[Update: public service announcement to all those googling 'et and others': the phrase you're looking for is 'et al'. Have a nice day.]
I haven't had time to blog much of late, and that's about to get even worse, as I'll be away in the U.S. till early April. There's plenty to read elsewhere, of course, as the latest Philosophers' Carnival demonstrates. But I like to keep this site active, so I've lined up a couple of guest contributors for while I'm away. I'll let them introduce themselves when they're ready...
In the meantime, any readers are welcome to use this post as an "open thread", and comment on whatever topic happens to be interesting you at present.
P.S. if you're not about to rush off overseas, do consider writing something for the upcoming Citizens' Symposium on free speech! The deadline is March 24.
Friday, March 16, 2007
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I'll get us started with a question. Following some interesting discussions on Leibniz's cosmological argument, I've been pondering the issue of infinity and the possibility of infinite regress. It seems that there are some interesting philosophical problems both with accepting the existence of actual infinites but also with rejecting them. So how about it? Do or can physical infinites exists?
ReplyDeleteThat was interesting. I had just written about how with relativized quantification instead of the standard ones (here) one can dispense with infinite sets altogether. (This is just based on Mycielski's work [1, 2).
ReplyDeleteI am inclined to say that infinities are useful tools and can be asumed in order to solve problems, rather like all the other simplifying assumptions we make - but in reality if you look very close you will never find an actual example of an infinity except questons that the universe effectively classifies as nonsense like "how many 0's are there in 100".
ReplyDeleteGNZ
There is a view, as I have read in some perspectives on quantum gravity, that if space and time were fundamentally quantal, then there would be no physical infinities (e.g. infinitely divisible time or space or zero-point "singularities"). So much for Zeno's worry.
ReplyDeleteit appears we all agree - we need harder questions!
ReplyDeleteIt's amazing that we all agree, because when this argument came up in a discussion on Leibniz's cosmological argument on my blog, there was strong disagreement with the concept of a finite universe.
ReplyDeleteThere is a difference between discreteness and finiteness. Even if the universe had some fundamental quantum-gravitational discreteness (a very contentious topic) such that in any finite region of spacetime there were finitely many discrete elements, it could still be spatially infinite (and the total number of elements would be then aleph_0). Present-day cosmology models the universe as spatially infinite, and I see no philosophical reason against this.
ReplyDelete> Present-day cosmology models the universe as spatially infinite
ReplyDeleteit does? I understood that it considered space to NOT be infinite. (then again maybe it MODELS it as being infinite, while knowing it isn't - which might be reasonable in most context)
It is possible to argue that an infinite universe would look similar to our universe using the old 'observable universe' issue but does the orthodoxy really believe that?
Philosophically, why would space be the one plce the universe decided to throw in an infinity? And why would it prop up what appears to me to be a inelegant solution compared to the goold old big bang is everything aproach.
I'm also inclined to think even if you take multiverse view then even THAT is not infinite - just a very large number.
GNZ
I'm no physicist (are any of the commentators?), but I see no reason to rule out physical infinities a priori. It at least seems possible that space is infinitely large, divisible, or has existed for an infinite amount of time. (Hell, if there are additional dimensions, those could be infinite too)
ReplyDeleteGNZ,
You ask why the universe would only throw in infinities with respect to space. I tend to think that this is false, as I state above, and further, it's no surprise that we don't often perceive infinity in numerous places, since it's not even clear that we could comprehend infinity if we came across it.
Infinities hypothesis is kind of like the god hypothesis except without pascal's wager.
ReplyDeleteIt is a "cool concept" at first glance but; it is unessercary to explain anything, is a complex and "ugly" way to describe the world (ie it doesn't give us simple answers like the big bang sort of hypothesis where everything colapses nicely) and wherever you look you fail to see them (afterall there are NOT infinite grains of sand, people don't live forever and neither do atoms).
It seems to me that just as people used to say nature abhors a vacuum it appears to abhor a contradiction and an infinity.
having said that - if you define infinity as somthing that I can't comprehend I can't rule it out I guess, but then I don't even know what it is that I am not ruling out.
GNZ
It is curious that we are so fond of modal notions, especially if we as Alex avers says that we can have no comprehension of infinity, which against his above claim seems to reason against physical infinity a priori. This is somewhat funny given the phrase "a priori" and then thinking of how Kant layed out our perception and sensibility of space and time, and then delimiting knowledge of things transcendental.
ReplyDeleteThis comment has been removed by the author.
ReplyDeleteAlejandro, I do not understand why you are insisting that present day cosmology models the universe as infinite. There seems to be many other possible shapes given: spheres hyperbolic parabaloids, etc. Present day cosmogony also on the whole acknowledges the big bang which would necessity a spatially finite universe.
ReplyDeleteIf there is only space-time and space is finite because of the big bang then only infinite time could exist (even though time is sort of an allusion in the concept of space-time). But the problem of infinite regress is very difficult for the concept of infinite time.
I have given some philosophical reasons against an infinite universe here and here.
Alex, infinite divisibility is different than the existence of an infinite universe or time.
Furthermore, the argument that we cannot comprehend infinity if we come across it makes the concept itself meaningless. You're setting up something that cannot be argued against (or argued for, for that matter).
