Here is a fun similarity between the liar paradox:
- Claim (1) is false
- Claim (2) is true.
- false(false(false(...)))
- true(true(true(...))),
- false(false(false(false(false(false(...))))))
For a long time I've thought the two sentences were closely related, but it didn't occur to me that they're related this closely.
Sorry for subjecting everybody to this set of ruminations on the liar paradox. Blame Mike Almeida. His post on prosblogion on Grim set me on this line of thought. :-)
22 comments:
Alex - really? lim((-1)^n) = lim(1^n) as n goes to infinity? Are you sure about that? Are you sure, in other words, that lim((-1)^n) = 1? Because that is what it means to say that (3) is logically equivalent to (4) - this is just the linguistic equivalent of an infinite product. I haven't taken this particular limit in a few years, but I'm pretty sure I'd remember if it were 1. As I recall, the product diverges, which is exactly what makes this a paradox.
Put differently, you're saying that (3) is infinitely repetitive, which must mean that false(3) = (3) (because adding one to infinity still equals infinity). But false(4) =/= (4), is it? In other words, there's a function you can apply to (4) that doesn't change its truth value, but applying that same function to (3) does change its truth value. So (3) has a property that (4) lacks, which means the two can't be identical.
I see something to what you say, but an infinite expression isn't the same as a limit.
But the limit is a useful analogy here, and what you say does worry me.
On the other hand, there is my argument, which seems good.
But maybe my argument is not good. The argument makes use of the principle that
(*) if p and q are logically equivalent, then one can replace p with q (in a large class of expressioons).
But that principle is not enough to yield the equivalence of (3) and (4). To get the equivalence of (3) and (4), I need to simultaneously make an infinite number of replacements. And maybe this is not allowed?
Certainly not all infinite expressions are good analogies to infinite products, but this one definitely is: it's a series that infinitely applies the same function such that a value toggles between two binary opposites. But if you would prefer that I not take that road, I could instead say that your argument implies that infinity is an even number: using false^n(p) = true(p) iff n is even, correct? But (4) = false^(infinity)(p), so for you to say that (4) = (3) = true(p)* implies that infinity is even. But we know that's not correct - infinity is neither odd nor even.
If you can specify what you mean by "a large class of expressions," I can probably make this even more clear. Your (*) seems very much not sufficient for your argument, but it might be exactly what I used in the second paragraph of my first comment. If false(p) is included in your large class of expressions, then we have:
If (3) and (4) are logically equivalent, then false(3) iff false(4).
Except it's not the case that false(3) iff false(4), apply modus tollens, etc. On the other hand, if false(p) isn't included in the large class of expressions, then how shouldn't you not be able to run your argument at all?
*Obviously, true(p) = true^n(p) for any n >= 1.
(Sorry - drop the "how" from that last question. Typed that out a bit too fast.)
I agree that you can argue that (3) and (4) are not equivalent. So now we have an argument that (3) and (4) are equivalent, and an argument that they are not. That's perfectly fine--since they're both nonsense.
Er...wha? Now I'm really lost. Is it fine because all nonsense is equally nonsense, or...?
I mean, I agree that both are paradoxical, but surely not all paradoxes are qualitatively the same. The liar's paradox is paradoxical, in some sense, because assigning it a truth value is logically impossible - the truth-teller's paradox is paradoxical because assigning it a truth value would be logically arbitrary. It seems to me that this is a real distinction and not one that our language is tricking us into making - do you not agree?
Please feel free to delete this comment after reading it as I do not want to foist this unto your site.
My name is Mariano, I blog on “Atheism is Dead.”
I have been very busy collecting data for an Atheism is Dead project entitled “Answering Atheism” which lists online resources for countering atheism.
The project consists of lists of hyperlinks to:
Websites/blogs that specialize in countering atheism.
Websites/blogs that have partial relevant content.
Particular essays, article, posts.
Audio, video and books.
I have included your website in the partial relevant content category.
If you are so inclined, I would appreciate it very much if you would announce the launch of project Answering Atheism. You can also go to Atheism is Dead and copy an image/button that you can hyperlink and add to your website - please hyperlink it as:
http://atheismisdead.blogspot.com/2008/12/answering-atheism-welcome-introduction.html
Also, please do spread the word amongst your sphere of influence.
Thank you so much for your time, attention, consideration and ministry.
aDios,
Mariano
PS: Stats—thus far the lists consists of 5 pages of video, 5 pages of articles/essays/posts, 10 pages of audio, 8 pages of books, 9 pages of partially relevant content, and one and a half pages of specialized content.
Hi, I'm glad I'm not the only brilliant philosopher-mathematician to have made such a mistake with infinite sums! I made such a mistake with the Grandi series in 1999, quite subtly - subtly enough to turn it into a publishable supertask (my first paper, in 2003) so I wouldn't be surprised if there wasn't something in your FF = T idea too...
Larry:
On reflection, I think my argument fails. Substitution rules cannot be applied infinitarily.
