From: Brian Holtz [brian@holtz.org] Sent: Saturday, March 23, 2002 11:12 PM To: alt.atheism.moderated Subject: Re: finite number of sentences "Paul Holbach" wrote: > I have no idea how there could be something real which is not measurable > because itīs quantity transcends all possible rational numbers serving > as units of measurement I don't know precisely how there could be such either, but I don't know that there couldn't be such. > Itīs nonsensical to say that > there is a "possibility beyond all possible possibilities". Those aren't my words, so please don't quote them as such. > > > > > Only if there were an explicit, > > > > > non-self-contradictory definition of "actually infinite" > > > > > > > > Example: space or time is "actually infinite" if and only if > > > > it has no end. > > > > > > Thatīs a non-contradictory but fairly tautological pseudo-definition > > > > Every definition is a tautology -- by definition. > > [..] a definition [is] not identical with "tautology" I didn't say it was. So again, the above is "an explicit, non-self-contradictory definition of 'actually infinite'". QED. > > Yes, and you seem to be asserting that "to have no end" is > > is obviously self-contradictory for existent things. I disagree. > > Now, thatīs a strawman of yours because Iīve said many times that a > strictly potentialistic interpretation of infinity is fully > acceptable. By "end" I don't just mean "procedural stopping point", but also limit, endpoint, or terminus. You assert that to have no such end is obviously self-contradictory for existent things. I disagree. > "actual" means "being really there > altogether, completely, finalized, here and now" Can you define "complete" and "finalized" in a non-question-begging way? Also, I dispute the criterion of "here and now". There is nothing metaphysically privileged about any point in spacetime. > there are people who claim that > time has no beginning and that, therefore, the number of past instants > constitutes an actual infinity. To that I could object that the past > instants are not here right now and that, therefore, they cannot be > an actual whole Again, you simply *define* 'actual' as "here and now" and thus never spatiotemporally infinite. That is hardly a convincing argument that spatiotemporal infinities can never be actual. It is rather argument by assertion/definition. > but then you would certainly blame me again for an > allegedly tendentious stipulation. Yes. So why did you do it? > it doesnīt make sense to say that a past event is actual Yes it does: it's called the "B-series" understanding of time, in contrast to your "A-series" understanding which holds that the present is somehow privileged. It isn't. > a past event has actually happened but now itīs no longer actual > itself (thatīs why itīs called "past") No, it's called the "past" because it is in the set of all events that could potentially be seen as influencing (but not being influenced by) the events you currently call "present". > > But to assume that the only lengths are finite > > lengths is just to assume your premise. > > An infinite geometrical line is elongated but it has no length. It has an infinite length. You again here just assume lengths must be finite. > > If you define a "table" as having two "ends", then an "infinitely > > long table" has an infinite distance between the two "ends", > > and it is an illicit appeal to inapplicable intuition to speak > > of "the carpenter" who can go from one "end" to the other in > > a completable journey. > > I thank you for this argument against the reality of an actual > infinity. No, it's merely an application of the definition of 'infinity', and only argues against carpenters ever being able to complete an infinite journey. > > > He then walks all along the infinitely long table and > > > fixes the other two legs on the other end so > > > that his work is eventually finished. > > > > Again, it's contrary to the definition of an infinitely long table > > to assume that the carpenter can finish a walk from the beginning > > of the table to the end. > > Yes sure, but if actual infinities existed, a table like the one Iīve > described would be a perfectly possible thing. You here agree that an infinite table cannot be walked, but then say a walkable infinite table (i.e. "the one I've described") would be "perfectly possible". That's a blatant contradiction. > Many people think that 100000 is closer to positive infinity than 1, > but thatīs nonsense because the "distance" between any number n and > "infinity" is always infinite itself. And how does this demonstrate that actual infinities are impossible? > > > Iīve been asking for an acceptable definition > > > of "actual infinity" and not of "potential infinity"! > > > > I would define an "actual infinity" as an existent infinity. > > Depending on your definition of "complete", including "complete" > > in the definition of "actual infinity" either begs the > > question (and thus demonstrates nothing) or leaves the > > question open (which is how I contend it is). > > If completeness is not essential for a set, Nothing like completeness is mentioned in the formal definitions of 'set' that I've seen. > then itīs up to you to explain what an open, > incomplete yet actual set is and how to construct it. I assume by "open, incomplete" you mean "not able to have its members listed in a finite number of steps". As for "how to construct it", you probably really mean "how to finish constructing it", to which I would say there is nothing in the definition of 'set' that requires its construction or enumeration to be completable in a finite number of steps. > If you give me a complete and explicit list of all natural numbers If by "complete" you mean having a definite (i.e. finite) number of members or having a last member, then of course this is impossible by the definition of "natural number". > the best way to show that something is possible is to actualize it! If an actual infinity could be shown to already exist, then you wouldn't bother defending your position. Similarly, if an actual infinity could be shown to be logically impossible, then I wouldn't bother defending *my* position. :-) > > that the measuring process cannot end is in fact what qualifies > > the thing as "infinitely heavy". > > But how could we conceive of a real quantitative effect that renders > the accurate measurement of its own cause impossible? Easily: by dismissing the unproven assumption that any quantity can be measured in a finite number of steps. > How can a real "thing" have an infinite quantitative property > such that the effect of that property eludes all scales? One way might be if the effect is not a linear function of the property but rather approaches an asymptote. > > You can certainly start; you just can't end. > > You canīt start measuring what doesnīt exist. "what doesn't exist": yet another restatement of your premise. Do you really think you're going to figure out how to restate your premise in a way that I won't be able to spot? :-) How long are we going to continue this exercise? > > there doesn't seem to be a single > > philosophy reference work that asserts that an actual infinity is > > logically impossible > > Itīs just a footnote and not intended to be a substantial argument. What is "it" here? > Nevertheless, listening to a > scientific authority is not always the worst thing to do! Huh? Don't you mean "not always the best"? Otherwise, why aren't you listening to the authorities' lack of a consensus that actual infinities are impossible? -- brian@holtz.org http://humanknowledge.net