you disagree
with Wittgenstein?
> >>
> >> Yes. Almost everything he wrote, in fact.
> >
> >Do you disagree with Aristotle
too:
>
> Yes.
Come on Pat, you can't be serious.
Of course Im serious.
These guys
are about the most insightful in the
history.
Aristotle was very good for his time, but that was a very long
time ago. Im afraid I think Wittgenstein was an over-rated, borderline
schizophrenic ignoramus.
> >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'.
So, once you have specified some size n, there is always
a larger size.
Right.
If you can specify n, it is not
arbitrary.
If you specify n+1, it is not arbitrary either. If you say
that for all natural numbers x(n > x), then n cannot be
a natural number... Do you mean that arbitrary is
something
that we weak humans cannot type
down?
The word 'arbitrary' is not a descriptive adjective: it has
adverbial force. "Pick an arbitrary number" means
"Arbitrarily pick a number". No actual number is
'arbitrary', any more than any particular person is 'average' or
'typical'
> >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?
You first take the Aristotelian potential infinity.
I don't know what that is either. What is it?
But, because we humans always can only
ever take
only finitely many things, one after another, the
potential infinity can in this sense be reduced
into finitism. How much can we ever take, that is
the vague finite border.
This holds for it: type a number down in decimal
notation.
Why decimal? Why not allow floating-point notation? Silicon chips
understand it, after all. Why not allow Conway's arrow-chain
notation?
It is either below the border, or it is
the border.
Of course, the border depends on the version of finitism.
Because Ack(Ack(Ack(5,5),5),5) cannot be understood
properly
What do you mean by this? I claim I do understand it. How
will you refute me?
I do not see why it should be qualified
it as a natural
number. What does 'natural' mean for
you?
In the context 'natural number' it means a nonzero integer.
> >> 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?)
You can imagine a textual string Ackerman(Ackerman(5 5) 5),
but you can imagine njnbaöbnjaöjdöbfk as well.
Im not imagining the text string: I'm reading the text string and
imagining the number. I do this, if you wish to know the details, by
using a special number which is the current best guess of the ratio
between the diameter of the known universe and the Plank length - so I
have a vision of a 'packed-full universe' - and then imagining a
super-universe which has one of those in each cell, and iterating from
there.
So, what is
the use of postulating Ackerman(Ackerman(5 5) 5)? Application
and utility are central argumants for finitism: postulate
only
those numbers that have a possibility of
application.
But we can never know what this will be. And in any case, Im
against any proposal to reduce mathematics to currently applied
mathematics. The history of science suggests that this would be a very
bad strategy indeed.
As said before, finitism gives a
different answer to
''what are real numbers?'' than Dedekind-cuts.
I imagine it would have to, yes. So, what answer DOES it
give??
So, can you by now settle with the
potential infinity?
Nope, as I have no idea what it is. And also because it seems to
me that I have a reasonably robust intuition of actual infinity, and I
can work happily with it, and have no reason to change my ways.
> >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.
So, you consider above the process of generation,
like, having 3, we have also 3!=6, and having 6,
we have 6!, and so on. And this is nothing more
than 1,2,3,...
You don't seem to have understood my point. It *is* more than
1,2,3, because it grows faster at an ever-increasing rate, so it meets
the boundary sooner. Of course this is meaningless for a infinitarist
like me, but it should worry a finitist like you. What does Factorial
mean when N is a finite integer but Factorial(N) is too big to be
considered finite?
I have tried to answer in the best way I
can, and I can't see any other
reason for your transfinitism than education. And you have given
no
other reason.
My reason is that transfinitism all seems fairly obviously true,
and the arguments against finitism seem absolutely conclusive and have
never been responded to. You havn't responded to them.
Unlike Wittgenstein, I don't see the ZFC
axioms as a joke,
but I see them as a measure of what is required in order to
make discrete into continuous, as a measure of how crazy
constructions do we need in order to make the point
continuum 'intelligible'.
Which of the ZFC axioms do you find so crazy? They all seem quite
obvious to me, except possibly Choice, which we all now know to be
take-it-or-leave-it, and the axiom of Foundation, which we now know to
be optional.
> > 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
>
> Unbelievable. Ok, tell you what. In ZFC, proposition
3 is provably
> false. So you obviously don't
accept ZFC.
If you hold on to ZFC, then you perhaps
can prove that 3. is wrong, but
I do not hold on to ZFC, and therefore 3. makes sense to
me.
Given as choice between ZFC and all of 20th-century mathematics,
on the one hand, and proposition 3. above, on the other, I
unhesitatingly choose the former. I would likely do that even if 3.
seemed intuitively true: but as it seems obviously, in-your-face,
blatantly false, the choice really is a no-brainer.
> > The conception of arbitrary is
severely rotten. It is supposed to be
> > a number randomly selected from within the set of all
natural
> > numbers. First of all, the selection process itself is
impossible.
> > How would you do it? Arbitrary, like the whole
transfinitisim, is
> > given an existence simply by saying ''hey, it exists''.
>
> Rubbish. To say a set of numbers contains arbitrarily large
members
> is completely clear and precise. It means simply that there
is no
> largest member; for any natural number in the set, there is a
larger
> one in the set. That's it. It has nothing to do with
"random
> selection" or any sort of "selection process".
There is no largest member? Isn't that the same as the definition
of an inductive set? And that is nothing else that the definition
of complete induction. So you say, if there is an inductive set,
such as {1,2,3,...}, then there are arbitrary numbers in the
set.
See above. All numbers are equally arbitrary: it is not a word
which indicates a classification.
I have clearly argued that potential infinity is totally enough
for the needs of the man kind,
Actually you havn't argued this at all. You have asserted it
several times, but you have given no *argument*. And you havn't said
what it means.
Pat
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