Why Not be a Nominalist about Sets?

The resident nominalist comments:

Nominalists say that the conception of an actual infinity of natural numbers depends on there being a set of all such numbers. But Ockhamists do not believe in sets. They say that the term ‘a pair of shoes’ is a collective noun which deceives by the singular expression ‘a pair’. Deceives, because it means no more than ‘two shoes’, and if there is only a pair of shoes, then there are only two things. But if a ‘pair’ of two things is a single thing, there are three things, the two things and the pair. Ergo etc.

I agree that there cannot be an actual infinity of natural numbers unless there is a (mathematical as opposed to commonsense) set of all such numbers. But of course this holds for all numbers, rational, irrational, transcendental, etc. Indeed, it holds for any category of item that is actually infinite. If there is an actual infinity of propositions, for example, then there must be a set of all propositions. I would point out however that there is nothing nominalistic about our friend's opening remark.

Nominalism kicks in with the claim that there are no sets.  What there are are plural referring devices such as 'a pair of shoes' which fools us into thinking that in reality, i.e., extralinguistically, there are three things, a left shoe, a right shoe, and the pair, when there are only two things, the two shoes.  The same goes for the following seemingly singular but really plural phrases: a gaggle of geese, a pride of lions, a parliament of owls, a coven of witches, etc.   

This all makes good sense up to a point. When I put on my shoes, I put on one, then the other. It would be a lame joke were you to say to me, "You put on the left shoe and then the right one; when are you going to put on the pair?" To eat a bunch of grapes is to eat each grape in the bunch; after that task is accomplished there is nothing left to do.  The bunch is not something 'over and above' the individual grapes that I still need to eat.

Consider now the Hatfields and the McCoys. These are two famous feuding Appalachian families, and therefore two pluralities. They cannot be (mathematical) sets on the nominalist view.  But there is also the two-membered plurality of these pluralities to which we refer with the phrase 'the Hatfields and the McCoys' in a sentence like 'The Hatfields and the McCoys are families  feuding with each other.'

If, however, a plurality of pluralities has exactly two members, as in the case of the Hatfields and the McCoys -- taking those two collections collectively -- then the latter cannot themselves be mere pluralities, but must be single items, albeit single items that have members. They must be both one and many. That is to say: In the sentence, 'The Hatfields and the McCoys are two famous feuding Appalachian families,' 'the Hatfields' and 'the McCoys' must each be taken to be referring to a single item, a family, and not to a plurality of persons. For if each is taken to refer to a plurality of items, then the plurality of pluralities could not have exactly two members but would many more than two members, as many members as there are Hatfields and MCoys all together. Compare the following two sentences:

1. The Hatfields and the McCoys number 100 in toto.

2. The Hatfields and the McCoys are two famous feuding Appalachian families.

In (1),'the Hatfields and the McCoys' can be interpreted as referring to a plurality of persons as opposed to a mathematical set of persons. But in (2), 'the Hatfields and the McCoys' cannot be taken to be referring to a plurality of persons; it must be taken to be referring to a plurality of two single items.

Or consider the following said to someone who mistakenly thinks that the Hatfields and the McCoys are one and the same family under two names:

3. The Hatfields and the McCoys are two, not one.

Clearly, in (3) 'the Hatfields and the McCoys' refers to a two-membered plurality of single items, each of which has many members, and not to a plurality of pluralities. And so we must introduce mathematical sets into our ontology.

My conclusion, contra the resident nominalist, is that we cannot scrape by on  pluralities alone. (Man does not live by manifold alone! He needs unity!) We need mathematical sets or something like them: entities that are both one and many.  A set, after all, is a one-in-many. It is not a mere many, and it is not a one 'over and above' a many.  The nominalist error is to recoil from the latter absurdity and end up embracing the former.  The truth is in the middle.

