Some remarks of Peter Lupu in an earlier thread suggest that he does not understand the notions of potential and actual infinity. Peter writes:

(ii) An acorn has the potential to become an oak tree. But the acorn has this potential only because there are actual oak trees . . . . If for some reason oak trees could not exist, then an acorn cannot be said to have the potential to become an oak tree. Similarly, if the proponent of syllogism S1 thinks that actual infinity is ruled out by some conceptual, logical, or metaphysical necessity, then he is committed to hold that there cannot be a potential infinity either. Thus, in order for something to have the potential to be such-and-such, it is required that it is at least possible for actual such-and-such to exist.

This is a very fruitful misunderstanding! For it allows us to clarify the different senses of 'potential' and 'actual' as applied to the analysis of change and to the topic of infinity. First of all, Peter is completely correct in what he says in the first two sentences of the above quotation. The essence of what he is saying may be distilled in the following principle

If actual Fs are impossible, then potential Fs are also impossible.

But this irreproachable principle is misapplied if 'F' is instantiated by 'infinity.' If an actual infinity is impossible, it does not follow that a potential infinity is impossible. For when we say that a series, say, is potentially infinite we precisely mean to exclude its being actually infinite. A potentially infinite series is not one that has the power or potency to develop over time into an actually infinite series, the way a properly planted acorn develops into an oak tree. On the contrary, it is a series which, no matter how much time elapses, is never completed. An actually infinite series, by contrast, is complete at every instant.

Consider the natural numbers (0, 1, 2, 3, . . . n, n +1, . . . ). If these numbers form a set, call it N, then N will of course be actually infinite. A set is a single, definite object, a one-over-many, distinct from each of its members and from all of them. N must be actually infinite because there is no greatest natural number, and because N contains all the natural numbers.

It is worth noting, as I have noted before, that 'actually infinite set' is a pleonastic expression. It suffices to say 'infinite set.' This is because the phrase 'potentially infinite set' is nonsense. It is nonsense because a set is a definite object whose definiteness derives from its having exactly the members it has. In the case of the natural numbers, *if* they form a set, then that set will have a transfinite cardinality. Cantor refers to that cardinality as aleph-zero or aleph-nought.

But surely it is not obvious that the natural numbers form a set. Suppose they don't. Then the natural number series, though infinite, will be merely potentially infinite. What 'potentially infinite' means in this case is that one can go on adding endlessly without ever reaching an upper bound of the series. No matter how large the number counted up to, one can add 1 to reach a still higher number. The numbers are thus created by the counting, not labeled by the counting. The numbers are not 'out there' waiting to be counted; they are created by the counting. In that sense, their infinity is merely potential. But if the naturals are an actual infinity, then they are not created but labeled.

Or consider a line segment. One can divide it repeatedly and in principle 'infinitely.' But if one does so is one creating divisions or recognizing divisions that exist already? If the former, then the infinity of divisions is merely potential; if the latter, it is actual.

Peter seems worried by the fact that no human or nonhuman adding machine can enumerate all of the natural numbers. But this is no problem at all. If there is an actual infinity of natural numbers, then it is obvious that a complete enumeration is impossible: the first transfinite ordinal omega has aleph-nought predecessors. If there is only a potential infinity of naturals, then as many enumerations have taken place, that is the last number created.

Peter seems not to be taking seriously the notion of potential infinity by simply assuming that the naturals must form an infinite set. He doesn't take it seriously because he confuses the use of 'potential' in the context of an analysis of change, where change is the reduction of potency to act, with the use of 'potential' in discussions of infinity.

But now I'm having second thoughts. I want to say that from the fact that a line segment is infinitely divisible, it does not follow that it is actually divided into continuum-many points. But what about the number of possible dividings? If that is a finite number, one that reflects the ability of some divider, then how can the segment be *infinitely* divisible? But if the number of possible dividings is a transfinite number, then it seems we have re-introduced an actual infinity, namely, an actual infinity of possible dividings. In other words, infinite divisibility seems to require an actual infinity of possible dividings. Or does it?

Round n+1,

1) It is worth emphasizing first that in my original post (both of them) I have anticipated that Bill will balk at my invoking the acorn/oak-tree example in connection with the notion of potential infinity. So I certainly agree with Bill when he says that “A potentially infinite series is not one that has the power or potency to develop over time into an actually infinite series, the way a properly planted acorn develops into an oak tree” and I have said this much in that post. The point I was trying to get across in the example of developing-potentialities (e.g., acorn/oak-tree) was precisely that the notion of “potential infinity” is in some sense a misnomer. But in what sense is it a misnomer? Well, since the model of developing-potentialities cannot be the right interpretation of the phrase ‘potential infinity’, I aimed to press Bill to produce an alternative. As it turns out he did and then again he did not.

2) Second, it is also worth emphasizing that any alternative account of “potential infinity” to the one exemplified by the acorn/oak-tree example must satisfy the requirement that there is no largest natural number. Bill in one of his previous posts explicitly agrees with this requirement.

3) Bill correctly emphasizes the anti-realist element in the position of those who reject actual infinity. He points out that those who reject actual infinity do not think of the natural numbers as being “'out there' waiting to be counted; they are created by the counting.” And yet it is precisely this concession that Bill is willing to make that creates several tension in his position.

4) Tension #1: According to the position Bill currently defends, the natural numbers are not out there in the world. Instead they are created by us. But, now, the following question arises: how do we know that there is no greatest natural number? Clearly, we do not know this fact by grasping a truth about an antecedent reality that exists prior to our creation, since such a reality is denied by the present view. Do we know this fact by some self-reflection on our own powers of creating natural numbers? But, surely, a serious reflection on our own powers of creation will deliver exactly the opposite: i.e., since we are finite beings, we most likely will create only finitely many numbers. Hence, at some point in time we will cease creating new numbers (i.e., when we cease to exist) and so the last number we created is the greatest number. So self-reflection should undoubtedly lead to the rejection of the principle that there is no greatest number. Yet Bill insists that there is no greatest natural number. I do not see how he gets to know this based upon the present view he holds.

5) Tension #2: Bill addresses a concern I have expressed in previous posts about the idea that some finite being or thing (e.g., a human being, the human race, or a super-computer) “can always add 1” to a number it produced. My concern has to do with the fact that any such being is obviously finite and at some point it will cease to exist. Hence, it is not the case that such a being “can always add 1”. Bill maintains that this is not a problem. He says: “If there is only a potential infinity of naturals, then as many enumerations have taken place, that is the last number created.”

Well, but if the last enumeration is the last number created and numbers do not exist beyond such creations, then the last enumerated number is also the largest number (since there are no numbers independently from the enumeration and, hence, there is no number greater than the one that was enumerated last). So contrary to Bill’s acceptance of the principle that there is no largest number, he appears to be denying it.

6) Tension #3: Bill says: “What 'potentially infinite' means in this case is that one can go on adding endlessly without ever reaching an upper bound of the series. No matter how large the number counted up to, one can add 1 to reach a still higher number.”

But this explanation of potential infinity surely conflicts with the quote I discussed in (5) above. In that quote Bill allows that due to the finite nature of any being that creates numbers, there is in fact a last number enumerated (and, hence, the largest number). Yet in the present sentence Bill appears to hold that the enumeration can go on “endlessly without ever reaching an upper bound…” I think we are here very close to an outright contradiction.

