Belette asks about the history of the continuum problem. I'm not an expert, and the subject is huge, but there are a couple of interesting books I recommend. One is Paolo Mancusu's Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century, which covers a lot of the history of the 'indivisibles' question in the seventeenth century and before. The other is Ewald's excellent source book From Kant to Hilbert which covers the period in the nineteenth century when a lot of the advances were made, both in the theory of the continuum and in mathematical logic (although the two subjects overlap considerably at this point).
In the fourteenth century and afterwards the main debate was not so much about whether the continuum could be composed of indivisibles (points), but whether indivisibiles could exist at all. Was the continuum composed of indefinitely divisible lines alone, or a mixture of lines and points? Ockham's discussion of the continuum is here in chapter 45 of part I of the Summa, where he argues against the existence of points, lines etc.
On Cantor's contribution, the idea of transfinite number is often mentioned, but I believe Frege predates him with (The Foundations of Arithmetic). Cantor's main contribution was the idea that the number of the reals was different from the number of the natural number. His argument for this, as I commented here, is unusual and remarkable – possibly the most unusual and remarkable thing in all logic and mathematics - in that nothing appears to predate it.
4 comments:
> the main debate was not so much about whether the continuum could be composed of indivisibles (points), but whether indivisibiles could exist at all.
How odd. Indivisibles - points - ie, when abstracted, numbers - exist if you say they do.
Or had they not worked out whether they were talking about the real world or the abstracted world?
Did they ever define "the continuum"?
>>How odd. Indivisibles - points - ie, when abstracted, numbers - exist if you say they do.
Argh. So
(A) S says that X exists
(B) Therefore X exists
is a valid argument?
>>Or had they not worked out whether they were talking about the real world or the abstracted world?
Like me, they only acknowledged what is real.
>>Did they ever define "the continuum"?
Well they invented the word. "Continuum" is neuter of the adjective 'continuous' which means continuous, unbroken. It means 'that which is continuous'.
> Argh. So... is a valid argument?
Not necessarily. But in maths-world, we're dealing with the objects of our imagination (which need to bear some relationship to the real world in order to be useful, but that is a secondary matter).
If I *define* a point as, err, well, as a number, then it exists as the thing I've defined.
>>Or had they not worked out whether they were talking about the real world or the abstracted world?
> Like me, they only acknowledged what is real.
Ah, that rather re-winds many of the earlier comments - when I abstracted the problem to the real line (different use of the word "real" of course), you made no complaint. *now* you say you (and A) are talking about the "real" world, whatever that might be. And indeed you yourself talked about Cantor and his proofs. Cantor's proofs only apply to maths-world.
>>Did they ever define "the continuum"?
> Well... 'continuous' which means continuous, unbroken. It means 'that which is continuous'.
Circular. Sounds to me that neither you nor A is actually capable of defining "continuum". In fact I'm not sure I'm capable of it either.
Quick test of the meaningfulness of *your* definition of continuous: are the rationals continuous (under your definition)? If not, why not?
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