What I have given is  an argument from ordinary language to mathematical sets. But there are also mathematical arguments for sets. Here is a very simple one. The decimal expansion of the fraction 1/3 is nonterminating: .33333333 . . . . But if I trisect a line, i.e., divide it into three equal lengths, I divide it into three quite definite actual lengths.  This can be the case only if the the decimal expansion is a completed totality, an actual infinity, not a merely potential one.  An even better example is that of the irrational number, the square root of 2 -- it is irrational because it cannot be expressed as a ratio of two numbers, the numerator and the denominator of a fraction as in the case of of the rational 1/3.  If the hypotenuse of a right triangle is   units of length, that is a quite definite and determinate length.  How could it be if the decimal expansion however protracted did not point to a completed totality, an actual infinity?

 

REFERENCES

Max Black, "The Elusiveness of Sets," Review of Metaphysics, vol. XXIV, no. 4 (June 1971), 614-636.

Stephen Pollard, Philosophical Introduction to Set Theory, University of Notre Dame Press, 1990. 

Comments

[oz]

We discussed Hatfields and McCoys a few years ago, and Ockhamists replied.

You say "'the Hatfields and the McCoys' cannot be taken to be referring to a plurality of pluralities; it must be taken to be referring to a plurality of two single items". I do not follow. "The Hatfields" refers to all and only the Hatfields, "the McCoys" refers to all and only the McCoys.

Is your point that we must interpret 'the Hatfields and the McCoys' as referring to two groups, ergo the number two must enter into all of this somehow?

[oz]

Sorry, first comment posted too soon. Your point is that there are two families. I.e. the word ‘two’ enters into the discussion by way of the number of families.

But we can equally say ‘two pairs of shoes’. We mean a pair of shoes and another pair, which is equivalent to “these two shoes the these other two shoes”. ‘Pair’ here functions here like ‘families’ in the Hatfield example.

Is your point (1) that once we get more than one pair we are forced into a non-nominalist account? Or that (2) the word ‘family’ does the heavy work, i.e. you are happy with the nominalist account of ‘pair’, but you think the word ‘family’ compels us to introduce a special sort of entity?

[Michael Brazier]

For one thing, "a pair of shoes" means more than "two shoes" - a pair of shoes is one left shoe and one right shoe that are nearly mirror images, intended to be worn at the same time. Similarly a "family" is more than a bunch of people; all the people in it must be related in certain ways to each other, e.g. blood, marriage or adoption. It's the relations among the collected things imputed by these words - relations which exist outside our language - that create a problem for nominalism. Further, while "pair" includes a specific number in its definition, "family" does not, making it impossible to construe it as a mere plural in any context.

[Brian Bosse]

Hello Bill,

I am over my head here; so, moderate accordingly.

3. The Hatfields and the McCoys are two, not one.

Clearly, in (3) 'the Hatfields and the McCoys' refers to a two-membered plurality of single items, each of which has many members, and not to a plurality of pluralities. And so we must introduce mathematical sets into our ontology.

It seems you are saying that someone who asserts (3) is thereby asserting the existence of a set made up of two families - the Hatfields and the McCoys. If so, are you assuming some Indispensability Argument to come to this conclusion?

Brian

[oz]

The decimal expansion one is trickier

[BV]

Brian,

If truth be told, I am over my head too. I am neither a mathematician, nor a philosopher of mathematics strictly speaking. You know my motto: Nescio ergo blogo!

There is a typo is what you quoted me as saying. It should read:

Clearly, in (3) 'the Hatfields and the McCoys' refers to a two-membered plurality of single items, each of which has many members, and not to a plurality OF PERSONS. And so we must introduce mathematical sets into our ontology.

My point is that only in some cases can a plural referring phrase such as 'the Hatfields and the McCoys' be taken to refer to nothing more than 100 persons (or whatever the number is). In other cases, this same phrase is used to refer to two families. How many families are involved in the feud? Two. How many persons? 100.Therefore, a family cannot be reduced to a number of persons; it is not a mere manifold, or a mere many, it is is a one-in-many. And so the nominalist reduction fails in some cases.