7) In my view Bill’s position is subject to an inescapable inherent tension. Bill wishes to accept certain obvious constraints such as that there is no largest natural number. So we can now ask: What has to be the case about mathematical reality in order for this proposition to be satisfied? While it is true that the proposition that there is no largest natural number in itself does not entail that there are infinitely many natural numbers, it is equally clear that if there are infinitely many natural numbers, then this proposition is going to be satisfied. The question arises whether this constraint can be satisfied by a “potential infinity” as interpreted by Bill. And it is at this point that the inherent tension arises.

On the one hand, Bill needs to set up the notion of “potential infinity” so as to satisfy the constraint that there is no largest natural number. So we have someone or something (only God knows precisely who or what) that is able to keep counting endlessly (and yet miraculously counting endlessly is not tantamount to an infinite series). On the other hand, since whatever it is that does the counting must be human-like and therefore finite, this something or someone will eventually stop counting (that is, when it ceases to exist) and thereby produce the last number, which according to this account will necessarily be the last number. So in order to satisfy the constraint that there is no largest natural number, Bill is willing to posit a creator of numbers that spins out numbers *forever*. Yet when it comes to describing who or what such a creator of numbers could be, he concedes that it has a finite lifespan and hence will eventually create a last number. And that number will inevitably be the greatest natural number. So Bill needs to have things both ways and so far he failed to convince me that having things both ways is a coherent position.

Posted by: Account Deleted | Wednesday, August 04, 2010 at 09:52 PM

Bravo! You are making some points that I've been trying to make for a while. I have stated it similar that "the numbers are not 'out there'", but I really like your point that "The numbers are thus created by the counting". I have long looked at numbers as a system of "naming".

For illustration imagine a limited universe where there are at most lots of three or items. You named the first one "Huey", the second "Duey" and the third "Louie". You would have a comparative sense of how many there were by the last name assigned. You could say of your sheep, I had "Louie" yesterday, but today, I only have "Duey".

I have also thought that a set of natural numbers with a greatest number N will have a maximum magnitude of N + 1. This is a property of finite sets. Cantor shows that the set of natural numbers itself has no greatest number, but the way he showed this was an induction. If we have a set { 0, 1, ..., N } we have another number N + 1 which is the magnitude of the set. This showed him that there will never be a greatest number. However, Russell in his 1902 Principles of Mathematics argues that "transfinite" sets do not obey induction. But the way we get to the proof is an induction, yet in assigning a pseudo-magnitude to the set, they seem to be arguing

throughinduction. It is somewhat trivial proof through induction, that all individual sets derived by from Cantor's patternobeyinduction. Thus "Cantor's set" either does not exist in the chain or obeys induction. So if that can't be said about an endless expansion of the pattern, then induction is at odds, or, I think, Cantor and Russell took the logical complement too far. Not having a greatest number is not having a greatest number, whether it applies a meta-magnitude is problematic if we pass a threshold where induction no longer holds. If we have a difference in type and no way to reason it through induction, then we really don't have a case for any attributes that might be imputed through induction.I very rarely refer to "infinity" anymore because of its colloquial use as a sort of number. I have learned to recast many sentences to reflect that meaning. I'll say that something "increases without bound" or an "unbounded" sequence.

Posted by: John Cassidy | Wednesday, August 04, 2010 at 10:03 PM

Thanks, John, but please bear in mind that I am not endorsing potential over actual infinity, but trying to explain the distinction clearly, not that I succeeded entirely above. In addition, I am claiming that it is not obvious that there are actual infinities, and I am looking for arguments for them.

>>I have also thought that a set of natural numbers with a greatest number N will have a maximum magnitude of N + 1.<< What does that mean? What do you mean by 'magnitude'?

Posted by: Bill Vallicella | Thursday, August 05, 2010 at 10:53 AM

Peter,

Ad 1. Is 'potential infinity' a misnomer? I suppose that that is arguable. But whether it is a misnomer, one has to understand what has been mean by it by philosophers and mathematicians from Aristotle on.

Ad 2. Your point is obvious. Why say it?

Ad 4. It seems to me that you don't understand the distinction. So I am trying to explain it to you. But that is not the same as endorsing the view that there are no actual infinities. One of my points is that it is not obvious whether there are or aren't. I want to know if there are kncok-down arguments for actual infinities.

>>But, now, the following question arises: how do we know that there is no greatest natural number? Clearly, we do not know this fact by grasping a truth about an antecedent reality that exists prior to our creation, since such a reality is denied by the present view.<<

This is not the right question. Of course there is no greatest natural number. Anyone who denied that I wouldn't bother talking to. The question is: does this obvious fact entail that there is an actual infinity of natural numbers? If it does, then we have a very quick and simple proof of the actual infinite from a self-evident premise! Can it be that easy? Not that everything must be hard, but I don't think this question an be resolved so quickly.

But if there is no actual infinity of naturals, then what does 'There is no greatest natural number' mean? It means something like this: the computational process which generates a successor for any given n can be iterated indefinitely. As opposed to what? As opposed to a computational process that cannot be iterated indefinitely, e.g. the process of reducing a fraction to its lowest common denominator, e.g. 4/8 to 2/4 to 1/2.

Why must the natural numbers pre-exist the computational process of adding 1? If each number is created by the computational process, then 'there is no greatest natural number' says something about that process: it says that the process is iterable indefinitely.

It seems to me that Peter is simply begging the question. He is assuming that that 'There is an actual infinity of natural numbers' is logically equivalent to 'There is no greatest natural number.'

Posted by: Bill Vallicella | Thursday, August 05, 2010 at 11:33 AM

Peter,

Ad 1. Is 'potential infinity' a misnomer? I suppose that that is arguable. But whether it is a misnomer, one has to understand what has been mean by it by philosophers and mathematicians from Aristotle on.

Ad 2. Your point is obvious. Why say it?

Ad 4. It seems to me that you don't understand the distinction. So I am trying to explain it to you. But that is not the same as endorsing the view that there are no actual infinities. One of my points is that it is not obvious whether there are or aren't. I want to know if there are kncok-down arguments for actual infinities.

>>But, now, the following question arises: how do we know that there is no greatest natural number? Clearly, we do not know this fact by grasping a truth about an antecedent reality that exists prior to our creation, since such a reality is denied by the present view.<<

This is not the right question. Of course there is no greatest natural number. Anyone who denied that I wouldn't bother talking to. The question is: does this obvious fact entail that there is an actual infinity of natural numbers? If it does, then we have a very quick and simple proof of the actual infinite from a self-evident premise! Can it be that easy? Not that everything must be hard, but I don't think this question an be resolved so quickly. But if there is no actual infinity of naturals, then what does 'There is no greatest natural number' mean? It means something like this: the computational process which generates a successor for any given n can be iterated indefinitely. As opposed to what? As opposed to a computational process that cannot be iterated indefinitely, e.g. the process of reducing a fraction to its lowest common denominator, e.g. 4/8 to 2/4 to 1/2. Why must the natural numbers pre-exist the computational process of adding 1? If each number is created by the computational process, then 'there is no greatest natural number' says something about that process: it says that the process is iterable indefinitely. It seems to me that Peter is simply begging the question. He is assuming that that 'There is an actual infinity of natural numbers' is logically equivalent to 'There is no greatest natural number.'

