## Tuesday, May 09, 2023

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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.

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.

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

Here is one:

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

— Joe

"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.

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.

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?

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

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.

>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.

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.

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