[BV]

Brian et al.:

Suppose the Hatfields have the McCoys surrounded. That cannot mean that each Hatfield has one or more McCoys surrounded. Simpler case: suppose the Hatfields have one McCoy surrounded. That cannot mean that each Hatfield has that one McCoy, Gomer for example, surrounded. That suggests to me that there is more to the Hatfields (and any family or group) than their mere membership. The nominalist reduction fails.

But I hasten to add that it also makes no sense to say that an abstract object -- which is what math sets are -- has Gomer surrounded. My point is negative, not positive: I am maintaining that a collectivity such as a family cannot be reduced to its members taken distributively, i.e., one-by-one. The problem seems to be this: how can we account for the unity of a one-in-many without hypostatizing the unity?

Our nominalist friend, sober Englishman that he is, sees the danger of hypostatization, but, as it seems to me, goes to the opposite extreme.

[BV]

Michael Brazier,

Excellent comment. I agree.

You are right to bring up relations. How will our nominalist accommodate relations? Will he try to reduce them to their monadic foundations in concrete individuals?

A set theorist can try to reduce relations to sets of ordered n-tuples, but that option is not available to our nominalist.

[BV]

Oz writes: >>Is your point (1) that once we get more than one pair we are forced into a non-nominalist account? Or that (2) the word ‘family’ does the heavy work, i.e. you are happy with the nominalist account of ‘pair’, but you think the word ‘family’ compels us to introduce a special sort of entity?<<

If I have one pair of shoes, then I have two shoes, and thus two spatiotemporal particulars. Obviously, the pair is not a third spatiotemporal particular. That is blindingly evident, and it is part of what you are saying. I am happy with that part of what you say. But you go further: you think that 'a pair of shoes' does not refer to ANY third item; it refers precisely and only to the two shoes. And so the phrase cannot refer to a set. I take it that you view is that 'a pair of xs' cannot refer to anything in reality outside the mind and outside language except the two xs taken distributively.

What I am saying in the main post is that in some cases the nominalist reduction of a plurality to its members works, but in other cases it doesn't.

Suppose you say to your wife, "Hand me my shoes." She hands you one of yours and one of hers. She has handed you a pair of shoes. Now what is the difference between the pair that she has handed you and the pair that you can wear? Each is a pair of shoes. There has to be some difference in reality, no? What does that real difference consist in, given your nominalist reduction of pairs to their members?

[oz]

>I am maintaining that a collectivity such as a family cannot be reduced to its members taken distributively, i.e., one-by-one. <

Correct, which is why we need plurals, and plural quantification.

People above have objected that 'pair' means more than than just two. OK, then try 'couple'. The point is that we should be misled by the singular case of a collective noun. The singular indefinite article in "A dozen eggs" does not signify any single thing over and above 12 eggs.

Re your tricky decimal argument, I have the answer.

Suppose there is a line of length 2. We can split the line into two parts, each of length 1. Then suppose that any (plural quantifier) parts of the line can be divided into further parts. This supposition is what gets us a potential infinity, but not an actual one. For if we could speak of 'all' parts of the line, these would be 'any' parts, and further subdivision will be possible, hence there cannot be 'all' parts.

>How will our nominalist accommodate relations?

Simple, by the means of n-place predicates. "- is married to -" is a 2 place predicate. CJF Williams (a nominalist) was keen on that.

[oz]

>Now what is the difference between the pair that she has handed you and the pair that you can wear?

The difference is that in one case each shoe is mine, in the other case, one of them is hers.

Wittgenstein: "Philosophical clarity will have the same effect on the growth of mathematics as sunlight has on the growth of potato shoots. (In a dark cellar they grow yards long)."

[Michael Brazier]

>Re your tricky decimal argument, I have the answer.

