Here's another argument* against the continuum being composed of indivisibles. An indivisible has a magnitude of zero. Thus adding the magnitudes of indivisibles will always result in a magnitude of zero. For, obviously, zero plus zero is zero. But anything which does have a magnitude, can only be composed of things which have magnitude when added.
Someone objects that this is only true when there are finite additions, or merely countably infinite additions. I don't understand enough of the subject to reply.
*Philosophers always refer to their arguments as 'arguments' and never as 'proofs'. This is because there is nothing in the entire, nearly three thousand year history of philosophy that would count as a proof of anything. Nothing.
17 comments:
Are you sure that an indivisible is of "zero" magnitude? Could it not be of an infinitely small magnitude, one infinitieth of a finite magnitude? Perhaps an infinite number of such infinitely small magnitudes would constitute a finite magnitide.
In cardinal arithmetic c.0 = 0. Certainly. But why is 'adding up the magnitudes' relevant here? 'Adding up' c points one way gets you an interval of length 1. Adding them up another way gets you an interval of length 2. Adding them up yet another way gets you the whole real line. Cantor again. But given the right diagram Euclid would have seen this!
Though he might not gave believed it.
Is addition even a well-defined operator when talking about this?
A few commenters have pointed out possible differences between how the 'real world' is and how the 'mathematical world' is.
Wouldn't the question remain even if the physical world were actually discrete, and space were atomic? E.g. assume that you cannot physically divide space after a certain point, and that a finite addition of such spaces resulted in a spatial magnitude. Then the 'indivisible' space would really have a small magnitude. We could still conceive of this magnitude being divided, even though this were physically impossible.
What's an indivisible? How do we define the magnitude of it?
>> We could still conceive of this magnitude being divided, even though this were physically impossible. <<
Agreed. It is a conceptual issue.
"assume that you cannot physically divide space after a certain point"
What does it mean to not be able to physically divide space? How does one "physically divide space"? What is "space"?
Space is a relationship between entities, not an entity.
>> >> We could still conceive of this magnitude being divided, even though this were physically impossible.
>> Agreed. It is a conceptual issue.
Concepts are abstractions of reality.
Or, concepts are abductions to explain reality but need not "correspond" to it in any way other than as an explanatory function.
Where's an astrophysicist or cosmologist when you need one?
You can't add up infinitely many things unless you've clearly defined what you mean by that.
>>Space is a relationship between entities, not an entity.
Obviously a nominalist.
>>Is addition even a well-defined operator when talking about this?
When adding magnitudes, surely yes? As long as the magnitudes are magnitudes of the same kind of thing.
The continuum is or has a magnitude. And it is composed of continua which have magnitudes which you can add together to get the original magnitude. E.g. a ruler that is 12" long can be broken into a ruler of 7" and another of 5". So you can give meaning to 'composed' and 'added together' in a way that makes sense.
But understood that way it doesn't make sense that the continuum is composed of solely of things which have no magnitude.
>> So you can give meaning to 'composed' and 'added together' in a way that makes sense. <<
Yes. But. And it's a big But. An uncountably big But. The 'adding together' only makes sense for at most countably many elements. The sum of 1, 1/2, 1/4, 1/8, 1/16,... is 2 and it doesn't matter what order we add up in, we always get 2. Any finite or countable set of points has measure (length, magnitude) zero. No surprises here I think. But to get a set of non-zero measure (ie, something that might contain an interval) we need to assemble uncountably many points. And we can get a set of any measure you like from zero to infinity (or indeed a set that can't be assigned a sensible measure at all) depending on how the points are assembled. So there is a deep chasm between finite or countable assemblies of intervals/points and uncountable assemblies of points. The latter is not a limit case of the former. There's no, er, continuity, here.
Readers can try the Wikipedia entry on measure theory but it's rather technical I'm afraid.
>> And we can get a set of any measure you like from zero to infinity (or indeed a set that can't be assigned a sensible measure at all) depending on how the points are assembled.
This is completely beyond my comprehension.
>> >> Space is a relationship between entities, not an entity.
>> Obviously a nominalist.
Is that what it means to be a nominalist? I haven't been able to come up with a good definition.
There is certainly a sense in which relationships exist. For example, the distance between the top of my bookshelf and the bottom of my bookshelf exists, and it is not an entity, but a relationship (I believe it would be called a "trope").
Trope theorists are, it seems, categorized as nominalists, but I'm not sure by what definition of nominalism they qualify as such.
>> This is completely beyond my comprehension. <<
Sorry, Ed, this would only make sense if you know some measure theory. For a picture of what I'm getting at see here.
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