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Monday, August 02, 2010

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I am just as Peter Lupu inclined to accept (3) over (1). I see a more fundamental problem here though. I have a strong suspicion that (3.) is logically prior to (1.) in the sense that all attempts to establish either (1.) or ~(1.) implicitly assume (3.) or ~(3.). I will try to show that dr. Vallicella's critique of arguments for ~(1.) implicitly assumes ~(3.).

To make (1.) believable, dr. Vallicella begins by claiming that it is not clear that there exist infinitely many numbers. What are numbers though? If my memory doesn't deceive me, the following definition is due to Russell: numbers are equivalence classes of sets w.r.t the relation of bijectivity (footnote 1). To make the definition not explicitly dependent on (3.), we can weaken it to:

(*). Numbers are equivalence classes of Williamesque collections of concrete objects w.r.t. the relation of bijectivity.

I claim that (*) captures the essence of what we call 'numbers'. Number twelve is that which is common to twelve apostles, twelve turnips and the Dirty Dozen. Note that 'twelve' in 'twelve apostles' is not the abstract number that I am defining. 'Twelve apostles' is an expression used to point to a particular collection of people. Here's the rub. If sets exist, then clearly by former definition there are infinitely many numbers. If sets do not exist and consequently the best definition of numbers is (*), then there are finitely many numbers because there are finitely many concrete objects.

The second issue is that of propositions. Dr. Vallicella anticipates and criticises an argument purporting to prove that there are infinitely many propositions if there is at least one proposition. The argument goes as follows. Let T denote an 'it is true that' operator from the class of propositions to the class of propositions. That is, for any proposition p

T(p) := 'it is true that p'.

If there exists a proposition p, we construct an infinite family of propositions indexed by natural numbers by iterating the operator T:

p_n := (T^n)(p).

Dr. Vallicella claims that (p_n) may be infinite only potentially. I claim that the question reduces to the question of the existence of an infinite collection of natural numbers. For if there are infinitely numbers 'out there', then there is an infinite collection of operators (T^n) which gives rise to an infinite family (p_n). If, however, numbers need to be computed to become real, so have the operators T^n and consequently propositions p_n. The question of existence of the natural numbers has been shown to be posterior to the question of the existence of sets.

I suspect every argument purporting to establish ~(1) is similar to the one described above. That is, it starts with one object of some kind and then iterates a procedure that 'creates' one new object with every step. It reduces to the question of the existence of sets in the way described. It seems to me that there is no easy way out; we need to tackle the question of truth of (3.) head on.

Footnote 1: This actually defines cardinal numbers. To get the natural numbers we need to restrict the definition to finite sets. We may define finiteness for sets without using the concept of numbers in many ways. For example, a set S is finite iff it is not bijective with a union of two sets bijective with S (loosely speaking, if two copies of S contain a different number of elements than S).

Bill,

As you know I'm somewhat 'modally challenged' so please forgive this interjection if it's completely wrong-headed.

Taking (3) to mean 'sets exist in the actual world', (3) and (2) together imply that the number of objects in this world is infinite. But (1) could still hold if there were a possible world in which no sets existed. So I'm not sure that the triad *is* inconsistent.

David,

You are right. (3.) is meant to say 'sets exists necesarilly'. It at least plausable to assume that if mathematical object exist, they do so necesarilly.

Response to Bill,

(I) Bill maintains that the aporetic triad of (1)-(3) is collectively inconsistent, yet each proposition included is individually plausible. He makes two stipulations:

(S) The modal notions included in (1) and (3) should be interpreted in a “broadly logical” sense;
(S*) The notion of ‘set’ is to be understood in the context of ZFC.

(i) Stipulation S raises the following issue: what is the background logic relative to which we interpret the modal notions involved in (1) and (2)? If the underlying logic is classical logic including the excluded middle principle, then clearly there is no reason to reject classical mathematics and then (1) is false. On the other hand, if the background logic is intuitionistic logic (or any of its allied approaches), then the background mathematics will be intuitionistic as well and (1) may very well turn out to be true, in which case we must reject (3). However, the later approach also commits us to reject the excluded middle principle. This approach then has a price well beyond merely accepting (1) and rejecting (3).