>Suppose there is a line of length 2. We can split the line into two parts, each of length 1. Then suppose that any (plural quantifier) parts of the line can be divided into further parts. This supposition is what gets us a potential infinity, but not an actual one. For if we could speak of 'all' parts of the line, these would be 'any' parts, and further subdivision will be possible, hence there cannot be 'all' parts.

That's not an answer, it's a nonsequitur. The point is that the decimal expansion of 1/3, an infinite series of fractions, is quite genuinely equivalent to the fraction 1/3 itself; you can substitute one for the other in any computation and get the same result. If infinite sets couldn't be completed this would be impossible - an error of the same kind as dividing by zero. For how can a sum that can't be computed, and thus can never concretely exist, be equal to anything at all?

> >How will our nominalist accommodate relations?

>Simple, by the means of n-place predicates. "- is married to -" is a 2 place predicate.

But what kind of entity is a binary predicate? It clearly isn't either of the things it's predicated of. In fact it can't be any concrete individual at all. So for the nominalist it must be a concept, and thus exists only in language, with no referent in reality. But that would entail that "oz is in London" is neither true nor false - even though oz and London are concrete individuals. Or is there a way to define a proposition as "true" if it doesn't correspond to anything real?

[oz]

Brazier:
>If infinite sets couldn't be completed this would be impossible

What does 'completed' mean here?

Do you mean that, on the nominalist account, we can't speak of every natural number? Or every member of an infinite expansion?

>But that would entail that "oz is in London" is neither true nor false

I don't see why that would follow. The 2-place predicate '- us in -'can be satisfied by oz and London respectively.

[BV]

Oz,

Michael B's objections are solid. Isn't it obvious that 1/3 of a line segment is not equal to .3 of a line segment? 1/3 is not equal to 3/10. Nor is a it equal to .33 of a line segment, or .333, etc. 1/3 is equal to .333333 ... where the nonterminating decimal expansion is actually infinite. These ideas are of course not mine; they come from Georg Cantor.

The other objection is solid as well, but it needs to be put more clearly. Later.

[Michael Brazier]

To expand on the decimal expansion: we can't compute a sum until we have all the summands in hand. A decimal expansion is an infinite sum, therefore to compute it we must have an infinite set of summands. And this has to be an actually infinite set - as long as any summand is still potential, we can't do anything with it, so the sum is not complete.

On a nominalist account, then, we can't coherently speak of the sum of an infinite sequence; it can't be computed, so it doesn't exist, so nothing can be said about it. Calculus disappears in a puff of logic, and Newton and Leibniz were fools wrapping themselves in webs of linguistic confusion.

[Paul Smith]

Is a left shoe a single, irreducible thing? It too consists of a sheer unmanageable degree of compositional complexity, which we often abstract away by calling the thing a "shoe". A single shoe exhibits a degree of physical connectedness and persistence that a pair does not, though in principle we could tie the shoes together to ensure they always appear as a pair.

It is a common procedure in practice: combine a large quantity of properties -- pairs of real numbers, epsilons, deltas, relations and quantifiers such as "less than, "for all", "there exist", etc. -- and use the entire bundle to define an object called, in this case, for example, a "continuous function". The defined object can then be conceived of as a point, that is, a featureless and finished thing, in a function space, which itself consists entirely of such points. If you're willing to accept, say, a real number -- which is designated by an equivalence class of certain sequences of rational numbers, one element of which class is its decimal representation -- as an existing object, then pairs and correspondences of them should cause no additional difficulty. Similarly, if you can accept "zero", then the set of natural numbers, that is, a timeless object that is described by the Peano Axioms, but not constructed or created by them, can be accepted as well.

It is often the case that we observe a certain regularity: numeric, geometric, probabilistic, relational, physical -- and after compiling a list of some of those items that exhibit that regularity, we look for the common features. Thereupon the mathematician can define a consistent logical system that encodes those features and the relations between them, and proceed to investigate the necessary consequences. For heuristic purposes, he can refer to the original physically observable regularities. With enough skill he can, by "observing nature", be inspired to formulate theorems that should be proveable in the system. The abstract objects described by the formalities are not waiting for us to name or observe them before they come into existence. God knows them all, so to speak.

