John, Pat, Chris, (01)
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. 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. John
is a by-stander who does not see the debate very important. (02)
I have responded to all your questions, so please do continue
if you still disagree. (03)
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. (04)
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. (05)
JS:
> CM> ... which of course means that there are infinitely
> > many finite integers, and hence that there is a set that
> > contains them...
>
> Wait! A 19th century mathematician would agree with the first
> point, but restate it without using the phrases "infinitely many"
> or "set of integers". As examples of "correct" 19th century
> mathematical English, one could say:
>
> 1. There is no largest integer.
>
> 2. Any nonempty set of integers has a largest integer. (Note that
> this is true because all early-19th-century sets were finite.)
>
> 3. There is no largest set of integers. (Follows from #1 and #2.)
>
> CM> ... and hence that there is a set that contains them, hence
> > a power set of that set, and off we go down the Cantorian bunny
> > trail! You may not like where that leads, but it is very hard
> > to argue that there is a nonarbitrary point at which you can stop
> > that line of reasoning.
>
> No, there is a natural stopping point: only those sets that can be
> constructed in a finite number of steps. (06)
JS:
> I'm not saying that I'm advocating the 19th c. position, but it is
> quite coherent, and I can sympathize with people who feel uneasy
> about infinite sets. There is no reason why they must accept them
> in their ontology if they don't want to.
>
> However, even Euclid would accept statements #1, #2, and #3 above.
> It's important to note that Euclid does not imply Cantor. (07)
PH:
> Its reasonably coherent but it breaks down at the edges. (08)
Up till this point, I haven't seen it breaking down in any way.
One way to look at the finitism-transfinitism debate is ''some
people think like this and some like that and let them just think
how they do for the rest of their lifes''. I believe that
consensus will be achieved. Every scientist is a finitist by
nature. Cantorism is only the cream cake that they have taken
because it was given to them when they were young. (09)
JS:
> >, and I can sympathize with people who feel uneasy
> >about infinite sets. There is no reason why they must accept them
> >in their ontology if they don't want to. (010)
And there is no real reason for anyone to accept them, other
than that people tend to hold on to what has been taught to them. (011)
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. (012)
So, why not take this as the finitist limit: what you can
type down. It will alway be finite. 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. (013)
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. (014)
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 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 (015)
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'. (016)
> > I am not interested in the ideology in itself, but only how the
> > ideology should affect mathematics and all science for that matter. (017)
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. (018)
What does it make impossible? Give one single example of something
infinite that the human kind needs in some way? (019)
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. (020)
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. (021)
> > It is currently a minor ideology, but will be (hopefully soon) also
> > the general mathematical ideology. (022)
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. (023)
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. (024)
> > It is not just an ideology like some religion. It is evident that
> > everything paradoxical should be pruned off from logic. (025)
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. (026)
Fair enough. What about the Burali-Forti paradox. All Cantorist sets exist
as 'being', not as generating. So, 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. (027)
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. (028)
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. 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. (029)
> > 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).
> > -Ludwig Wittgenstein: Philosophical Remarks XII.129
> >
> > Do you disagree with Wittgenstein? I don't consider all his stuff
> > good, but his critique against transfinitism is enjoyable. (030)
PH:
> Except W. never actually provides any critique in the sense of a
> coherent, logically precise argument. Just a rambling series of
> pictures, metaphors, and loose assertions like the one above. This is
> the inevitable currency of the mathematical skeptic. (031)
A large part of Wittgenstein's work is metaphoric, and I would also
prefer that he had written sharp, clear, and systematic totalities.
But he would have had to do so much work that there was no time in
his life to do this. His finitist remarks are quite clear and
understandable. You should really see LW as a proponent of good
instead of bad. A finitist does not try to cause troubles, but
only to purify mathematics of transfinitism. (032)
By the way, Aristotle and LW are on the same finitist lines.