Posted by: Bill Vallicella | Thursday, August 05, 2010 at 02:52 PM

Like, I suspect, many readers, I've been struggling to understand what Bill is trying to do here, so I'm reduced to picking holes in specific statements. Bill says

Surely this can't be right? It's true that 'there's no greatest (natural) number' *implies* that such a computational process carries on indefinitely, but this is not how we *understand* the proposition. If it were we would be waiting for ever to find out if it were true, yet we already know it to be true. Speculating a little, I doubt it's possible to explain negative existential statements about numbers in terms of computations, for we will always come round to explaining 'going on forever', and the only resource we have for this are the numbers themselves. Any such explanation will be circular. Speculating rather more, this might be the basis for an argument for actual infinities. Potential infinities are closely related to initial segments of the natural numbers which themselves are closely related to computations. If computations fail to account for our understanding of negative existentials then perhaps so do potential infinities.Posted by: David Brightly | Friday, August 06, 2010 at 04:46 AM

Hi David,

Peter was trained by contemporary analytic philosophers and you, if I am not mistaken, are a mathematician. Both of you, I suspect, dogmatically assume that the actual infinite is the only game in town. You were never exposed to anything else. The pot/act distinction was never raised. And for pure mathematicians it needn't come into their actual work in any case.

In the case of Peter, I don't think he understands what the distinction means, or else he thinks it doesn't have a tolerably clear meaning. So all I am trying to do is explain what the issue is between those who maintain that there are actual infinities and those who maintain that infinity is potential only.

How you understand

P. There is no greatest nat'l number

depends on whether you think there are actual (completed) infinities or not. Of course, (P) will be granted by everyone whether or not they are aware of the pot/act inf issue. But now we are going a step further and asking what exactly (P) could mean if there are no actual infinities. Well, what does it mean if there are actual infinities? Something like this

P.* There are all the natural numbers that there could have been. They form a set, an infinite set N. Each n has a successor which is an element in N. The elements of N do not depend for their existence on us or on any being or computing machine.

If there are no actual infinities, then (P) means something like this

P**. Numbers are constructed or generated. They do not exist in themselves independently of a process of construction. The computational process which generates a successor for any given n can be iterated indefinitely. In this sense the infinity of natural numbers is potential only, not actual.

That seems quite clear to me!

Posted by: Bill Vallicella | Friday, August 06, 2010 at 11:33 AM

David writes, >>If it were we would be waiting for ever to find out if it were true, yet we already know it to be true.<<

Perhaps your argument is this:

a. We know that there is no greatest nat'l number.

b. If the nat'l numbers did not form an infinite set, then we would not know this

ERGO

c. The nat'l numbers form an infinite set.

I grant (a), but I don't grant (b). We know that there is no greatest natural number simply by understanding concepts such as these: nat'l number, immediate successor, addition, closed under addition.

Posted by: Bill Vallicella | Friday, August 06, 2010 at 11:48 AM

I would like to offer the following observations:

I agree that the phrase "actually infinte set" has the same meaning as "infinte set" as that is defined in mathematics. A set is infinite (in this sense) if and only if it is not finite. Then by the principle of the excluded middle, every set is either infinite, or finite. If the set in question is any set whose members are natural numbers, to say that it is finite is equivalent to saying that it is bounded. Therefore, if the set of natural numbers is not actually infinite, it is bounded. This conclusion seems undesirable.

Note that this argument does not depend upon the existence of infinite sets. In light of that, there seem to be three possibilities:

1. The set of natural numbers is infinite.

2. The principle of the excluded middle is to be denied, allowing the possibility of a third mutually exclusive category called "potentially infinite".

3. The natural numbers do not form a set.

Regarding 2, I can say that the notion of "potentially infinte" seems imprecise at best, and not sensible at worst. Usually, and in fact in this post, "potentially infinite" refers to some sort of procedure which "goes on forever". What does this mean? I don't claim to know, but a similar question arises in the context of the theory of infinite sequences, which theory of course assumes the axiom of infinity. The problem is to give precise meaning to symbols like {x1, x2, x3, ...}, where the "..." is read "and so on". The way this is done is to state that the members of the sequence are equinumerous with the set of natural numbers. I am not aware of any other coherent formulation of this concept; I challenge proponents of "potential infinity" to devise one which does not depend, as this does, on the axiom of infinity.

Regarding 3, that the natural numbers do not form a set, I must ask, what then should be the criteria for set formation? If we take ZF without the axiom of infinity, we do not arrive at a contradiction by considering the set of natural numbers. If that were the case, then the axioms of ZF including the axiom of infinity would form an inconsistent set, and it is generally held that this is not the case. So if the axioms of set theory do not disallow the existence of the set of natural numbers, why should there not be such a set?

As a sort of aside, I would like to mention that the number of possible divisions of the real line (or a segment of it) is equal to the power of the continuum. In fact more is true: the set (if it exists) of possible divisions of the real line is isomorphic (with appropriate and frankly obvious operations defined) to the set of real numbers.

My conclusion is this: the question of the existence of infinte sets is one that can and should be formulated within the context of the axioms of set theory, whatever we take them to be. To discuss the question in terms of whether these objects are "out there" or are "created" by human beings seems suspicious to me. If I were to adopt the non-Platonist view, and say that the natural numbers do not exist independently of the minds which apprehend them, I would not be committed to the denial of "actual infinity" for a very simple reason: there is no need to restrict the alleged creation of natural numbers to the operation of the successor function; one can instead claim that the natural numbers are created "all at once" by the adoption of the axioms which define the properties of that set.

Posted by: Phoenix Kyle MacGregor | Friday, August 06, 2010 at 05:54 PM

Welcome, Phoenix.

>>Therefore, if the set of natural numbers is not actually infinite, it is bounded. This conclusion seems undesirable.<<

That doesn't follow. If the set of nat'l numbers is not actually infinite, then what follows is that there is no set of all natural numbers. It is perfectly obvious that if there is a set of natural numbers, then it cannot have a finite cardinality. If a child asks, "How many nat'l numbers are there?" It would be absurd to answer 4,569,935 or any finite number. What is not obvious is that there is a set of natural numbers. Can you prove to me that there is?

It is not obvious because the infinity of natural number may be a merely potential infinity as Aristole and so many through the centuries have thought.

>>Regarding 3, that the natural numbers do not form a set, I must ask, what then should be the criteria for set formation?<<

Part and parcel of the naive conception of set is the notion of Unrestricted Comprehension: for any condition (predicate, property, propositional function, etc.) there is a corresponding set. This leads to Russell's Paradox. One way to avoid it is via the iterative conception of set which may be a way to answer your question about the criteria of set formation. See George Boolos, "The Iterative Conception of Set." Very rough illustration:

Suppose at stage 0 you have two individuals (nonsets, Urelemente)a, b. Form all the sets you can from these materials. At Stage 1 you have a, b, {a}, {b} {a,b} { }. Then form all possible sets from these six objects, and so on. One will never in this way arrive at a set of all sets, or a set that has itself as a member.

Posted by: Bill Vallicella | Friday, August 06, 2010 at 08:04 PM

(I) The disagreement between Bill and I centers around the following question: Is there an adequate account of the notion of ‘potential infinity’ (PI)? This question in turn reduces to the question about the conditions that an *adequate account* of PI will have to satisfy. I think that Bill and I can secure an agreement on the following two conditions:

(a) PI does not presuppose nor entail the possible existence of *any* actual infinity;

(b) PI entails that it is not the case that there is a largest natural number.