(ii) Stipulation S* also raises certain serious problems. If our very notion of a set is to be considered *only* within ZFC, then we are liable to view the axioms of ZFC as somehow capturing all the truths there are to be captured about the notion of a set. I think Bill’s suggestion can easily slide into such a view. But due to various independence results; that is results pertaining to undecidable propositions such as for instance the Continuum-Hypothesis which is independent from the axioms of ZFC, we already know that ZFC is too weak to entail all the truths even about the very existence of certain kind of sets (e.g., are there infinite sets between the countable and the continuum?). I think that one reasonable conclusion we may draw from the existence of independence results is that no one single set of axioms (including ZFC) exhausts all truths about sets. In fact an even stronger conclusion may be entertained: perhaps, the axiomatic system itself cannot capture all the truths about sets. So I would be reluctant to agree to accept S* as a stipulation without further reflection.

(II) Bill’s post includes two responses to some things I have said in previous posts:

(A) Those who opt to accept syllogism S1 can accommodate most of our intuitions, including intuitions about mathematics, by rejecting *actual infinity* in favor of some notion of *potential infinity*.

(B) Even if there are abstract objects, it does not follow that there isan “actual infinity of them (i.e., of abstract objects).”

(III) Reply to (A).

(i) There are those who object to the notion that an actual infinite collection of natural numbers exists on the grounds that the very concept of an ‘infinite collection’ is somehow incoherent. But, then, they face the burden of explaining how adding ‘potential’ to an already incoherent concept yields a coherent concept. Of course, Bill might reply that it is the notion of an *actual* infinite collection that is objectionable, not the notion of infinity per-se. Hence, if we are willing to substitute ‘potential’ for ‘actual’, then the resulting notion of ‘potential infinite collection’ is fine. But this reply seems on the face of it odd. After all, by analogy of reasoning one might just as well say that while nothing can *actually* be both a square and a circle simultaneously, this fact does not rule out that something could *potentially* be both a square and a circle simultaneously. But, surely, such a claim is incoherent.

(ii) An acorn has the potential to become an oak tree. But the acorn has this potential only because there are actual oak trees or at the least because it is possible for there to be actual oak trees (so as to take into account novel hybrid trees or plants). 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.

(iii) Bill most likely will accuse me of completely misinterpreting the intended meaning of ‘potential infinity’. The notion of ‘potentiality’ in the present context is not the same as when we say that an acorn has the potential to become an oak tree. Since there are actual oak trees, an acorn has the potential to evolve into one. But since there cannot be an actual infinite collection, say of the natural numbers, no finite sequence of them can be said to somehow gradually unfold and become an infinite series.

(iv) Instead, Bill might offer the following explication of the notion of a potential infinite collections, say of the natural numbers:

“So someone who denies that there are infinite sets can say something like this: for any n that you have counted up to, you can alway add 1, and the result will be a nat'l number. We will all agree that the natural numbers are closed under addition. But as far as I can see, it does not straightaway follow that there is a math. set of natural numbers.” (Saturday, July 24, 2010 at 06:21 PM)

So you imagine yourself spending your life counting numbers. Perhaps we extend this image so that the whole human race does nothing but count numbers. To say that the set of natural numbers is *potentially infinite* is to say that for every number so far counted by you or the whole human race, you or the human race “can always add 1”. Thus, according to this interpretation there is no largest natural number because for any given number already counted, someone can always add 1 in order to obtain a larger number.

But this proposal just won’t do. No finite being or even thing (e.g. computer) can satisfy the “can always add 1” requirement. If by ‘always’ we mean the whole infinite sequence of the time series, then it is not the case that a physical thing (an individual human, the whole human race or a computer) can always add 1 for there comes a time when an individual, the whole human race, or a computer will perish and once it perishes, clearly it cannot add 1. And if by ‘always’ we mean a finite segment of the time series, then there is going to be some number that is the largest number counted.

