In this blog, Alex helpfully explores Cantor's Thesis (CT): 'the potential infinite entails an actual infinite'. As I did before, I'll correlate my sections topically with how Alex divided his blog topically.
1. Introduction - Alex follows Morriston in thinking there's a symmetry between an infinite temporal regress and an infinite temporal progress. Indeed, Morriston argues that if the future is potentially infinite (in the sense that you'll never 'arrive at a time at which you've reached infinitely many praises [this is his Gabriel/Uriel thought experiment]), it still remains that the number of events that 'will occur' is actually infinite (I will note some semantics distinctions later below). Alex notes Craig's notion that a temporal 'progress' of events is potentially infinite only (in line with Craig being an A-theorist regarding the nature of time). Alex will now motivate a version of CT, consider the possibility that a beginningless past is a potential infinite, and consider why Craig's symmetry breaker (tense) is inadequate. This last thesis should bear on me and Alex's conversation and should be helpful.
2. Potential infinite implies actual infinite - Alex helpfully points out that if CT does have universal applicability, then Craig's distinction between potential and actual infinities won't do the work he wants.
3. A potential infinite that's not an actual infinite - I agree with everything here. A potentially infinite number of set hierarchies can't be an actual infinite without Russel-type paradoxes involves with, say, naive set theory.
4. A potential infinity that is also an actual infinity - This section was interesting. Alex considers the natural numbers. The activity of counting all the natural numbers is potentially infinite; the set of all natural numbers is actually infinite. One gets the picture of walking down an infinitely long corridor. My walking is potentially infinite; the corridor is actually infinite. The set of all natural numbers is like the corridor (this is what I was intimating in my water/riverbed analogy in the talk); the walking is like the counting. You'll never finish counting just as you'll never finish walking. The counting/walking amounts to Cantor's 'variable magnitude'; the set/corridor denominates the 'range'. The range's cardinality never changes; the magnitude changes with each count/pace. But here I run into an ambiguity with the way Alex applies this notion to Craig.
As pointed out in Part 1 of the series, there are at least three different senses according to which Craig could espouse actual infinities as they pertain to the natural numbers: espousal of an actually infinite number of discrete quanta of reality (Platonism), avowal of an actually infinite number of discrete propositional quanta (an actually infinite number of future-tense propositions), and a denial of the first option and a denial of the second option (in terms of a potentially infinite number of propositions being a byproduct of human intellection in an analysis of the extent of God's knowledge vs. a solitary proposition functioning as a blind truth ascription of a description the content of which is non-propositionally known, being a mode of God's knowledge contingent, not on the nature of God's omniscience, but on God's abilities - see Part 1 of the series for details).
5. Conclusion - Alex asks us to consider the following conditional (call it C): if x is a potential infinity, then x is an actual infinity. I must confess that Alex's example with natural numbers isn't persuasive to me. There doesn't seem to be parity. Agreed that the series of natural numbers is actually infinite; agreed that the counting is - and always will be - potentially infinite. But this example doesn't motivate C. In order to motivate C, your 'x' has to be univocal. In this case, 'x' isn't univocal. In one sense, it's the activity of counting the natural numbers; in the second sense, it's the natural numbers themselves. In this case, C turns into something like C* and C**:
C*: if x is a potential infinity, then x will always be a potential infinity.
and
C**: if y is an actual infinity, then y is an actual infinity
which is trivially true and doesn't motivate an example CT.
In these cases, x = the activity of counting the numbers, and y = the series of natural numbers.
A true example of CT (it seems to me) will involve a conditional whose variable is univocal from antecedent to consequent. In the case of C, that doesn't seem to be the case: the act of counting the natural numbers isn't the same as the series of natural numbers itself - and it surely doesn't seem to me to follow that given the antecedent, the consequent follows. It seems to me that the consequent could go on serenely being the case even if no one began the activity of counting. The problem is that Alex bases his entire case against Craig's potential/actual distinction (in his response to Morriston's Gabriel/Uriel thought experiment) on the fact that C is a genuine example of the universal applicability of CT. But without C, it seems to me that Craig can marshal the said distinction without worry. The burden seems to be on Alex to marshal a genuine example of CT and that until he does, Craig's distinction provides us with sufficient reason for thinking that the future is endless in the sense that the number of future events that will be is potentially infinite.