The ability to mentally translate back and forth between the "thingification" of the defined object, and the object's defining individual properties, is a fundamental element of the mathematical handicraft. Regardless of the status of reality or existence, it would be prohibitively tedious to have to name all the parts of any item before discussing it. And if we find the abstract theory useful, and carry on long enough to accustom ourselves to its nomenclature and methods, then we begin to believe in the existence of its objects. As von Neumann said: "In mathematics, you don't understand things. You just get used to them."

[oz]

>we can't compute a sum until we have all the summands in hand.

All we need is every summand. Do you agree there is a difference between “all the Fs” and “every F”? Let’s start with that question.

> Isn't it obvious that 1/3 of a line segment is not equal to .3 of a line segment?

Of course. And using plural quantification (perfectly consistent with nominalism) we can speak of every member of a decimal expansion. See my objection to Brazier above.

[Michael Brazier]

oz, you are the one being misled by grammar now. No, the only distinction between "all Fs" and "every F" I see is grammatical, and whatever distinction you see in it makes no difference to the argument. I think you're trying to express the difference between first-order logic (which allows only quantifiers over elements) and higher-order logics that allow quantifiers over sets; but that doesn't help you here. Your problem is to calculate the sum of an infinite series - you can't do that without first producing every term of the series, which is an actual infinity of numbers even if you don't choose to collect them into a set.

Also, you can't even describe the real numbers without quantifying over sets, so if only first-order logic is meaningful the real numbers are indescribable - but I suppose nominalists would be willing to go that far, and live within the algebraic numbers. The reality of transcendentals is already cast into doubt by nominalism, after all.

[Brian Bosse]

Hello Michael,

Thank you for your corrective comments regarding the relationship between functions and sets. I think there is an interesting discussion to be had concerning the construction of the natural numbers via recursion and the ZF Axiom of Infinity. It is my understanding that the recursive construction (definition) used for natural numbers is insufficient to prove the existence of the set of all natural numbers. This is why the set's existence is postulated.

To expand on the decimal expansion: we can't compute a sum until we have all the summands in hand. A decimal expansion is an infinite sum, therefore to compute it we must have an infinite set of summands. And this has to be an actually infinite set - as long as any summand is still potential, we can't do anything with it, so the sum is not complete.

If I am not mistaken, such "computations" are accomplished by taking the limit of a summands formula representing such decimal expansions. In other words, no addition is actually taking place; rather, a proof is provided showing that the epsilon-delta definition is met. No actual infinite set of summands is needed.

On a nominalist account, then, we can't coherently speak of the sum of an infinite sequence; it can't be computed, so it doesn't exist, so nothing can be said about it.

If the sum of an infinite sequence is defined in terms of limits, then it seems to me that nominalists can speak of such things.

Calculus disappears in a puff of logic, and Newton and Leibniz were fools wrapping themselves in webs of linguistic confusion.

I am over my head in these discussions, but I think the Calculus of Newton and Leibniz utilized infinitesimals. The limit was the formalization of this concept. As such, I don't think Calculus disappears for the nominalist.

Kind Regards,

Brian

[BV]

Oz,

>>What does 'completed' mean here?<< It means actually infinite. This is standard set-theoretical talk.

The limit of 1/n as n becomes larger and larger = 0. This is a case of what Cantor calls the "improper infinite." The proper infinite is actual or completed. The hypotenuse of a right triangle with sides equal to 1 unit = the square root of 2 units. This is a determinate length. If a line segment is composed of dimensionless points then there must be an actual infinity of them in that hypotenuse.