Trouble begins when the following principle of potentiality is introduced:

POP. If actual Fs are impossible, then potential Fs are also impossible;

(Note: POP was proposed by me in a previous post and subsequently was restated in its present much better and more succinct form by Bill (in his “On Potential and Actual Infinity”, August 04, 2010 at 07:15 PM).

(II) Clearly, if POP is without exceptions, then no account of PI can satisfy condition (a). So Bill claims that POP is “misapplied if ‘F’ is instantiated by ‘infinity’”. But, why is POP “misapplied” precisely in the case of ‘infinity’? Does Bill offer any principled grounds that distinguish those cases where POP is an “irreproachable principle” from cases, such as the case of ‘potential infinity’, where its application is inappropriate? I will examine some of the considerations Bill offers in defense of this exemption.

(i) One of the ways Bill justifies the exemption in regards to ‘potential infinity’ is as follows:

“If an actual infinity is impossible, it does not follow that a potential infinity is impossible. For when we say that a series, say, is potentially infinite we precisely mean to exclude its being actually infinite.”

The second sentence in the above quotation is intended to offer the grounds for the first, which in turn simply reiterates the claim that ‘infinity’ should be exempt from POP. And I certainly agree with the second sentence. The trouble is that the second sentence cannot support the first for the following reason. While the first sentence calls for an exemption from POP specifically in the case when ‘infinity’ instantiates ‘F’, the second sentence holds regarding all instances of ‘F’. For the second sentence is simply an instance of the following exclusionary principle of potentiality (EPP), which holds for every instance of ‘F’:

(EPP). For every property F, individual x, and time (t): if x is potentially F at t, then x is not actually F at t.

Since EPP holds for all suitable instances of F, including the contested case of ‘infinity’, it cannot justify excluding the case of ‘infinity’ from POP.

(ii) However, Bill offers a somewhat different justification for exempting ‘infinity’ from POP, a justification that does not reduce to EPP. He says:

”A potentially infinite series is not one that has the power or potency to develop over time into an actually infinite series, the way a properly planted acorn develops into an oak tree. On the contrary, it is a series which, no matter how much time elapses, is never completed.”

There are two different claims that are made here regarding a potentially infinite series:

(1) A potential infinite series S does not have the *power* or *potency* to develop into an actual infinite series;

(2) A potential infinite series S is *never completed*.

There are two major difficulties with both (1) and (2) and these difficulties are intimately related. First, both (1) and (2) share the defect that they tell us what a potential infinite series lacks, but they fail to tell us in what being a potentially infinite series consist. For instance, (1) tells us that a potential infinite series S lacks the power or potency to “develop” or become an actual infinite series. Similarly, (2) tells us that a potentially infinite series S is *never completed*. The trouble is that this tells us very little in what specifically the potentiality of such a series consist. And I do not think that any account of ‘potential infinity’ is satisfactory unless it ascribes some power or potential to an object or process.

Second, neither is a sufficient condition to justify exempting ‘infinite’ from being a proper instance of POP. For if they were, then many proper instances of POP would also be exempt. Consider the following analog of (1):

(1*) A potentially round-square object does not have the power or potency to develop into an actually round-square object.

In (1*) the property of ‘…being a round-square’ instantiates the ‘F’ in POP. Now, clearly, if (1) is true, then so is (1*). Yet clearly the fact that (1*) is true does not exempt the relevant property (in this case the property of being a round-square) from properly instantiating ‘F’ in POP, for the relevant instance of POP for this property is

(POP*) If actual round-squares are impossible, then potential round-squares are also impossible;

Clearly POP* is a proper instance of POP and moreover it is true. Furthermore, POP* together with the fact that actual round-squares are impossible explains why potential round-squares fail to have the power or potential to develop into actual round-squares: namely, since actual round squares cannot exist, potential round squares cannot exist either. And non-existent objects do not have any powers or potencies. Thus, the fact that a given property (‘being an infinity series’; ‘being a round square object’) satisfies something like (1) or (1*) is not a sufficient condition for it to be exempt from POP.

Similar examples defeat (2) as a justification for exempting a property from being a proper instance of ‘F’ in POP. Consider the example of the property of ‘…being a perpetuum mobile machine’. The analog of (2) in this case would be

(2*) A potentially perpetuum mobile machine is *never completed*.

Now, clearly, if (2) is true, then so is (2*). Moreover, the fact that a machine that has the potential to be a perpetuum mobile is never completed cannot be a sufficient condition to say that the property ‘is a perpetuum mobile machine’ is an inappropriate instantiation of ‘F’ in POP. On the contrary: the appropriate instance of POP regarding a perpetuum mobile machine together with the fact that the laws of physics rule out the possibility of the existence of an *actual* perpetuum mobile machine explain why a potential perpetuum machine was never, is not, and never will be completed.

(III) The above considerations suggest that there is something systematically wrong with Bill’s argument. The trouble is that a potentially infinite series may lack the powers or potencies to become an actual infinite series or be ever completed because no potential infinite series is possible in the first place or because the phrase ‘potential’ is intended to modify not the property ‘…is an infinite series’ but rather some other closely related property such as ‘…is an indefinite series’. If we adopt this change, then none of the above arguments prevent the opponents of actual infinity to maintain that actual infinity is impossible, while also maintaining that the impossibility of actual infinity does not rule out that a potentially indefinite series is possible, provided of course that an actually indefinite series is possible. The burden of explication now falls upon the coherence of the notion of an ‘indefinite series’ that also satisfies conditions (a) and (b) above. I shall leave this question to another occasion.

Posted by: Account Deleted | Friday, August 06, 2010 at 10:59 PM

Bill, thanks for having me here, and for your response. I have a few comments.

>>That doesn't follow. If the set of nat'l numbers is not actually infinite, then what follows is that there is no set of all natural numbers.

I am fairly certain that it does indeed follow that if any collection of natural numbers is not infinite, then it is bounded. This is a simple application of the definitions, and no more is intended by it than to try to formulate the question with some precision. The result, as I said, does not assume an infinite set, but it does assume that the natural numbers in fact form a set. So one possibility is that there is no such set.

This possibility is reasonable, because the axioms of ZF without the axiom of infinity do not require such a set. It is also reasonable because there are other formulations of set theory, such as, as you point out, the iterative conception of a set. The construction you suggest will clearly never produce a countably infinite set.

On the other hand, the possibility that the natural numbers do not form a set, while reasonable, is not inevitable. Again, if it were, then ZF would be inconsistent.

I would have to investigate this to be sure, but at a glance I suspect that the sets which may be formed from the aforementioned construction are more or less the same sets which may be formed from ZF without the axiom of infinity. Certainly the former should form a subset of the latter. That's interesting; probably it is known, but not by me.

In any event, I of course not prove that there is a set of natural numbers, nor have mathematicians more clever than I. Analogously, no one is required to believe in the uniqueness of a line parallel to a given line. Again, I mostly wish to clarify the issue and attempt to put it into a meaningful context. To this end I must ask again for a precise definition of "potentially infinite"; without one, I cannot know how to respond to the phrase "the infinity of natural numbers may be merely a potential infinity".

One other thing I would be interested to learn is, if we deny the existence of an infinte set of natural numbers, what becomes of the definition of a real number, which depends explicitly on the existence of infinite collections of rational numbers?

Posted by: Phoenix Kyle MacGregor | Friday, August 06, 2010 at 11:15 PM

Bill,

I just read that paper by Boolos. Very clear and very illuminating.