(v) In order to avoid the above objection, Bill might propose (he did) that the entity which does the counting in his formula is some sort of an “ideal mind” or some such thing that is not perishable and therefore it is not subject to the limitations to which all physical entities are subject. And what exactly are the limitations we wish to avoid? Well, as noted in (iv) above, we want an entity the existence of which does not begin at a given time and does not end at any time. Entities such as these have a name; they are called “abstract objects.” So Bill proposes to avoid the above objections by stipulating that some kind of an abstract entity, call it X, (X is perhaps an ideal mind or, as my friend Phoenix suggested, an abstract algorithm) is such that for any natural number it counts, it can always add 1.

(vi) The above dialectic led Bill to concede that the proponents of S1 must admit into their ontology at least one abstract object. So now we must proceed from claim A to claim B.

(IV) Reply to (B).

(i) Let me here restate my claim regarding abstract objects and infinity:

(C) If there is at least one abstract object, say of the sort X, then there is at least one infinite collection of objects (not necessarily of the same kind as X).

Bill maintains that it makes sense to deny (C). That is, it makes sense to hold that:

(~C) There exists at least one abstract object, say of the kind X, & it is not the case that there exists an infinite collection of objects of any kind.

(ii) To say that X is an abstract object is to say that it is not in space and time. But this formulation won’t do because ‘in’ is a spatial notion. So what is really meant by denying that abstract objects are *in* space and time is that an abstract object is not subject to physical change, it has no physical properties, and its existence cannot be mapped into any finite time segment. The last point can be put as follows. Assuming an infinite time sequence, the existence of an abstract object is isomorphic to the whole time sequence.

(iii) Let us now return to our X. X’s existence is isomorphic to the whole infinite time sequence; for if it is not so isomorphic, then its existence has a limited time span and, therefore, X is no longer abstract. Since X’s whole existence is to spin out numbers so that for every number it spins out, it will always spin out another number greater by 1, we can safely conclude that X will produce an infinite series of natural numbers corresponding to its lifespan which in turn is isomorphic to an infinite sequence of temporal points. Hence, if an abstract of object exists, then there must be some infinite sequence of objects. Therefore, (C) is true and (~C) is false.

(iv) Bill will very likely balk at the notion that the existence of an abstract object should be viewed as isomorphic to an antecedently given infinite sequence of temporal points. But, then, he needs to explain what does it means to say that X is abstract. He makes the following claim:

“The following three claims are consistent: (i) Declarative sentences express propositions; (ii)Propositions are abstract; (iii) Propositions are man-made. Karl Popper's World 3 is a world of abstracta. It is a bit like Frege's Third Reich (as I call it), except that the denizens of World 3 are man-made.”

How can (iii) above be true of any abstract object, be that a proposition, a set, a number, an algorithm, a form (in Plato’s sense) or whatever? If an abstract object is manmade, then it came into existence at a particular time: i.e., the time at which it was made by someone. But the very concept of an abstract object mentioned in (ii) precludes any possibility that an abstract object came into existence at a particular time. I have no clue what Popper’s World 3 is, but it is certainly not the sort of abstract object that is said not to be “in” space and time. Perhaps, what Popper means by his World 3 is that this world was created by *abstraction*. But whatever we get by the process of *abstraction*, that thing is not the same as an abstract object. So contrary to Bill, I do not think that claims (i)-(iii) are consistent. Claim (iii) entails that propositions (or whatever) came into existence at a given time: i.e., the time they were created by human beings. Whereas claim (ii) entails that propositions are abstract and, hence, by the very nature of abstract objects, they cannot come into existence at a particular time. If Bill is willing to deny this later entailment, then I have no clue what he means by the notion of an ‘abstract object’. It is now his burden to tell us what he means by this notion.

