Notes on Infinite Series

The resident nominalist writes,

Your post generated a lot of interest. What I have to say now is better put as a separate post, rather than a long comment. Feel free to post.

1) Plural reference provides a means of dealing with numbers-of-things without introducing extra unwanted entities such as sets. Even realists agree that we should not have more entities than necessary, the disagreement is about what is ‘necessary’.

BV: We agree that entities  should not be multiplied beyond necessity, i.e., beyond what is needed for explanatory purposes. The disagreement, if any, will concern what is needed.

2) Using plural quantification we can postulate the existence of an infinite number-of-things. We simply postulate that for any number-of-things, there is at least one other thing. That gives a larger number-of-things, which itself is covered by the quantifier ‘any’, hence there must be a still larger number-of-things,  etc.

BV: You give no example, so let me supply one. Consider the series of positive integers: 1, 2, 3 . . . n, n + 1, . . . . Given 1, we can generate the rest using the successor function: S(n) = n + 1.  I used the word 'generate' since it comports well with your intuition that there are no actual infinities, and that therefore every infinity is merely potential.

3) In this way we neatly distinguish between actual and potential infinity. Using plural quantification, we can prove that there is no plural reference for ‘all the things’. For that would be a number-of-things, hence there must be an even larger number-of-things, which contradicts the supposition that we had all the things.

BV: Your argument is rather less than pellucid. Here is the best I can do by way of reconstructing your argument:

a) If the plural term, 'all the positive integers,' refers to something, then  it refers to a completed totality of generated integers. But

b) There is no completed totality of generated integers.

Therefore, by modus tollens,

c) It is not the case that 'all the positive integers' refers to something.

Therefore

d) There is no actually infinite set of positive integers.

If that is your argument, then it begs the question at line (b). One man's modus tollens is another man's modus ponens.  If the above is not your argument, tell me what your argument is. So far, then, a stand-off.

4) In this way we also avoid the pathological results of Cantorean set theory. If there is a set of natural numbers, then this is also a number, but it cannot itself be a natural number, so it is the first ‘transfinite number’. The nominalist approach avoids such weird numbers.

BV: But surely polemical verbiage is out of place in such serene precincts as we now occupy. You cannot shame Cantor's results out of existence by calling them 'pathological' or 'weird.' Most if not all working mathematicians accept them, no?

5) The problem for the nominalist arises when in trying to explain the sum of an infinite series, e.g. 1 + ½ + ¼ + ⅛ …  The realist wants to argue that unless this series is ‘completed’, we don’t have all the members, so the sum will amount to less than 2.

BV: Note that the formula for the series is 1/2ⁿ where n is a natural number with 0 being the first natural number.  Recall that any number raised to the zeroth power = 1.  (If you need to bone up on this, see here.)

Question for our nominalist: what does '1/2ⁿ' refer to? Can't be a set! And it can't be a property! Does it refer to nothing? Then so does '1-1/2n.' How then explain the difference between the two formulae (rules) for generating two different infinite series?

Or more simply, consider n. It is a variable. It has values and substituends. The values are the natural numbers. Only the ones we counted up to, or generated thus far? No, all of them. The ones we have actually counted up to in a finite number of countings, and the rest which are the possible objects of counting. The variable is a one-over-the-many of its values, and a one-over-the-many of its substituends, which are numerals, not numbers.  Numerals bring in the type-token distinction.  And so I will ask the nominalist what linguistic types are. Are they sets? No. Are they properties? No. What then?

6) It’s a difficult question for the nominalist, but here is my attempt to resolve it. Start with the notion of non-overlapping parts. Two non-overlapping parts have no part that is part of the other. Then there can be a number of non-overlapping parts such that there is no other such part, i.e. these are ‘all’ such parts.

BV: OK.

7) Then suppose we have a method of defining the parts. Start with a line of length 2. Note that the nominalist is OK here with the existence of lines, because lines are real things and not artificially constructed entities like ‘sets’. And suppose we can divide the line into two non-overlapping parts of equal length, i.e.,  a part of length 1, and another part of the same length.