For instance,
(*) 0 = 0 + 0 + 0 + ...
It is tempting to infinitarily apply the rule that a + 0 = a + 1 - 1, thereby transforming the above into:
(**) 0 = 0 + 1 - 1 + 1 - 1 + 1 - 1 + ...
But while (*) is true, (**) is simply false.
Good to see you've come around! Have a nice New Year's Eve, and a safe one.
Apologies if this is a really dumb question, but why is (**) false?
0 = 1-1+1-1...
=> 0 = 1 - (1-1+1-1...)
=> 0 = 1-0 = 1
But 0 =/= 1, so it can't equal that infinite sum either.
But couldn't I respond by saying that if
0 = 1 - (1-1+1-1...)
then since
0 = 1 - (1-1+1-1...) - 0
it follows that
0 = 1 - (1-1+1-1...) - 1 - (1-1+1-1...)
in which case
0 = 0
hence the paradoxical nature of all this stuff? Or doesn't maths work like this?
Apologies. I missed out the brackets: I meant
0 = 1 - (1-1+1-1...) - (1 - (1-1+1-1...))
but you get the general idea I guess
At best, that would establish that 0 = 1 and 0 = 0, so that's not particularly helpful. Infinite sums are not paradoxical, though - in cases like this, they're said simply to diverge (in other words, you can't say 1-1+1-1... is equal to any number in particular). I'm guessing there are some philosophical theories of language that have a similar concept of truth-divergence, but maybe Alex doesn't subscribe to one?
I see. So (**) is false because the left-hand side is well-defined but the right-hand side isn't? Something like that?
That's pretty much correct, yeah.
But as I say, I think that infinite sums can be paradoxical. You see, the '0' and the '1' in them are zero and one. And those are terms with real meaning. They aren't just formal entities. And the infinite conjunction of such terms would express something meaningful if actual infinite sums were possible. So yes, one side has a convergent sequence of partial sums and the other doesn't, but both sides are infinite conjunctions, and zero does equal one minus one.
A non-paradoxical instantiation of the Grandi series occurs within infinite space, if such is possible. The vacuum of the space fluctuates quantum-mechanically, with virtual particle-antiparticle pairs appearing randomly. If a sequence of such pairs appeared, then they might annihilate as follows:
0 = 0 + 0 +... (the vacuum)
= (1 + -1) +... (the virtual pairs)
= 1 + (-1 + 1) +... (annihilation)
=1. That is, creation ex vacuum of ordinary matter. Then it's a short step to something paradoxical perhaps...
Er, so ought we not just conclude then that infinite space isn't possible? Or else that the oddity isn't really due to the Grandi series but rather a probabilistic feature of quantum mechanics (i.e., that the 1s and -1s are really simplified expressions of a more complicated statistical quantum process), or a strained analogy between the physical process of particle/antiparticle matching and the associative property (i.e., that physics doesn't allow the arbitrary pairing of particles in the way that mathematics allows the arbitrary pairing of terms in a sum)?
Hmm... why ought such to be the case, though, rather than some other possibility, e.g. the one I mentioned? (I don't see how one could conclude that space couldn't possibly be infinite, for example, if only because of the other possibilities that you mention:)
The most natural way for me to strengthen my case would, I think, be my exhibiting a variety of more paradoxical scenarios, which varied in their physical presumptions but had the mathematics of infinity in common. (My explicitly non-paradoxical application of the Grandi series was just to indicate, relatively simply, that such appliations were conceivable:)
And aside from the length of such an exhibition, there would still arise the tricky philosophical job of getting others to accept as justified some assertion about what such an exhibition could possibly, and furthermore what it probably does, show. (Still, if you're interested, much that would then be exhibited is easily available online nowadays, although I think its analysis remains obscure:)
By the way, it seems to me that if absurdity follows from the conjunction of quantum mechanics and infinite space, then all one learns is that in a world where space is infinite, quantum mechanics cannot hold. One does not even learn that in our world space is finite, since quantum mechanics, probably, doesn't hold.
Apologies - Alex has it phrased right. I was working under the assumption that quantum physics held, because otherwise the conversation about the Grandi series representing a quantum event doesn't make sense, but technically he has it right.
The point - and this isn't true for just philosophers or mathematicians - is that paradoxes and inconsistency in laws is something that people aim to eliminate. Sure, infinity introduces significant problems into systems of physical equations, but the traditional response (and I think the right one) is to admit that infinities in physics represent things we don't truly understand and need more work, rather than to theorize that we do really understand what's going on and what's going on is paradoxical.
Because, after all, what could it possibly mean to say that you understand a paradox? What would there be left for us to do as thinkers, having accepted a paradox? Etc. and so on - basically, the reason to go from
(A & B) -> paradox
to
~(A & B) (i.e., ~A or ~B)
is because ~A or ~B, no matter what A and B are, at worst demands a paradigm shift. Paradox, on the other hand, essentially demands the abandonment of paradigms altogether.
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