(v) Let me now approach matters from a slightly different angle. Suppose we accept into our ontology one abstract object such as X. What consideration will prevent us now from allowing infinitely many objects such as sets, numbers, propositions, or whatever? Certainly these considerations will not be epistemic in nature, for any epistemic qualms against infinite sets of numbers, propositions, or whatever has been already undercut by our acceptance of at least one abstract object. So if we allow a realm of abstract objects, then I say we might as well allow infinitely many of them. Moreover, once our epistemic timidity is lifted by allowing abstract objects, there is really no good reason to give in to intuitionistic restrictions or accept (1) on some other grounds. Bill will have to explain to me whether there are grounds other than epistemic grounds to reject (3) and accept (1).


David and Jan,

Jan is right. 'Sets exist' is short for 'Necessarily, sets exist' which means that, necessarily, there are sets. That is not to say, however, that every set exists necessarily. The unit set consisting of me is as contingent as I am.

The triad is clearly inconsistent.

Peter and Jan,

Jan raises the question whether numbers are sets. Paul Benacerraf in a seminal article "What Numbers Could Not Be"argues that they cannot be sets. He also argues that numbers cannot be objects at all.

Suppose he is right. Then I have yet another way of rebutting Peter's rebuttal of (1). Peter assumes that numbers are objects. But if they are not, then his main objection to (1) collapses. Benacerraf: "Arithmetic is therefore the science that elaborates the abstract structure that all progressions have in common merely in virtue of nbeing progressions. It is not a science concerned with particular objects -- the numbers."

Peter,

Thank you for all the rich commentary.

You rightly point out that talk of 'broadly logical' modality presupposes a background logic. But it might presuppose only a small fragment of standard logic. For example, we might only need LNC (and not LEM) to pin down BL-possibility. (To be precise, more is needed because of the 'broad' part: certain nonlogical necessary truths.)

To keep it simple, let's discuss just narrowly-logical possibility. We define: S is NL-possible =df S does not contravene LNC.

It looks as if we don't have to worry about LEM for purposes of explaining NL and BL possibility and their duals.

Bill,

(a) Background logic.

(i) Clearly I will object to have as a background logic a fragment that fails to include the excluded middle principle, for then you simply presuppose some version of intuitionistic logic and, thus, help yourself to something that favors (1) over (3).


(ii) But, suppose you do so. Then first it is no longer the case that every proposition in the set {(1), (2), (3)} is either true or false. One or more could have no truth value, unless we have a proof for it. Second, how would you interpret the modality in (1)-(3).

(b) Benacerraf's Structuralism.

(i) I am not sure I understand Benacerraf's Structuralism even in a rudimentary way. In any case, what are these progressions progressions of? Physical objects? If so, then each of these progressions will be a progression of a determinate number of physical objects. Does that mean there is a largest natural number? Moreover, if so then progressions will have in common many properties that are not relevant to the study of arithmetics.

(ii) How many progressions are there? Or to put it differently: "all progressions" quantifies over how many progressions: infinitely many progressions or only finitely many of them?

Bill,

(a) Background logic.

(i) Clearly I will object to have as a background logic a fragment that fails to include the excluded middle principle, for then you simply presuppose some version of intuitionistic logic and, thus, help yourself to something that favors (1) over (3).


(ii) But, suppose you do so. Then first it is no longer the case that every proposition in the set {(1), (2), (3)} is either true or false. One or more could have no truth value, unless we have a proof for it. Second, how would you interpret the modality in (1)-(3).

(b) Benacerraf's Structuralism.

(i) I am not sure I understand Benacerraf's Structuralism even in a rudimentary way. In any case, what are these progressions progressions of? Physical objects? If so, then each of these progressions will be a progression of a determinate number of physical objects. Does that mean there is a largest natural number? Moreover, if so then progressions will have in common many properties that are not relevant to the study of arithmetics.

(ii) How many progressions are there? Or to put it differently: "all progressions" quantifies over how many progressions: infinitely many progressions or only finitely many of them?


This is very strange. Twice now I posted something and it fails to appear. So I post it again and suddenly both appear. Wonder why?

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