[BV]

Brian sez: >>If the sum of an infinite sequence is defined in terms of limits, then it seems to me that nominalists can speak of such things.<<

The limit of 1/n = 0 as n approaches ∞. The symbol ∞ as far as I know is not used in set theory. In any case, in the example I just gave, it makes some sense to speak of potential infinity. starting with n = 1, and adding 1 we get 2, and so on indefinitely. For each successive n, the fraction becomes smaller and smaller: 1/1, 1/2, 1/3, . . . 1/989785 and so on such that the series converges to 0 'at infinity.' This 'infinity' is not a number cardinal or ordinal; it is more like an endless task. Cantor calls it an "improper infinite."

The proper infinite is actual or completed. The hypotenuse of a right triangle with sides equal to 1 unit = the square root of 2 units. This is a determinate, definite, actual length. It is not an indefinite or potentially infinite length. If a line segment is composed of dimensionless points then there must be an actual infinity of them in that hypotenuse.

There is another problem with what you said. How can a nominalist admit numbers at all, and sequences thereof, whether potentially or actually infinite? The Three Stooges and the Pep Boys (Manny, Moe, and Jack) are each three in number. So the number 3 cannot be identified with either threesome. The two threesomes are each just three particular dudes. The number, however, is universal, and that is exactly what a nominalist cannot admit.

So I ask: what could a number be for a nominalist?

It could only be a word or phrase. But words are subject to type-token ambiguity, and what is a type if not a universal, a repeatable item? Beside a word needs to have a sense (Sinn) and sense is not a word-type or token but an abstract object. If our nominalist says that sense are mental, how then does he account for the fact the two or more word tokens of the same type have one and the same sense for different minds?

[oz]

oz, you are the one being misled by grammar now. No, the only distinction between "all Fs" and "every F" I see is grammatical, and whatever distinction you see in it makes no difference to the argument. <

Not at all. Let me walk through plural quantification again. “The Fs” is a plural phrase that refers to all the Fs. Do we agree that no separate existent is required other than the Fs referred to? Plural quantification simply extends the notion of plural reference to first order but plural quantifiers. See this article.

>I think you're trying to express the difference between first-order logic (which allows only quantifiers over elements) and higher-order logics that allow quantifiers over sets; but that doesn't help you here. <

The difference is between quantifiers over single elements, and quantifiers over plural elements.

>Your problem is to calculate the sum of an infinite series - you can't do that without first producing every term of the series, which is an actual infinity of numbers even if you don't choose to collect them into a set.<

We can produce every element of the series using plural quantification, as I remarked above. All you need then is a relation ‘part of’ which expresses the relation between part and whole.

>Also, you can't even describe the real numbers without quantifying over sets

That’s a different and more difficult question.

[oz]

>what could a number be for a nominalist?

A number is a number of things. Note the collective noun that combines the singular 'a number' with the plural 'of things'.

The dispute between the realist and the nominalist is whether the (grammatically) singular term signifies a singular that is different from any of the 'things'.

[David Brightly]

In the sentence,

At dawn some of the Hatfields surrounded three of the McCoys but at noon the McCoys broke out and escaped,

the second 'the McCoys' obviously refers to the trio of McCoys previously surrounded, not to the McCoys plurally or the McCoy family singly. The syntax of the language around mathematical sets with its ∈, {}, and so on, is intended to avoid this kind of context-sensitive ambiguity. But its semantics must still rest on that of ordinary language referring terms, including plural terms. When we write,

Let S = {0, 1, 2},

we are not picking out and naming some kind of object. Rather we are introducing a new plural referring term. Likewise, the set operations like ∪ and ∩ generate new referring terms from old in order to serve an exposition of some sort. This seems to be how the language of sets is used in ordinary mathematics. Set theory itself could be seen as an applied mathematics of referring terms.

Bill says,

I agree that there cannot be an actual infinity of natural numbers unless there is a (mathematical as opposed to commonsense) set of all such numbers.

Why are we linking the potential/actual infinity issue with the question of the nature and existence of sets? Likewise with decimal expansions, irrational line lengths, etc? I don't see any connection.

I have just read the Max Black article on JSTOR. He seems to be seeking a middle way through this thicket.