Even Russell got convinced by wittgenstein that his project was
no good. Russell was too wise to hold on to crazy ideas, even
thought they were precious to him. Russell abandoned them, similarly
as you should abandon transfinitisim. But no, you just hold on to it,
even though you don't even need it. I can't understand why, and
that's why I keep asking. (033)
> >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. (034)
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". (035)
> > Having a set {1,2,3, ..., n}, its order type is n
CM:
> That is an unordered set. It has no order type. (036)
Of course a set such as {1,2,3} has an order type: 3,
and is also ordered! This means that you don't even
need the axiom of choice to well-order it : ) (037)
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. (038)
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: (039)
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 (040)
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. 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. (041)
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. (042)
PH:
> This is vanishingly unlikely. It is just easier
> to be a Platonist when doing actual mathematics. (043)
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. (044)
> >It is not just an ideology like some religion. It is evident
> >that everything paradoxical should be pruned off from logic. (045)
PH:
> Of course. But one can work paradox-free without adopting strict
> finitism. (046)
And this is the point: being paradox-free. Transfinitism is the very
swamp of contradictions, as I have shown above, while finitism manages
to do all that needs to be done, without paradoxes. (047)
> >PH:
> >> BTW, you make
> >> one logical error in an earlier post. An infinite
> >> series of finite things can grow indefinitely
> >> without any one of them actually becoming
> >> infinite. This is true even with a strict
> >> finitist understanding of "infinite".
> >
> >What you call a 'logical error' is that I do not accept
> >Cantor's subjective interpretation of induction.
>
> No, the error (repeated below) is to assume that
> an infinite set must contain an infinite object.
> The natural numbers are an obvious counterexample. (048)
As explained above, this is a question of subjective
interpretation, something that is built to supplement
logic. How big natural numbers there are then, if not
infinitely big? Arbitrary? (049)
> >Do you disagree with Wittgenstein?
>
> Yes. Almost everything he wrote, in fact. (050)
Do you disagree with Aristotle too: (051)
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.
-Aristotle, Physics, book 3, chapter 7. (052)
> >If the set has infinitely many members, there should be infinitely big
> >memebers too. (053)
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. (054)
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''. (055)
Consider that you select a number randomly from within
the set {1,2,3,4,5,6,7,8,9,10}. The average is 5. When
the set grows, the average number grows. When the set is
infinite, also the average should be infinite. Arbitrary
is only invented in order for a mathematician to be able
to talk about random numbers. A Finitist mathematician
can talk about arbitrary number, but arbitrary means
that the mathematician has to give intelligible, even
though vague, borders from where to select from. (056)
> >Also appealing to intuitonism does not help. The free generation has a
> >problem: if the series is generating, it has to have some speed of
> >generation.
>
> What kind of series are you talking about? (057)
Both division 1.5, 1.25, 1.125, ..., and expansion 1,2,3,... (058)
> >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. (059)
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. (060)
> >CM:
> >> The view isn't even coherent. If Ackerman(5 5) exists, why not
> >> Ackerman(Ackerman(5 5) 5) -- a massively larger number? And of
> course
> >> if *that* number exists, well, you get the idea.
> >
> >Yes, I get the idea, and that is the classical rebuttal of
> >finitism, but it has flaws in it. I omitted the following
> >from Jean Paul Van Bendegem: Why the Largest Number Imaginable
> >is Still a Finite Number.
>
> Of course it is: ALL natural numbers are finite.
> But what this argument tells me is that the
> notion of 'the largest imaginable number' is just
> as meaningless as 'the largest number', and
> indeed for the same reason. (061)
For me it teaches that there is no use and no sense
in complete induction committed on natural numbers,
because this makes them 'unnatural'. (062)
Suppose that Imagines(x, n) is an abbreviation for ”person x is capable
of imagining the numeral n”, then it is claimed that both (V = forall): (063)
(1) Imagines(x, 1)
(2) Vn(Imagines(x, n) -> Imagines(x, n + 1)) (064)
are extremely plausible. After all, (2) is nothing but a reformulation
of the idea that the next numeral can always be imagined. But, given (1)
and (2), the conclusion (065)
(3) Vn(Imagines(x, n)) (066)
follows immediately by mathematical (complete) induction and that is
nonsense. (067)
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. (068)
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. (069)
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? Your move. (070)
PH:
> It seems pretty easy to write down
>
> A(A(A(5,5),5),5)
>
> or for that matter
>
> 10|(10|(10|100))
> (071)
And again, how will you intend to apply those numbers,
and do you understand them? (072)
> >PH:
> >> Its not enough to just announce as an obvious doctrine
> >> that infinity is wrong; still less to seem to link conventional
> > > mathematics to some kind of dark political conspiracy.