Posted by: Account Deleted | Saturday, August 07, 2010 at 04:34 AM

Some Metaphilosophical Observations,

(A) Let me make some observations regarding Bill’s stand in the present controversy. Bill himself does not hold the beliefs that there is no actual infinity; that there are no sets; that there are no infinite sets; that there is no infinite set of NN; and so on. Like me, and others, Bill believes the contrary of all of the above. Moreover, Bill believes that there are weighty arguments in favor of his beliefs that there is at least one (if not several) actual infinite series; that there are sets; that there are infinite sets; that there is an actual infinite series of NN.

However, he also believes that there are equally weighty arguments in favor of rejecting actual infinity of any kind and so on. So Bill thinks that if the position of each side makes sense within their own conceptual framework (premises) and the arguments in favor of each side are more or less as weighty as the arguments on behalf of the other side, then there are no *rational grounds* for choosing one position over the other. This does not mean that we ought to refrain from adopting one side or the other or that if one does adopt one view or the other, then doing so is *irrational*. Rather, I suspect that Bill thinks that such a choice would be *non-rational* in the sense that it is not based on purely rational grounds that outweigh in a demonstrable way the considerations in favor of the contrary view.

(B) So one should not construe what Bill has been doing in the last few posts as defending his own beliefs that there are no actual infinities; that there are no infinite sets; and so on. He himself does not hold any of these beliefs. Rather one should view what Bill has been undertaken to do in the last few posts as defending a certain Metaphilosophical view that the aporetic triad (1)-(3) (or some equivalent thereof) cannot be dispensed with rationally by means of our currently available knowledge and conceptual resources.

(C) It is important to note that it is part of Bill’s Metaphilosophical view regarding aporias such as (1)-(3) (as I understand it) that there is a fact of the matter out there in the world which determines whether there exists an actual infinite series of NN, for example, or not. Thus, the world contains suitable truthmakers which determine whether (1) or (3) is false and, therefore, this is not a matter of mere taste or personal preference analogous to a preference between chocolate ice-cream versus vanilla ice-cream. Holding that there is a fact of the matter about which proposition in an aporetic triad is true and which one is false means that the truth or falsity of such propositions is itself *not reducible* to any epistemic decision procedure or epistemic norms which we might employ in choosing which propositions to believe, if any. Thus, Bill’s Metaphilosophy is that even if we do not currently possess and perhaps will never possess decisive rational means to decide which proposition to reject and which one to adopt in any given aporetic set of propositions, this should not be taken to mean that there is no fact of the matter which one of these propositions is true and which one is false.

Posted by: Account Deleted | Saturday, August 07, 2010 at 05:58 AM

Thanks again, Peter. I do appreciate your hard work. You say >>I think that Bill and I can secure an agreement on the following two conditions:

(a) PI does not presuppose nor entail the possible existence of *any* actual infinity;

(b) PI entails that it is not the case that there is a largest natural number.<<

I would strike 'possible' from (a) and replace (b) with this: That the only infinities are potential infinities is consistent with it being the case that there is no greatest natural number.

One of the interesting features of this discussion is that Peter is doing something that I (perhaps naively) supposed no one would ever do, namely assimilate or perhaps conflate the senses of 'potential' and 'actual' as they figure in Aristotelian analyses of change in material substances with the senses of 'potential' and actual' as they figure in discussions of infinity. I just assumed (perhaps naively) that it would be obvious to people that these terms were being used in different ways in the two contexts.

I admit, though, that the two contexts are not totally unconnected. If a material substance actualizes one or more of its potentialities over time, then this reduction of potency to act is what an Aristotelian calls change. Now the media of change are space and time, and both of these 'media' give rise to questions about infinity. For example, is a line segment composed of points? If yes, how many? Finitely many? (This could be the answer if there are space atoms.) Or infinitely many? Is there an actual infinity of points in a line segment?

But despite the fact that the two contexts are related, it does not follow that 'potentiality' is being used in the same sense in both. In fact, it is not, as Aristotle himself says at Physics 206a18-25, where he says that we needn't assume that all potentiality must be understood along the lines of the potentiality of bronzew to be sculpted into a statue.

But I must leave it here for now. I have to get ready for a little vacation I am taking. I will return to this discussion near the end of next week.

Posted by: Bill Vallicella | Saturday, August 07, 2010 at 06:53 PM

Phoenix,

Here is the argument you gave: >>A set is infinite (in this sense) if and only if it is not finite. Then by the principle of the excluded middle, every set is either infinite, or finite. If the set in question is any set whose members are natural numbers, to say that it is finite is equivalent to saying that it is bounded. Therefore, if the set of natural numbers is not actually infinite, it is bounded. This conclusion seems undesirable.<<

It is highly undesirable, indeed absurd. But you are assuming that the natural numbers form a set which is precisely the issue.

I'll return to this when I get back from a short vacation.

Posted by: Bill Vallicella | Saturday, August 07, 2010 at 07:01 PM

Peter,

Your last comment was truly outstanding. I am tempted to exaggerate a bit and say you know my mind better than I do. You are an alter ego.

I would only add that my commitment to aporetics is tentative. After all, there is the following metaphilosophical aporetic triad to contend with:

1. Philosophical problems cannot be solved by reason (or reason plus empirical input)

2. There is nothing beyond or other than reason for resolving them.

3. Philosophical problems are soluble.

I am inclined to accept (1) and (3) and reject (2). Things get very interesting if it is maintained (and I am open to the idea) that each limb is equally defensible.

Posted by: Bill Vallicella | Saturday, August 07, 2010 at 07:18 PM

Bill,

Indeed an interesting aporetic triad. I got to think about it. I think Godel entertained a similar sort of a problem regarding mathematics.

Have a wonderful vacation. Will contact tomorrow.

peter

Posted by: Account Deleted | Saturday, August 07, 2010 at 08:00 PM

To all,

While Bill is enjoying an undeserved (just kidding!) vacation, I invite people to continue posting on this thread. We of course will not benefit from Bill's insightful commentary for the duration, but this should not stop us from engaging the topic.

Posted by: Account Deleted | Sunday, August 08, 2010 at 04:44 AM

Reply to Some of Bill’s Recent Comments & Objections,

1) Bill Vallicella | Saturday, August 07, 2010 at 06:53 PM:

(i) Re. adequacy condition (a): I agree to strike ‘possible’ from condition (a).

(ii) Re. adequacy condition (b): Bill recommends replacing adequacy condition (b) which requires that an account of PI (=potential infinite series) *entail* the proposition that no greatest NN exists with the following: “That the only infinities are potential infinities is consistent with it being the case that there is no greatest natural number.”

I do not think consistency will suffice. Let Q = There is a greatest NN. Suppose that a certain account of PI does not entail that ~Q, but it is consistent with it. Then such a PI will also be consistent with Q: i.e., such a PI will be consistent with the existence of a number that is the greatest NN. But, now, if potential infinities are the only infinities and the existence of such infinities is consistent with both Q as well as ~Q, then it follows that Q as well as ~Q are both undecidable relative to the axioms of such a PI. But this is an extremely unwelcome result for the proponents of potential infinity. The proposition that there is no largest NN should be decidable and entailed by any account of mathematics. I thought this much we agreed upon all along.