Thus, Alex's question (which potential infinite is the future?) is nipped in the bud.
Alex notes Aristotle's position that time involves a beginning, middle, and endpoint, this involves the idea that time can't have a first moment (if I'm understanding Alex here), since the first moment will have a beginning, middle, and end, setting off an infinite regress of beginnings subdividing each and ever forever. Alex doesn't press the point, but, suffice it to say, Craig is aware and has addressed this idea in his Tensed Theory of Time: A Critical Examination (2000).
Nevertheless, Alex mentions a comparison between (1) the totality of numbers not having a highest number, and (2) the totality of time not having a final time. But again. This comparison, while imaginatively suggestive, begs the question against Craig's distinction. The question is: is the extent from the present to the future like the series of natural numbers? If the future is potentially infinite, then it's not like the series of natural numbers. It'll be like the counting of the natural numbers. And as we saw above, the only reason Alex gave us for thinking that the potential infinity of counting could imply an actual infinity is that there's an actual infinite number of natural numbers. But I don't see how that follows. And what I need to follow is that a potential infinity of counting implies an actual infinity of counting, something I see no reason to believe so far. Of course, Craig will have no issue with the locution 'the totality of time'; it's just that the totality will quantify over a growing edge (the progressing present) forever progressing into the future a potentially infinite number of times. In this sense, the totality of time is always followed by another time, and this in a way that's not analogous to the 'set example'.
So far, it seems that Alex hasn't provided any reason to reject Craig's distinction.
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Edit: 2/25/19
Alex has graciously corrected some of the points I made above.
1. CT and the natural numbers - Alex argues that CT is true for the natural numbers. I responded by pointing out what I thought was a disanalogy between counting and the natural numbers, and the temporal progress of events and the actually infinite series of future events. Alex responded by pointing out an alleged symmetry between the following pairs of claims:
*Pair A
1. The number that have been counted.
2. The numbers that will be counted.
*Pair B
3. The times that have become actual.
4. The times that will become actual.
Let's fit these four claims into the schema of the CT-condition:
C: If x is a potential infinite, then x is an actual infinite.
*Pair A = If (the numbers that have been counted) is a potential infinite, then (the numbers that will be counted) is an actual infinite.
*Pair B = If (the times that have become actual) is a potential infinite, then (the times that will become actual) is an actual infinite.
I still see a change in the variable as we transition from the antecedent to the consequent. Thus,
In *Pair A,
x = The number that have been counted.
and
y = The numbers that will be counted.
and
In *Pair B,
x = The times that have become actual.
and
y = The times that will become actual.
Thus, it's not that the potential infinity of 'x' implies the actual infinity of 'x'; it's that the potential infinity of 'x' implies the actual infinity of 'y' (in both cases). The next question is: is the 'x' - in both cases - just a potential infinity? In *Pair A, it's not clear as to whether we're beginning at 0 (demarcating the series of natural numbers and progressing, one number per second, say; in that case, the series isn't beginningless) or whether we've already been progressing from a beginningless series of 'integers' rather than merely natural numbers. Perhaps Alex is saying that he's just trying to locate an instance of CT and the series of natural numbers (having its mathematical beginning at 0) is such an instance. Fine. I still don't discern the 'necessity' of the consequent. It seems to me that it could still be the case that 'x' could forever be potentially infinite and 'y' be potentially infinite as well (finitism). Or, it could be that one could render the mathematical notion of an actual infinite coherent (a set some of whose proper subsets could have the same cardinality as the whole, say) without a commitment to an actually infinite number of mathematical truths or an actually infinite number of mathematical objects. Hence, 'x' might imply a kind of 'y' that has nothing to do with the kind of 'y' needed for the absurdities implied by its metaphysical instantiation.