> Mathematicians
> >> tend to be Platonists because they are driven to it by following
> >> chains of thought which seem to be inevitable and conclusive. If you
> >> want to announce an alternative, you have to tell us where the less
> >> travelled paths branch off the mathematical highway.
> >
> >Mathematicians want to say ''for all natural numbers''. But
> >all that a human needs
>
> For what purpose? Humans use Mathematica
> thousands of times every day to solve real
> problems in real engineering, and all of this
> would have to be abandoned if we were to take
> strict finitism seriously. Hilbert wasn't joking
> when he referred to 'Cantor's paradise'. (073)
As a finitist, you don't have to abandon anything, except useless
figments of imaginations such as the unnatural natural numbers,
that cannot be applied, that is, only with the probability 1. (074)
> >is the 'natural' part of the natural
> >numbers. Therefore, the 'for all' in ''for all natural numbers''
> >should be interpreted as e.g. ''for all those naturals that
> >can be typed within the known universe''. And why is this
> >important and better? It is better than having the Cantorist
> >conception, because the Cantorist conception is contradictory
>
> No, its not contradictory. You may dislike it, but it is consistent. (075)
A neverendig as a totality, unnatural natural numbers, undefinable
selection of arbitrary numbers, Burali-Forti paradox, and you
call it consistent. The same as a plagued person is said to be
healthy, because there are no contradictions is his dreams to
be healthy. Finitism suffers from none of these. Obviously you mean
that the axioms of ZFC do not contradict each other, but the
Burali-Forti paradox shows that the general way of interpreting
complete induction is in contradiction with the axioms of ZFC.
Of course, the axioms of Alice in the Wonderland are also
consistent, but this does not make the Wonderland intelligible. (076)
> >and leads to very obscure things.
>
> A subjective judgement. It provides a unique
> foundation for all normal mathematics which has
> not been surpassed, and overcomes a host of
> difficulties that late 19th-century mathematics
> was wrestling with (convergences of infinite
> series, accounting for the irrationals, etc.)
>
> >Nobody needs it,
>
> Mathematics needs it. (077)
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. Come on, in practice
the axioms of ZFC only say ''you can do whatever you want''.
They serve better as the measure of just how crazy things
are needed in order to make the idea of point-continuum work. (078)
Did anyone realize that point-continuum is probably the
worst idea in the history. Or at least in the top ten.
Anti-foundationalism is a long-standing alternative,
and will take the upper hand eventually because it
is an economical choice. (079)
> 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. (080)
What use can there be for a number that cannot even
be typed down? If there is no use, why postulate it? (081)
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. (082)
> >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. (083)
Do you think that a it requires any less explanation than the conception
of arbitrary? A good explanation of it is the following: (084)
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. (085)
JS:
> My recommendation to Avril would be to avoid trying to stop any
> mathematicians from following their own inclinations. A more
> useful approach would be to show how some approaches that have
> been overlooked have important applications. (086)
I don't disagree that some approaches that have been overlooked have
important applications, but I also think that it is highly useful for all
scientists to understand 1) the problems of transfinity, 2) that we
do no need it, and that 3) it can be seen as an implication of the
point-continuum. (087)
Also, it is almost impossible to change people's beliefs after they
have held them for decades. But if the general natural inclination is
the Cantorist one, we must at least try to prevent it being taught to
young people. (088)
Avril (089)
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