2) In one of my posts (I can’t remember which one) I challenged Bill to tell us how we know that ~ Q holds based on an account that maintains that only potential infinities exist. David Brightly in a comment (dated August 6, 2010, at 04:46 AM) argued along the same lines that if we assume that potential infinity is to be cashed out in terms of an *indefinite computational process*, then we simply could not tell whether ~Q is true before we actually see whether the computational process in question terminates at some natural number or does not. David Brightly generalizes this argument to negative existential. For instance, on such an account how would we know in advance that there is no largest even number; odd number; prime; etc? Bill’s response takes two forms:

(i) He says: “If there are no actual infinities, then (P) means something like this

P**. Numbers are constructed or generated. They do not exist in themselves independently of a process of construction. The computational process which generates a successor for any given n can be iterated indefinitely. In this sense the infinity of natural numbers is potential only, not actual.

That seems quite clear to me!”

But this response presupposes

(A) The notion of an *indefinite computation process* is coherent in the absence of any actual infinity;

(B) We know that a computation process “which generates a successor for any given n” it already generated will indeed go on *indefinitely* (i.e., never terminate). But how do we know that a computational process that never terminates is possible? After all, we have never actually encountered any computational process that can do that.

Problem (A) is a conceptual matter and I shall ignore it here. Question (B) involves an epistemological issue. It is a significance issue because the principal motivation for adopting a stand which rejects actual infinity and seeks to replace it with something like a potential infinity is typically epistemological. But now we can see that adopting such a position faces its own epistemological problems.

(ii) Bill reconstructs David Brightly argument in the form of the following syllogism:

“a. We know that there is no greatest nat'l number.

b. If the nat'l numbers did not form an infinite set, then we would not know this

ERGO

c. The nat'l numbers form an infinite set.

I grant (a), but I don't grant (b). We know that there is no greatest natural number simply by understanding concepts such as these: nat'l number, immediate successor, addition, closed under addition.”

I, and David I am sure, agree with Bill that we know that there is no greatest natural number based on our understanding of the concept of a natural number, immediate successor, addition, closed under edition, etc. However, at least I would respond that these concepts make sense only if there is an actual infinity and that our understanding of it, therefore, presupposes a grasp of such infinity. Moreover, Bill has not shown that all of these concepts can be given a reasonable sense within a conception that allows only potential infinities.

3) Let me now address the most fundamental issue: is the notion of a *potentially infinite series* coherent in the absence of any actual infinity?

3.1) I have argued in a previous post that the following principle of potentiality holds:

POP. If actual Fs are impossible, then potential Fs are impossible. (Bill’s formulation)

3.2) Clearly, POP holds for all the familiar substitution instances of ‘F’ such as when we ascribe the property of ‘having the potential to become an oak-tree’ to an acorn, or when we ascribe the property of ‘having the potential to become a great basketball player’ to a child, etc. It is also clear that if POP holds for the case when ‘infinity’ is a substitution instance of ‘F’, then one cannot hold simultaneously that an actual infinite series is conceptually, metaphysically, or logically impossible, but a potential infinite series is possible. POP rules out such a position.

3.3) Bill responded that there are two related but nonetheless different concepts of potentiality. One notion of potentiality—call it Potentility1—applies to all the familiar uses of the notion ‘potential’ mentioned above. The other notion of potentiality—call it Potentiality2—applies to ‘infinity’ and is exempt from POP. Bill expressed an astonishment that I either fail to understand the difference between Potentiality1 and Potentiality2 or that I challenge the coherence of the distinction so drawn.

3.4) I certainly challenge the distinction between two concepts of potentiality if the distinction is drawn only in terms of exempting one of these concepts from POP. Such a manner of drawing the distinction strikes me as completely ad-hoc. Moreover, since it fails to illuminate the pertinent characteristics of Potentiality2 that justify such an exemption, it begs the question in a severe way. Thus, anyone who maintains that there is a concept of potentiality that differs from the standard uses of this concept in a manner that exempts it from POP carries the burden of justifying such a claim by reference to special features of this alleged concept that explain the exemption and yet do not change the concept so radically that we no longer recognize it as a species of potentiality.

3.5) In the above post Bill made an effort to propose several substantive ways of drawing the distinction. However, I think that I have raised serious doubts about the viability of each of his proposals. The objections I have raised have a lot to do with the meaning of ‘potential’ and ‘actual’ and their respective relationship when they are used to modify the same property. This relationship is expressed by the combination of POP and EPP. It is this relationship between the two modifiers that renders my objections valid against the considerations Bill adduces on behalf of exempting the property of being infinite from POP when modified by ‘potential’ and ‘actual’. I shall argue that if a property is exempt from POP when modified by ‘potential’ and ‘actual’, then EPP is no longer applicable to the resulting respective properties. And if the two principles which express a fundamental aspect of the meanings of ‘potential’ and ‘actual’ do not apply when these are used as modifiers of a certain property, then it is doubtful that they are used in that context with their respective standard meanings.

3.6) Let us suppose that some property R is modified by ‘potential and ‘actual’, respectively. Let us also stipulate per Bill’s claim that R is exempt from POP when so modified. Then the following holds: actual Rs are impossible & potential Rs are possible. Now enter EPP, or a strengthened form thereof. The strengthened EPP says that when ‘potential’ and ‘actual’ modify the same property F, then the resulting two properties: i.e., ‘…is potentially an F’ and ‘…is actually an F’, rule each other out in the sense that they cannot be applied simultaneously to the same object. However, it is imperative to underscore that this fact; namely, that the two modified properties (i.e., ‘…is potentially an F’ and ‘…is actually an F’) cannot be simultaneously applied to the same object, is not a function of some general logical, conceptual, metaphysical, or even nomological truths that rule out one of them independently of the other. Rather being actually an F at t is ruled out by being potentially an F at t *because*, and *only because*, the object is potentially an F at t. Similarly, being actually an F at t rules out being potentially an F at t because, and only because, the object is already actually F at t; the potentiality as it were was already achieved.

3.7) Consider now what happens if R is a substitution instance of F. Since we have exempted R from POP, it follows that being actually an R is impossible. But this is not so because the object has the potential to be an R. The possibility of the object being actually an R is ruled out by consideration that have nothing to do with the object’s potential to be an R. Hence, EPP does not even apply to the case of R. The fact that it is impossible for an object to be an actual R has nothing to do with the fact that it is potentially an R. Therefore, EPP is irrelevant to any property that is exempt from POP. But now that we have determined that the property obtained by employing ‘potential’ to modify R is governed neither by POP nor by EPP, the two most important principles that govern the meaning of ‘potential’, I do not see what grounds we have to maintain that the meaning of the term ‘potential’ employed under such stipulations has anything to do with the familiar modifier ‘potential’. And now that we see this we can also see that those who deny the possibility or even intelligibility of an actual infinity cannot use ‘potential’ to modify the property of being an infinite series and mean to express the property of being a potentially infinite series. Instead they can maintain that actual infinity is impossible and therefore the best we can have is a *potentially indefinite series*. According to this proposal, the modifiers ‘actual’ and ‘potential’ modify different properties; i.e., the former modifies the property of being infinite, whereas the later modifies the property of being indefinite, and therefore POP and EPP do not apply for this obvious reason. They are now free to explicate the property of being an indefinite series quite independently of the impossibility of actual infinity. Again I postpone discussion of this later concept for another time.