[Michael Brazier]

oz, that article's claim that plural quantifiers are first-order expressions is not convincing in the least; and the argument from decimal representations depends on the number of existing terms in the representation, not on whether the representation is a single entity. Indeed, the whole point is that we must conceive of the representation as a single entity, for if we don't the claim that it's equal to a specific number is meaningless.

Before you reply again, I invite you to substitute in the argument, at every point BV or I mention sets, plural quantifiers, and see where the argument fails. To the best of my knowledge it won't - and then where will you be?

Brian: Yes, one doesn't really calculate limits by carrying out an infinite number of additions. The question is, though, why the process of taking limits leads to valid results. How is it that we can pass from "as I add more terms from this series together, I come closer to a particular number" to "the sum of every term of the series is that number" without any difficulty? (The epsilon-delta definition of a function's limit is more complicated than that, but it has the same issue, and it's easier to think about it with sequences.)

It's kin to the question of how to justify proofs by induction, which also involves an infinite process - once you show that 0 has a property, and that if n has it so does n+1, you say every n must have it, because if 0 has it then 1 has it, then 2 has it, and so on forever. But you don't actually go on forever, you just say what would happen if you did. Why does that work, exactly?

[Paul Smith]

The principle of induction -- if a subset of the naturals contains 0, and n+1 whenever it contains n, then it must contain all naturals -- does not imply the infinity of the set of natural numbers. The principle is rather a delimiting axiom, implying there are no wild strands of natural numbers out there that cannot be reached from 0 by the successor ("increment by 1") function. For example, the set of integers modulo 2 also satisfies the principle, though it contains only two elements. It is the successor function itself, which maps the set of natural numbers bijectively onto a proper subset of itself, that -- by definition -- guarantees infinity. There is no need "to keep doing something forever" here, or for that matter, to do anything at all, to call the natural numbers from the vasty deep.

[oz]

Brazier:
>Before you reply again, I invite you to substitute in the argument, at every point BV or I mention sets, plural quantifiers, and see where the argument fails. To the best of my knowledge it won't - and then where will you be?

As I have argued, positing an infinite number of things via plural quantification gives a different result from using set theory.

The question is whether we can express common mathematical concepts (such as the sum of a series) without set theoretical language.

[BV]

Paul Smith,

Welcome and thanks for the comments. Your comment @2:41 strikes me as correct. What say you, Oz?

[Michael Brazier]

Paul Smith, I said induction is needed, not to define the natural numbers, but to prove things about all of them. It's reaching the end of an endless proof - similar to how taking a limit reaches the end of an endless calculation. The question is how, if actual infinities can't exist even as abstract objects, reaching the end of endless processes is possible, or even conceivable.

(I'm a complete amateur is this field, but surely someone has thought of this before?)

oz: What difference does it make?

Seriously. When we take a limit, we are doing more than just positing an infinite number of things. We are asserting that these things, taken together, are related to a single thing. The limit of the series .333... is 1/3; that doesn't entail that the limit of .3 is 1/3, or the limit of .33 is 1/3. It's the difference between collective and distributive predicates again.

Now if a plural quantifier simply applies a predicate to each of the objects it refers to, a collective predicate is inexpressible and "taking a limit" is meaningless. If it applies the predicate to the plurality, it asserts that the plurality exists in a sense distinct from the existence of the particulars, in which case it might as well be a set and nominalism is false. I honestly can't see any way to avoid the dilemma.

The main rationale for plural quantification that I can see is to talk about sets without admitting that one is doing so.

[oz the ostrich]

>Welcome and thanks for the comments. Your comment @2:41 strikes me as correct. What say you, Oz?

It seems right to me. He says " It is the successor function itself, which maps the set of natural numbers bijectively onto a proper subset of itself, that -- by definition -- guarantees infinity."