BV:  You shouldn't say that sets are artificially constructed. After all, you think numbers are artificially constructed, no? They  are artifacts of counting. Your beef is with abstract objects, not artificial objects. Sets are abstract particulars. You oppose them for that reason. As a nominalist you hold that everything is a concrete particular. (Or am I putting words in your mouth?)

Second, you are ignoring the difference between a geometrical line and a line drawn with pencil on paper, say. The latter is a physical line, which is actually a 3-D object with length, width and depth. In addition to its pure geometrical properties, it has physical and chemical properties. It is a physical line in physical space. The former is not a physical line, but an ideal line: it has length, but no width or depth. Ideal lines are not in physical space. Suppose physical space, the space of nature, is non-Euclidean. Then Euclidean lines are obviously not in physical space. But even if physical space is Euclidean, Euclidean lines would still not be in physical space.

8. So the proposition “2 = 1+1” says that a line of length two can be divided into two equal non-overlapping parts. Then suppose that we divide the second part into two equal parts. Thus “2 = 1 + ½ + ½” says that the line can be divided into three non-overlapping parts, one of length 1, and the other two equal. Do the same again, thus 2 = 1 + ½ + ¼ + ¼. And again and again!

BV: An obvious point is that the arithmetical proposition '2 = 1 + 1' is not about lines only.  It could be about a two-degree linear cool-down of a poker. (I am thinking about Wittgenstein's famous poker-brandishing incident.)  It could be about anything. Two pins. An angel on a pin joined by another.

Besides, "2 = 1 + 1" cannot be about the non-overlapping parts of a particular line, the one you drew in the sand. It is about a geometrical line, which is an ideal or abstract object.  The theorem of Pythagoras is not about the right triangle you drew on the blackboard with chalk; it is about the ideal right triangle that the triangle you drew merely approximates to.

9. It is clear that for every such division, the parts ‘add up’ to the same number, i.e. 2.

10. Then consider what the proposition “2 = 1 + ½ + ¼ + ⅛ … “ expresses. Surely that every such series, however extended, has a sum of 2. Do we need the notion of a ‘set’? No.

BV: I don't see how this answers the question that you yourself raised in #5 above. What makes it the case that the series you mention actually has a sum of 2? The most you can say is that series potentially has a sum of  2.  The Cantorean does not face this problem because he can say that there is an actual infinity of compact fractions that sums to 2. No endless task needs to be performed to get to the sum.

Comments

[Joe Odegaard]

An infinite number of mathematicians walk into a bar.

The first mathematician says, "I'll have one beer."

The second mathematician says, I'll have half a beer."

The third mathematician says "I'll have a quarter of a beer."

The fourth Mathematician says, "I'll have an eighth of a beer."

Etc Etc Etc ∞

The Bartender puts two beers on the bar, and says, "You guys need to know your limits."

• • • • • • 

Math nerds have informed me that neither zero nor infinity are numbers.

[BV]

That's a good joke, Joe.

That there are transfinite cardinals and ordinals may be reasonably disputed.

That zero is a number cannot be reasonably disputed. I will write a separate post about this. I will grant you, though, that zero is a very strange number. Ask a mathematician what a positive integer divided by 0 is and he will say it is undefined. Cop out? But apart from this anomaly, 0 is indispensable for the simplest arithmetic. As you know, it can serve as the numerator of a fraction, as an exponent, as a limit, etc.

[Joe Odegaard]

"Numberphile" has many great videos about zero, and numbers generally.

Here is one:

https://www.youtube.com/watch?v=BRRolKTlF6Q

Enjoy ! My physics kid Brunel Odegaard introduced mr to Numberphile.

— Joe

[Michael Brazier]

"Start with the notion of non-overlapping parts. Two non-overlapping parts have no part that is part of the other. Then there can be a number of non-overlapping parts such that there is no other such part, i.e. these are ‘all’ such parts."

How many ways are there to divide one line into two non-overlapping, connected parts? Abstractly considered, there are an uncountable number of them - they are the Dedekind cuts of the rational numbers, which Dedekind used to define the real numbers. Geometric concepts go two steps into Cantor's transfinites; the poor natural numbers only need one.

"Then consider what the proposition “2 = 1 + ½ + ¼ + ⅛ … “ expresses. Surely that every such series, however extended, has a sum of 2."