Posted by: Account Deleted | Monday, August 09, 2010 at 05:11 PM

Bill is quite right. I'm a mathematician by training and do find it quite natural to think of a completed infinity of natural numbers. My problem with Bill's exposition of potential infinity is that I can't see how to apply logical quantifiers to statements about the elements of a potential infinity. My understanding of 'there is' and 'for all' requires a pre-existing domain of objects, which is why, perhaps, I have to think of the natural numbers as forming a set. Bill suggests that in the absence of such a domain 'there is no greatest number' has to be understood in a rather different way, as saying something about an open-ended process by which the numbers are constructed. I think Bill needs to tell us some more about how he sees this before we can usefully comment. But while he is away let me make a possibly bold assertion. The position on potential infinity that he is defending is equivalent to the denial of the principle of mathematical induction. Bill sees 'there is no greatest number' as equivalent to the potentially infinite sequence of propositions

but he refuses the inference to because the standard interpretation of this presupposes a completed infinity. Perhaps this is better seen through the traditional example of induction. Let S(n) be sum of the numbers from 0 to n, and let P(n) denote the proposition that S(n)=n(n+1)/2. P(0) says S(0)=0, P(1) says S(1)=1, P(2) says S(2)=2*3/2=3, etc. Bill will, I think, accept upto some number N which some process of verification might reach. He may also, after doing a little algebra, accept that for an *arbitrary* choice of n. But he baulks at the inference to for the same reason as before.Posted by: David Brightly | Wednesday, August 11, 2010 at 04:46 AM

I wrote my previous comment on Tuesday but didn't post it. Considering it a little more I think it reveals another difficulty for the 'only potential infinity' view. Bill will have to say that 'there is no greatest number' is equivalent to the potentially infinite sequence of propositions

But if, like the natural number sequence, this sequence is only ever partially completed, what is the status of the propositions, beyond, as it were, the latest one reached in the enumeration? If these propositions 'are not there' what grounds has BIll got for claiming that there is no greatest number?Posted by: David Brightly | Wednesday, August 11, 2010 at 05:03 AM

David has made some excellent points in those last comments. I'm glad we are in agreement that the notion of "potential infinity" needs more explication.

I'd like to draw attention to the paper "Completed versus Incomplete Infinity in Arithmetic" by Edward Nelson

www.math.princeton.edu/~nelson/papers/e.pdf

Posted by: Phoenix Kyle MacGregor | Wednesday, August 11, 2010 at 08:41 AM

Phoenix, thanks for the reference. I am preparing a post that will address David's suggestion, although I am willing to bet that Bill will not accept it.

Posted by: Account Deleted | Wednesday, August 11, 2010 at 02:24 PM

David, Phoenix,

DavidB makes the following suggestion:

“Bill suggests that in the absence of such a domain {i.e., a pre-existing infinite domain of NN} 'there is no greatest number' has to be understood in a rather different way, as saying something about an open-ended process by which the numbers are constructed. I think Bill needs to tell us some more about how he sees this before we can usefully comment. But while he is away let me make a possibly bold assertion. The position on potential infinity that he is defending is equivalent to the denial of the principle of mathematical induction.”

DavidB’s suggestion is indeed bold. However, Bill I am sure will decline the offer. Moreover, the denial of actual infinity does not require rejecting mathematical induction, although it needs to be suitably reinterpreted. For instance, intuitionism accepts a modified version of mathematical induction that conforms to their interpretation of the universal quantifier, which appears in the last step of mathematical induction, and the material conditional. The logical connectives according to intuitionism are not interpreted by reference to truth-values, but rather by reference to proofs. Thus, the intuitionistic interpretation of ‘(x) Px’, for instance, would go something like this: a proof of (x) Px is a proof which when applied to *any* number n, yields a proof of P(n). Note the asterisks around ‘any’. This is important since ‘any’ replaces ‘all’ and its cognates in many of the intuitionistic interpretation of the logical constants. Intuitionistic interpretation of ‘PQ’ goes something like this: a proof of P yields a proof of Q. Mathematical induction, thus, will be interpreted as a proof of P(0); and a proof that for *any* number n, a proof of P(n) yields a proof of P(n+1) (See Dummett’s , “Elements of Intuitionism”, pp. 12-14).

Similarly, those who are strict finitists need not reject mathematical induction (in my opinion), although they have to restrict it to finite domains. Thus, according to strict finitism, in the last step of mathematical induction the universal quantifier will have to be restricted to finite domains only: for every x in a finite domain D; (…Fx…) holds. However, since this is going to have to be applied to an indefinite number of finite domains, like a stacking of these formulas, each for a different and larger but still finite domain, the problem of indefiniteness, or unboundedness, remains.

Posted by: Account Deleted | Wednesday, August 11, 2010 at 05:44 PM

On Potentially *Indefinite* or *Unbounded* Series,

1) Thus far I have been battling with Bill over the intelligibility of the notion of ‘potential infinity’. In order to go beyond this issue we can perhaps stipulate that by ‘potential infinity’ we shall mean something like ‘potentially indefinite series’ or perhaps a ‘potentially unbounded series’. The task then is to make sense of these notions so as to satisfy both conditions (a) and (b) which I have formulated in a previous post:

(a) PI does not presuppose nor entail the existence of *any* actual infinity; (revised in accordance with Bill’s suggestion).

(b) PI entails that there is no largest natural number. (Note: As I stated in my previous post, (b) cannot be weakened so as to require merely that PI is consistent with there not being a greatest natural number).

2) Bill proposed several formulations of the notion of ‘potential infinity’ or ‘potentially infinite series’. I will quote some of these formulation in their original form, although I will examine them as if he is speaking of an *indefinite* or *unbounded* series of natural numbers.

(B1) August 04, 2010 at 07:15 PM:

“What 'potentially infinite' means in this case is that one can go on adding endlessly without ever reaching an upper bound of the series. No matter how large the number counted up to, one can add 1 to reach a still higher number. The numbers are thus created by the counting, not labeled by the counting. The numbers are not 'out there' waiting to be counted; they are created by the counting. In that sense, their infinity is merely potential.”

(B2) August 05, 2010 at 11:33 AM:

“But if there is no actual infinity of naturals, then what does 'There is no greatest natural number' mean? It means something like this: the computational process which generates a successor for any given n can be iterated indefinitely. As opposed to what? As opposed to a computational process that cannot be iterated indefinitely, e.g. the process of reducing a fraction to its lowest common denominator, e.g. 4/8 to 2/4 to 1/2.”

(B3) August 06, 2010 at 11:33 AM:

“Numbers are constructed or generated. They do not exist in themselves independently of a process of construction. The computational process which generates a successor for any given n can be iterated indefinitely. In this sense the infinity of natural numbers is potential only, not actual.”

Note: I have included (B2) and (B3) because in the former Bill uses the notion of a ‘computation process” that is indefinitely iterated in order to explicate what the phrase ‘There is no greatest natural number’ means, whereas in the later it is used to explicate the notion of ‘potential infinity’. These two tasks may not be equivalent. Hence, prima-facie we need to view (B2) and (B3) as addressing different problems.

3) All of the above formulations suffer from several systematic problems:

(3.1) They all seem to rely essentially on the modal claim that something *can be* done (all three quotations include the modal term ‘can’)

(3.2) All three contain either an explicit temporal reference (e.g., (B1) contains the term ‘endlessly’) or an implicit temporal reference (e.g., (B2) and (B3) contain implicit reference to some process that can be *iterated indefinitely*, where this last notion may be given a temporal interpretation).