Alex mentions Michael Dummett's Indefinite Extensionality to motivate the claim that the 'iterative conception' of time implies time's actual infinity, or, as Alex puts it, time's being 'totalisable'. Agreed that the totality of the natural numbers isn't a natural number (it's a transfinite number). Agreed that the totality of time is not a time (even though I question whether time being a totality implied the impossibility of adding time's to such a totality, which seems to be implied by an A-theory of time). I disagree that the totality of time (due to the aforesaid question) implied that the addition of more time to the ever-changing totality was impossible. Alex might object that I'm equivocating on 'totality'. On the one hand, I seem to be equating the totality to the ever-changing march of the temporal becoming of the 'present'. On the other hand, there may be a sense of 'totality' according to which such a 'totality' is the 'y' implied by the 'x' in *Pair B. That is, the totality of all that time that 'will be'. I would agree with this move if I had the motivation to disqualify my analysis of 'will be' in terms of a potential infinity that doesn't imply an actual infinity. That is, I think there is a marked distinction (motivated above and in the prior blog post) between a potentially infinite number of events that will be and an actually infinite number of events that are yet to be. This distinction seems to me to introduce a change to *Pair A and *Pair B:
*Pair C:
2. The numbers that will be counted.
5. The numbers that are yet to be counted.
and
*Pair D:
4. The times that will become actual.
6. The times that are yet to be actual.
This changes the conditionals a bit.
*Pair C: If 2 is a potential infinite, then 5 is an actual infinite.
*Pair D: If 4 is a potential infinite, then 6 is an actual infinite.
Due to reasons for the mathematical legitimacy of the actual infinite, I'm prepared to admit *Pair C (though perhaps not for the reasons, and without the implications, Alex might have/draw). The only reason I'm given to accept *Pair D is Dummett's Indefinite Extensionality (IE). First, the application of IE depends on the success of analogically mapping *Pair C onto *Pair D. I'm convinced that there is not an actually infinite number of times that are yet to be actual and that an appeal to *Pair C to motivate that there are is unpersuasive. There isn't an actually infinite number of future-tense propositions that map onto every event/number in terms of the events that 'will be' and the numbers that 'will be' counted. The series of natural numbers that 'will be counted' is not akin to an existent corridor that my counting implies; the series of events that 'will be' is not akin to an existent riverbed that time's progress implies. I'll get to why I don't think so in a second. Let's see what Dummett has to say about IE (from Dummett, M. 1963/1978. The Philosophical Significance of Gödel’s Theorem. In Dummett (1978), Truth and Other Enigmas. Harvard University Press. Dummett, M. 1996. What is Mathematics about? In Dummett, The Seas of Language. Oxford University Press):
’concept is indefinitely extensible if, for any definite characterisation of it, there is a natural extension of this characterisation, which yields a more inclusive concept; this extension will be made according to some general principle for generating such extensions, and, typically, the extended characterisation will be formulated by reference to the previous, unextended, characterisation’ (195-196).Hence, for the definite concept of 'number', there is a natural extension of the concept to the more inclusive concept 'infinity' by appeal to the general principle that, for any number you pick, there is a rule (n + 1, for example), according to which the extension is generated. The general principle appealed to to justify the extension is itself an 'unextended characterization' (since there isn't another general principle according to which that characterization is generated). Craig would, no doubt, agree with IE and affirm that 'infinity', per IE, has a meaning on that score. His affirmation would be in line with what he has said about the concept's mathematical legitimacy. But this will do nothing to avert the idea that it's metaphysically impossible for there to be an actually infinite number of 'countings' or 'future progressings'. The concept's mathematical legitimacy does nothing to undermine the metaphysical point. Counting and progressing are metaphysical verbs. Alex will retort that when IE is applied to all the countings/progressings that 'will' metaphysically unfurl, IE implies the metaphysical possibility of an actual infinite. But I deny that IE can be applied to future metaphysical unfurlings of countings/progressings because I deny that there can be an analogical mapping between the series of natural numbers (a mathematical thesis) and the growing series of countings/progressings (a metaphysical thesis).
I look forward to Alex's criticisms here.
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