Perhaps I should clarify the "cannot comprehend infinity" comment. GNZ implied (perhaps unintentionally) that if there were infinities in the universe, we'd be bumping into them all over the place. My point was simply that our perception doesn't work at a scale where we could see things that are infinitely small or large.
ReplyDeleteSo let's imagine that my keyboard divides into molecules, which are composed of atoms, which are composed of quarks (simplifying!), which are composed of X, which are composed of Y, which are composed of Z, and so on. Let's imagine that this process has no "bottom"; that things are infinitely divisible. This seems a priori possible, and would mean that we were bumping into infinities in our everyday lives. But it would not be the case that we'd be able to point to them, because it's not something our perception is able to pick up.
In short, my point was the negative one that it's not a good argument to suggest that we'd find more infinities about if there were any. I obviously caused some confusion: I didn't mean that we can't even hold a concept of infinity.
A couple more specific points:
Bryan,
Could you give some kind of argument to support the claim that the kind of infinite divisiblity I raise above is different from space which is infinite, as it were, in the other direction? (i.e. in total size)
GNZ,
Talk of the "infinities hypothesis" is most misleading. For that to be true, I would have to be the one asserting the existence of something that others reject. Now this matter is complex, but there's at least something to be said in favour of the claim that your position is the one making the positive claim. To claim that everything is finite is to claim that there are always boundries to everything. What does this hypothesis explain?
> Talk of the "infinity’s hypothesis" is most misleading.
ReplyDeleteReally it just points out that they may no exist (i.e. that there is at least one competing theory). I felt since infinity seems such an attractive hypothesis it needs a little taken from it.
> To claim that everything is finite is to claim that there are always boundaries to everything. What does this hypothesis explain?
Can't I just claim there are no infinities?
I don't mind boundaries being undefined (like the position of a photon or something) just as long as they are not infinite.
------
Anyway my point is that a lot of people have sought infinites and THOUGHT that they observed infinities, and those things have largely proven to not be infinities. That would by my experience and my interpretation of science's experience. A sort of induction from previous debates.
> our perception doesn't work at a scale where we could see things that are infinitely small or large.
Which does seem a bit like the sort of defense offered for other things that would usually meet with Occam’s razor.
I acknowledge that some infinities cannot be absolutely disproven (but posible to prove) - for example "there is an omnipotent god" is one as is "space is infinitely divisible" and one can retreat to those sorts of infinity aside from the occam's razor or induction osrt of arguments.
Aside from that I'll throw out my other backstop. I am inclined to think that, for example - I can't disprove that you can divide any unit of space in two, but I might be able to disprove that "space can be infinitely divided" and that that in itself eliminates infinity from the equation. I.e. the point being that we (the whole universe) need not concern ourselves with infinities that exist in contexts that can't occur.
GNZ
I reiterate, since I obviously wasn't clear:
ReplyDeleteOccam's razor cannot be applied uncontroversially here, since it's not clear that I'm adding another entity to my ontology. To claim that infinities may exist is to assert that I've yet to see evidence that everything is bounded. This position is the result of Occam's razor type considerations; not a position to which it can be applied.
Im not sure what the general rules are for where people think occams razor is applicable so I haven't actually proposed using it (well at least not in the last post), what I mean is that it is pretty much a perfect defence to say "you haven't proven there are no non-black crows" as they say.
ReplyDeleteThis is similar to the sort of defense that gets used to defend things that are in danger of occams razor (even if your example is not). I guess its a bit of debate based induction.
> To claim that infinities may exist
I dont reject they MAY exist - that wuld be a hell of a strong claim rather like rejecting god could exist or rejecting that there is a crow born with a flashing coca cola sign on his side.
But I can say that given the evidence it is likely those things don't exist. Or 'it would be reasonable to assume they don't.
Ill take 51%:49% if I have to.
....reasonable in most situations (of course that changes if the cost benefits are hugely out of alignment)
ReplyDeleteLast night I was wondering about 'infinity' after reading all the above. It occuured to me that i have carelessly used the word infinity in a lot of ( not public) writing that ponders the nature of the universe.
ReplyDeleteUpon reflection i was not able to actually conceive of infinity.
In my writings i had been in fact thinking of "immeasurably large beyond comprehension." But not infinite.
On the other hand there is one dimension of experience that one can imagine as infinite, and in fact I do think of it that way quite instinctively.
I mean the dimension of time.
this 'instinct' says that there was always time, and there 'always' will be.
One hears the notion that with a big bang event there ( may ) have begun time/event sequences as we know them, but none the less it appears the experience we have as mind/brain creatures seems to allow or maybe force the sense of time being infinite. Never ending.
I note also that for example when i attempt to think of infinite space, i imagine some objecct or point of view hurtling at maximum speed across a void of space for ever .
Without the sense of time being able to go on and on and on,and on, and on .... infinite space has no meaning.
come to think of it, do spatial dimensions have a meaning for us at all if we do not allow time?
just a thought.
david l.
I am not sure if that is a fundimental concept or if it is like how some people prefer to see thing, others to tough things, and others to read about them (ie learning styles).
ReplyDeleteMaybe some find space to be naturally infinite and find it hard to understand time as infinite... Or maybe not.
GNZ