Friday, August 01, 2008

A Platonistic potential infinity?

One would naturally think that, for any time in the future (and similarly, for any point in space, any collection in a hierarchy of collections and so forth), either it is the nth day from now, for some natural number n, or it is infinitely far into the future, so that an infinite future might be divided into the finitely and the infinitely remote, the former region being infinitely many (i.e. aleph-null) days long since otherwise there would be no nth day for some n. But maybe it is not the case that, for any time in the future, it is either the nth day (for some n) or not.
......If it was an nth day, for some n, it would be so objectively; but maybe that property, of being an nth day for some n, is sufficiently like an indefinite property (as a consequence of the endlessness of the natural number sequence, not because of anything fuzzy about units, additions or repetitions, nor because of anything specifically temporal) for it not to follow that an arbitrary future time would either be an nth day (for some n
) or not. (Such a possibility is indicated by how the natural assumption, of aleph-null, leaves us with a proper class of such cardinal numbers.)


Anonymous said...

Can you be more specific on how n could be an indefinite property. I'd buy it if we were working with something larger than aleph-null, but it seems since aleph-null is small enough that every ordinal is a successor and there are no limit ordinals that that would be avoided.

Enigman said...

Being more specific about that is my long-term goal; for now I'm wondering why it is regarded as impossible. So I'm interested in why you think that the smallness of aleph-null indicates impossibility (or implies unlikelihood). I've a sketch of an argument that it rather implies likelihood, if we take a Platonistic view (wanting more than formal consistency), via Cantor's paradox...

If the natural numbers all exist together as a set, in some quasi-spatial way, then we have aleph-null of them. And in that totality we have each subset, so we have all the subsets, so we get beth-one. And similarly beth-two and so on... Our Platonistic view always takes us further, and furthermore we can never stop with any totality of all such cardinals. So the paradoxical problem is, how can we not have the totality, if we have all the elements?

Having such a thing automatically was how aleph-null seemed so obviously number. The problem is that once we have started that process with aleph-null, we never reach a new kind of thing, one sufficiently different to explain why we get proper-classical behaviour with the totality. So it's explanatory to have such behaviour arising at the very start, with the simple endlessness of the natural number sequence. That was where things changed radically, from the finite to the endless (e.g. it was where ordinals and cardinals diverged).

There could be no finite model of how being a natural number could be an indefinite property; but that seems akin to the problem of understanding proper classes Platonistically. To me it seems as obvious as that there are laws of nature (of some form), that Cantor's paradox shows Platonists that simple infinities are potential infinities. Can you say more about why aleph-null being small means that it is very unlikely that simple infinities are potential infinities?

jeff said...

I'm too far away from my philosphical training (and never was much of a philosophy of math guy) to keep up with all these aleph-nulls and Cantor's parodoxes.
Therefore, I hope I'll be forgiven if I muddy the waters. And if I'm so far off base that what I say here has nothing to do with the conversation, they'll be no hard feelings if somebody says "Dude, what you say has nothing to do with the conversation."
Having said that:
I've managed to wrap my brain around the following:
1) We can produce an infinite sequence of numbers.
2) If we chose two whole numbers, say numbers two and three, we could theoretically provide a infinte list of all the decimals that fall between the whole numbers: e.g. 2.1, 2.01, 2.001, 2.0001...
3) All sorts of wierd mathematical ramifications result. A couple examples: Though there are an infinite number of whole numbers the total possible number of decimals is the number of whole numbers times the possible decimal numbers... Therefore, the total possible numbers (not just whole numbers) is something like infinity to the infinity power.

4) Now, all this stuff is quite dizzying enough. Probably, there's some problems in my line of reasoning by treating infinity like a number rather than a concept.
However, I noticed that in addition to some of this dizzying stuff, in your post, Enigman, you were focused on the concept of future days.
I wonder if this doesn't muddy the waters some. It seems like your trying to comment upon mathematical facts but by using the future as a model, there's all sorts of stuff about the nature of space-time, and the question of destiny versus an open future, etc...
So one of my questions is this: would this be easier to wrap our brains around (and still mantain the point your trying to make) if the focus was a number line or ruler or something rather than the future?
Second question: Would we Americans say "Platonic" where you all say "Platonistic" or is there some technical distinction between "Platonic" and "Platonistic"?

Enigman said...