[Brian Bosse]

Hello Bill,

An even better example is that of the irrational number, the square root of 2…If the hypotenuse of a right triangle is √2 units of length, that is a quite definite and determinate length. How could it be if the decimal expansion however protracted did not point to a completed totality, an actual infinity?

What exactly are we talking about when we speak of an irrational number like √2? To answer this question Georg Cantor made use of sequences of rational numbers called ‘Cauchy sequences’. Some Cauchy sequences converge to a rational number, and others do not…but they appear to converge to something. Given that different sequences can have the same convergence, a definition was introduced defining what it meant for sequences to be equivalent. Cantor’s idea was to define a real number to be the equivalence class of a Cauchy sequence. Here is one possible definition for √2:

√2 is the equivalence class of the Cauchy sequence defined recursively by s(0)=1 and s(n+1) = [s(n)+2/s(n)]/2 as n → ∞.

The first few elements in this sequence are:

s(0) = 1

s(1) = 3/2 = 1.5

s(2) = 17/12 = 1.41666…

s(3) = 577/408 = 1.414215686…

You state that the hypotenuse of a unit right triangle has a definite length. Based on this you ask how this can be if its decimal expansion does not point to a completed totality – an actual infinity. In answer to this question, the truncated decimal expansions of the √2 does form a Cauchy sequence of rational numbers. In fact, this sequence is in the same equivalence class as the infinite sequence defined above. At no point in this construction (defining) of √2 is there an assumed completed totality of the infinite sequences in this class. The “pointing” of these sequences is simply another way to speak of their shared convergence, which has a proper definition not requiring (or assuming) an actual infinity.

Bill, I am leaning more and more towards an anti-realist position when it comes to mathematical objects. When we speak of such idealizations like unit right triangles are we committed to the actual existence of such entities? Does our use of ordinary language necessarily require us to commit to an ontology?

Brian

[Brian Bosse]

Hello Michael,

When we take a limit, we are doing more than just positing an infinite number of things.

I do not agree that the taking of a limit posits an infinite number of things. Consider the recursive construction (definition) used for natural numbers. It is insufficient to prove the existence of the set of all natural numbers. This is why the existence of an infinite set is an axiom of set theory.

Brian

[BV]

Hi Brian,

I'm boning up 'bigly' on this stuff; my shallow pate is under terrible strain. I'll respond later. I reckon you won't be heading my way tomorrow. I shall miss you, the conversations, and the dialectic of the 64 squares.

[Michael Brazier]

Brian: Cantor's construction of the real numbers isn't the only one. Dedekind defined them as "cuts" of the rational numbers, divisions of the rationals into two sets such that every member of one set is strictly less than every member of the other. That construction is provably equivalent to Cantor's, in that for every Cantor real there is a unique Dedekind real, and the relations between Cantor's reals are preserved between their Dedekind counterparts. So Cantor's and Dedekind's constructions are both adequate representations of the real numbers.

But the two constructions are not the same. A set of rational numbers that's bounded above is not even one Cauchy sequence (because it isn't arranged in a sequence) let alone an equivalence class of them. And most Cauchy sequences aren't bounded above by their limits, so they aren't part of the cuts they correspond to. Therefore we can't just identify the real numbers with either construction. What we mean by "the real numbers" is the abstract structure that Cantor's and Dedekind's representations have in common (and were designed to express.)

[David Brightly]

Bill says, at 03:21 PM.

...a family cannot be reduced to a number of persons; it is not a mere manifold, or a mere many...

The same is true for a pair (of shoes) and other collective terms. They all imply relations of some sort between their members. But this is not so for a mathematical set. No such relations are implied. A set is a pure manifold. Bill also says, at 03:42 PM,

...it also makes no sense to say that an abstract object -- which is what math sets are -- has Gomer surrounded.

I dispute this: mathematical abstract objects have relations between their parts and sets qua sets do not, so sets cannot be abstract objects. These are the chief reasons for thinking that sets are pure plural referring terms. Artificial, structured, linguistic extensions engineered by us for clarity of mathematical thought and expression.