(3.3) If we replace the problematical ‘potential infinity’ with something like ‘potentially indefinite’ or ‘potentially unbounded’ and view Bill’s formulations as aiming to explicate the later notions, then (B2) and (B3) are circular. They explicate the notion of an ‘indefinite series’ with the notion of an indefinite process (of calculations).

4) It should be obvious that any explication of the desired notions that presupposes an antecedently existing infinite series of temporal points will violate condition (a) and for this reason will be unacceptable. The big question is whether such explicit or implicit references can be eliminated without reminder while preserving an account that satisfies condition (b). Bill gravitated lately towards such an account by speaking of a computational process that *can be iterated indefinitely*. So formulation (B1) is now rejected in favor of (B2) or (B3).

4.2) I have stated several times that the modal notion that is employed here needs to be clarified. Even if we think of the modal notion in connection to a “computation process” that can be iterated, the character of the computational process in question needs to be made explicit. A computational process or *algorithm* can be described abstractly or in terms of one or more of its physical realizations. The later will not work because of the obvious physical limitations on all such realizations. So Bill will have to retreat to a non-physical notion of a computational process or algorithm, perhaps along the lines described by Turing. And now several questions arise.

4.3) First, does the question of whether a computational process that is iterated *indefinitely* halts at a certain point equivalent to the famous halting problem? I am not skilled mathematically to answer this question. However, since I know that the halting problem is undecidable, then *if* the two questions are the same, then the question of whether a computational process that is iterated *indefinitely* halts at a certain point or does not is also undecidable. But then we cannot tell whether such a computational process will produce a last number and then simply halt. Yet if that should be the case, then such a computational process would violate condition (b) and produce the largest number and, worst yet, we would never know.

4.4) Second, if the computational process is viewed abstractly, then why can’t it compute all the actual infinity of natural numbers at once? After all, such a computational process is not temporally limited; it is not subject to any constraints on processing resources, limitation on memory storage, and so on. So why can’t it simultaneously compute all values of n: n is a natural number, all at once? I see no reason why we can’t simply describe the instantaneous production of all n, n+1, (n+1)+1,…etc., as simultaneous iteration. Moreover, since such an algorithm or computational process is not created by us (although it may be described by us) and it could produce all the actual infinity of natural numbers at once, the current antithesis (anti-actual-infinity) reduces to the thesis (pro-actual-infinity), albeit through the synthesis of an abstract computational process that is iterated indefinitely and, thus, produces all the infinity of natural numbers at once.

4.5) Third, there is nothing inherent in the notion of ‘iteration’ that tells us the number of times the process of iteration is performed. There are finite iterations, infinite ones, and processes of iteration that are not antecedently determined. Thus, when we attempt to explicate the notion of ‘indefinite iteration’, the burden of explication falls on the notion of ‘indefinite’. And this later notion packs several meanings: vague, indeterminate, unpredictable, open-ended, unlimited. Clearly, the first and third meanings are to be excluded for obvious reasons: the intended use of the term ‘indefinite’ is not in the sense of vague or unpredictable. The second sense; i.e., ‘indeterminate’ may also not be pertinent in the present context. For ‘indeterminate’ means that a given end results is not determined by initial conditions and perhaps some pertinent laws. But surely we are not examining here a purely physical process. So the only senses of ‘indefinite’ left are that the iteration of the computational process is open-ended or is unlimited by anything other than the inherent characteristics of the algorithm that specifies the computation. In this case, the algorithm is simple: start by computing 0; after that add 1 to every number previously computed. But, clearly, such an algorithm has no inherent characteristics that will inhibit the process of computation at any given number. Hence, since nothing external to the computation process inhibits its computations and nothing inherent to the algorithm inhibits the computation, we face once again the problem raised in (4.4): namely, why don’t we have here an actual infinite series of natural numbers that are produced all at once?

4.6) The open-ended conception of ‘a potentially indefinite series of natural numbers’ raises a serious problem about the identity of the series of natural numbers. Suppose we waive all of the above difficulties and we allow that at any given time, the computational process iterates the algorithm to produce the successor of the last number it produced. Then the phrase ‘the set or series of natural numbers’ becomes indefinitely ambiguous. Since sets or series are identical just in case their members are, a series of numbers produced at t1 and a series of numbers produced at t2 will be different, since they do not share *all* their members. Hence, we now have the natural number series at t0 which is different than the one at t1, which is different than the one at t2, …, and so on. Thus, there is no one series of natural numbers as we normally conceive of this series. Instead we have an indefinite number of such series each different from the rest. And how many members this family of different series of natural numbers contains? We simply do not have an answer to this question? And, perhaps, there cannot be an answer if the issue raised in (4.3) about the halting problem is pertinent. Yet even without the halting problem, our conception of the set of natural numbers is that there is one such set, not an indeterminate multiplicity of them. (The present argument is based on Shaughan Levine’s “Understanding the Infinite”; pp. 158-9).

5) Conclusion: I think that the notion of a ‘potentially infinite series’ is a misnomer. The natural replacement of this notion; i.e., the notion of a ‘potentially indefinite or unbounded series’ faces some serious obstacles that need to be addressed. I have tried to state some of these in the present post. It is worth mentioning that a Structuralist account along the lines initiated by Benacerraf faces some difficulties of its own and moreover is not directly applicable to the proponents of (1) in Bill’s original triad. First, such a structuralist account speaks about progressions. But, progressions of what? It can’t be a progression of physical objects since there are only finitely many of those and these progressions are allowed to be infinite. Hence, some proponents of this view refurbish Structuralism with a modal account. Second, Structuralism’s principal point is to deny that mathematics is about any objects at all, abstract or concrete. The issue here is less about infinity and more about whether mathematics leads to a commitment to a realm of mathematical objects. Hence, it is not clear to me that the proponents of (1), who are motivated by denying infinity but accept finite sequences of numbers, can find a natural ally in Structuralism.

Posted by: Account Deleted | Thursday, August 12, 2010 at 10:20 AM

Peter,

Regarding 4.3 of your latest post. The halting problem is a problem which refers to a single execution of a computer program assumed to be implemented on a physical machine. To begin to translate this statement into the language of 4.3, I would say that the problem applies to a single "iteration", not to an entire sequence of iterations. This observation would not seem to cast doubt on your basic argument. In fact, what it does is force us to distinguish between computer programs (a concrete, physical object) and algorithms ( an abstract object). Turing's result is, roughly, that we cannot, in general, decide in advance whether a given program will stop in a finite time. This would seem to presuppose that the algorithm of which the program is a physical instance is abstract in the sense of "not bounded in time". The account I am giving here no doubt raises several questions about the terms involved and the relationships among them which I am not prepared to answer, but I hope it sheds some light on that one point.

Posted by: Phoenix Kyle MacGregor | Thursday, August 12, 2010 at 05:28 PM

Phoenix,

Thank for your comment on the halting problem. I wonder whether it makes sense to speak of a computer program that is implemented on a physical machine when it is assumed that there are no temporal limitations and other limitations on resources such as memory are lifted.

Gleaning from your last few sentences in the post, I think that we basically agree on the description of the problem. My concern is to insure that the equivalence I proposed between the two questions holds and then determine precisely what are the consequences of the undecidability of the halting to an account of 'indefinite series' which Bill proposes, if the equivalence between the questions holds.

Posted by: Account Deleted | Friday, August 13, 2010 at 06:58 AM