1 is true, 2 is false (I think) and 3 is misleading. The number of decimals is (standardly) 2 to the power of the number of the whole numbers (I think it is the same number, a potential infinity).

Regarding 4, you can think of it spatially or temporally, or in some unimaginably abstract way. I just picked time for the sake of picking something concrete and commonly associated with endlessness.

Platonic is what I mean, but I want to add "more or less (I don't know much Plato)" so I add the "ist" to cover myself (but leave the capital "P" in to indicate that I meant Platonic alright:-)

jeff said...

I don't think that 2 is false and I don't think that 3 is misleading. Perhaps I'm misunderstanding what you say in the end of your reply. It seems to me there are a variety of strategies to demonstrate an infinite number of decimals between two whole numbers.
Perhaps the easiest is just this:
I could follow up a 2 with any number of 0's. The number of 0's could be none. It could be an infinite number. There are therefore an infinite numer of possibilities here. None of these numberals actually signify a diferent quantity, of course, however, if we followed these zeroes with a constant numeral (say, 5) each of these stands for a different amount. (i.e. 2.05 is different from 2.005 and these are different from 2.0005...)
So begin with the fact there are an infinite number of possibilities in this string of decimals ending in 5.
Then, multiply this number by 9. There are nine different numerals we can put at the end of these strings to signify different ammounts. (0 at the end of the string wouldn't signify a different ammount.)
Just by varying the last digit in the sequence, we end up with infinity times nine.
There are an infinite numbers before this, though, and each one could be a 0-9...
It seems like there must be something wrong with my logic, because it seems like we're getting into factorials of infinity or some other such incomprehensible number...

Enigman said...

Yeah, maybe 2 was just misleading (and 3 was fine as far as it went). I'm still not sure about the semantics of infinity...
2) If we chose two whole numbers, say numbers two and three, we could theoretically provide a infinte list of all the decimals that fall between the whole numbers: e.g. 2.1, 2.01, 2.001, 2.0001...
We can certainly have that endless listing, theoretically (i.e. abstracting from physical limitations), and given the axiom of infinity we can theoretically (e.g. within standard mathematics) give the list, with an infinite number of elements.
What I doubt is that if the axiom of infinity is false we should call that a number; and I think that that axiom is false (and that talk of truth - and not just formal consistency - makes sense for mathematics, since numbers are instantiated in the world).

It is certainly weird that 0.999... equals 1.000... but easily seen to be true since 1/9 = 0.111... as can be shown by long division: 9s into 10 go once with 1 left over, making the next 10 to divide by 9.

Jeff said...

That's an interesting distinction-- if I'm understanding you correctly, you're suggesting that the potential to create a number does not imply that the number actually makes it count as a number.

I wonder if there's some sort-of category mistake going on here. What use is it, to ask whether or not a number is real? It might make some sense to say that a numberal isn't real. But a number? I don't know.

I wonder if you'd run over the 1= .999... thing. It seems like if it's true it calls into doubt everything we know about math.
Is it relevant, in your whole argument, to consider the idea that when division problems appear to terminate (i.e. they don't go on forever) there actually is an endless string of zeroes at the end.
Though we say 20/4 = 5, in fact, we could just as easily say 20/4 = 5.000000000...

Enigman said...

My idea is more platonistic than constructivistic. The potential to create a natural number does make it a number, I think, but my suggestion is that there is no number of all of them because their totality is an endless sequence. That is, they are defined altogether by an endlessly reiterated process of adding one, and my suggestion is that that might mean that there is no totality, no "all of them" really.

One might think that if each is actually a number then they all are, but that is to presuppose that there is such a totality of all of them, I think, to presuppose that their definition is that definite (and although each one is definite, they are defined as a totality). A number is real if there could be that many things, I think; so my suggestion is that there cannot be some real things such that there is one of them for each natural number.

1/9 = 0.111... by long division, so multiplying both sides gives us 1 = 0.999... I'd guess that the '1' on the left needs to be followed by something like all those zeros, since we use them in the long division. And if that '1' is not something like a real number (if it can't be divided) then we could only say that nine into one does not go. But I don't see that this is much of a problem:

For me the endless sequences are indefinitely extensible things, so there is not really an equality in 1 = 0.999..., the right-hand side is just tending towards the left-hand side. And on the standard view the right-hand side is an infinite series, 0 + 9/10 + 9/100 + 9/1000 + ..., and such are defined to be their limits (where their limits exist), in this case 1.