Well, no, it doesn't say that at all. What it really says is that the sequence of partial sums 1, 1 + ½, 1 + ½ + ¼, 1 + ½ + ¼ + ⅛, … approaches 2, in the sense that as you generate partial sums you can get as close to 2 as you like; and that there's no other number that the sequence approaches in this way. That's the strict mathematical definition of a limit.

The issue, for a nominalist, is that "this infinite sequence has a limit" is true of the sequence as a whole, but is false for any member of the sequence, or even any finite set of its members. The technical definition of a sequence's limit includes, and can't avoid, a statement about all the terms in the sequence past a certain point - a universal quantifier, not a plural. Thus, if we are limited to plural quantifiers, we cannot even say that limits exist.

[Joe Odegaard]

If "zero" is taken to mean "nothing," but it is also a number, that seems awfully close to "A equals Not A," and how do you get around that?

I look forward to the philosophical answer.

[BV]

Joe,

'Zero' does not mean 'nothing.' Consider the following two contradictory propositions:

1) Something exists
2) Nothing exists.

Clearly, (2) is false. But the following is true:

3) Zero exists.

Therefore, 'zero' does not mean 'nothing. From the POV of set theory, zero is the null set. The null set exists; ergo, zero exists. Can I give a reason for the existence of zero? Well, if the natural numbers are closed under subtraction, then n-n (n minus n) = the number 0. Can I give a reason for the existence of the null set? Well, if intersection is defined for two disjoint sets, then the intersection is the null set!

In set theory the existence of the null set is simply posited in an axiom. Could it be proven as a theorem?

[BV]

Joe,

Here is another consideration. It is true that -1 < 0 < 1. Since 0 has the property of being strictly less than the positive integer 1 and strictly more than the negative integer -1, 0 cannot be nothing at all. An item that has properties cannot be nothing.

In the immortal words of Kris Kristofferson, "Nothing ain't worth nothing, but it's free." https://www.youtube.com/watch?v=8LaHPmD8nuc

[Joe Odegaard]

Well I could see the null set as being an event, that is, the event of "We looked but we didn't find anything." So that does have an existence, and I will grant "zero" with this existence, so in my mind zero is of the hook now. What a funny thing it is.

[oz]

>You give no example, so let me supply one.

No example is needed, although we need a definition of ‘number’. Suppose that a thing exists, and define ‘one’ as ‘a’. So if there is a thing, there is one thing. Then assume that for any things, there is at least one other thing. Then there is one thing and another thing. Define ‘two things’ as ‘one thing and another thing’. Then there must be two things and another thing, define ‘three’ as two things and another thing etc. The number terms ‘one’ to ‘nine’ can be defined this way. Then we can move to a system of generating number names like our decimal system. Note we generate names, not things.

>Your argument is rather less than pellucid.

On the contrary it is crystal clear. It is yours that is less than pellucid. You say “If the plural term, 'all the positive integers,' refers to something, then it refers to a completed totality of generated integers.” You have the undefined term “completed totality of generated integers”.

> If the above is not your argument, tell me what your argument is.

The above is not my argument. I shall repeat my very clear argument, which requires understanding plural quantification only. “we can prove that there is no plural reference for ‘all the things’. For that would be a number-of-things, hence there must be an even larger number-of-things, which contradicts the supposition that we had all the things.” This follows from the assumption that if there exist a number of things, there exists at least one other thing.

> What makes it the case that the series you mention actually has a sum of 2?

My emphasis. I say that the proposition “2 = 1 + ½ + ¼ + ⅛ …” is actually true, where its meaning is as I defined it.

[Michael Brazier]

oz: I say that the proposition “2 = 1 + ½ + ¼ + ⅛ …” is actually true, where its meaning is as I defined it.

But your definition isn't the one the mathematicians use. The mathematician does not say "if you break a stick in half, break one of the halves in half, break one of those in half, and continue as often as you like, the lengths of the fragments add up to the length of the stick you started with", which is what you said. The mathematician says "if you have an infinite bundle of sticks, where each stick is half the length of the previous stick, the lengths of all the sticks add up to twice the length of the first stick."

Here's a challenge for you: try to say "this sequence does not have a limit" without referring to an actual infinity.