But what does .9999999999… MEAN?

[aleph_0]

Clever try, but nope, the literature just wasn't convincing.

[Trebor]

Clever try, but nope, the literature just wasn't convincing.

When you say that it "wasn't convincing," I take your meaning as being infinite, not finite. I'll take everything you have to say as being infinite. How can I possibly know that the Law of Identity applies in every case?

[aleph_0]

I've seen no argument, from the premise that A equals A, to the conclusion that all quantities of things in reality are finite. It is not enough to assume that all things which exist, exist as finite beings. Nobody likes being circular, in any sense of the term. I claim the law of identity is compatible with the quantity of some thing being infinite. Indeed, the only argument I've ever seen to that conclusion has been one which attempts to appeal to induction, which might in some sense indicate a probability judgment that the universe is finite, though I don't think it does. In any case, it cannot even aspire to show that the concept of infinite quantity is inherently contradictory.

As a note, I have no idea what it could even possibly mean to say that you take everything I say as infinite. English words, but I don't have a clue what they mean in that order.

[Mindy]

And which quotient is that? I'm gonna lose it you come back with 1/1.

"Would" does not mean "is." I was tempted to stop there, but I'll add that non-terminating decimals are generally understood, by the layman, to be quotients. Few of us have memorized a list of the non-terminating decimals which can be produced by a ratio, and those which cannot. If that is the whole point, then this entire debate is about a trick question.

-- Mindy

[aleph_0]

As a minor point of correction, whether laymen are aware of it or not, most non-terminating decimals are irrational, not rational.

[Mindy]

I've seen no argument, from the premise that A equals A, to the conclusion that all quantities of things in reality are finite. It is not enough to assume that all things which exist, exist as finite beings. ... I claim the law of identity is compatible with the quantity of some thing being infinite.

Do you see that being infinite, in any respect, is contrary to having identity?

-- Mindy

[AlexL]

Most of mathematics doesn't really need the concept infinity.

This concept ("concept") is incompatible with arithmetic: operations with infinity are inconsistent and mostly undefined.

In calculus - to which the concept of limit belongs - the idea of infinity is also unnecessary: it is only used as a shorthand for a more precise, but much longer, formulations. For example, we write

as a shorthand for this very cumbersome phrase:

a is the limit of the sequence , that is for any there exists an integer N, so that for any .

Moreover: arithmetic and classical analysis operate with specific quantities, which is not.

Alex

PS: the equations were shamelessly copied from Wikipedia :-)

Edited July 30, 2010 by AlexL

[Marc K.]

I've seen no argument, from the premise that A equals A, to the conclusion that all quantities of things in reality are finite. It is not enough to assume that all things which exist, exist as finite beings.

The law of identity is properly designated as: A is A; meaning: a thing is itself. And the feature of reality that it illustrates is that everything that exists, exists as something, something specific. There is nothing in reality that only sort of exists or exists as nothing in particular. You have to know what a thing is in order to posit its existence and to know what it is means that you must be able to describe it or its properties. If those properties are nothing specific then you can't describe it or even know what to look for.

While useful for some types of calculations, the concept "infinity" does not describe any thing that exists since it is not something specific.

[Mindy]

The law of identity is properly designated as: A is A; meaning: a thing is itself.

I offer a different interpretation. "A is A" means: a thing is what it is. My statement addresses the issue that to exist is to be characterizable. That there is no separation between existing and having character. That is contrasted with specifying what it is that anything can be said to be identical with.

-- Mindy

[Nate T.]

"Would" does not mean "is." I was tempted to stop there, but I'll add that non-terminating decimals are generally understood, by the layman, to be quotients. Few of us have memorized a list of the non-terminating decimals which can be produced by a ratio, and those which cannot. If that is the whole point, then this entire debate is about a trick question.

Would be, but for what?

And whether layman accept it or not, there are no natural numbers p and q for which (p/q)^2 = 2. And don't even think about throwing me overboard for having mentioned it.

Incidentally, there is a criterion for determining whether a non-terminating decimal arises from a quotient. Such a decimal number is rational precisely if it repeats after a certain point. There's even a nice way, given a periodic decimal expansion, to recover the quotient it's equal to. That method also recovers 0.999... = 1.

[aleph_0]

Do you see that being infinite, in any respect, is contrary to having identity?

-- Mindy

Nope.

Most of mathematics doesn't really need the concept infinity.

Not all of it, but arithmetic has, as part of its definition, an infinite domain.

This concept ("concept") is incompatible with arithmetic: operations with infinity are inconsistent and mostly undefined.

Operations on "infinity" denoted by oo (the lemniscate) in the extended real numbers, are largely undefined, but that does not mean that infinite sets are ill-defined.

In calculus - to which the concept of limit belongs

Actually, that's topology. Limit elements of sets can be defined purely in terms of open sets, and open sets can be defined quite variously, so it's really a much more general concept. But your point about calculus still depends on the infinity of real numbers, and we still define limits at infinity, and in more advanced mathematics we discuss infinite-dimensional spaces. Infinity is really everywhere in mathematics, from the beginning and more and more as you develop it. That's not to say that mathematics ever claims an infinite quantity of anything, if that's your concern, but nobody has claimed that.

The law of identity is properly designated as: A is A; meaning: a thing is itself. And the feature of reality that it illustrates is that everything that exists, exists as something, something specific. There is nothing in reality that only sort of exists or exists as nothing in particular. You have to know what a thing is in order to posit its existence and to know what it is means that you must be able to describe it or its properties. If those properties are nothing specific then you can't describe it or even know what to look for.

While useful for some types of calculations, the concept "infinity" does not describe any thing that exists since it is not something specific.

This seems like you're arguing that an infinite quantity is somehow not defined.

Edited July 31, 2010 by aleph_0

[Trebor]

This seems like you're arguing that an infinite quantity is somehow not defined.

You're equivocating between the definition of "infinite quantity" and an infinite quantity.

The definition of an infinite quantity is a quantity without limit. But a quantity without limit, a limitless quantity, is not a specific, finite quantity. In that, later, sense, an infinite quantity is not defined; it's undefined as in unlimited.

Again, the concept of "infinite" is, as Peikoff said, "valid only when used to indicate a potentiality, never an actuality."

Edited July 31, 2010 by Trebor

[aleph_0]

A quantity without limit is still a specific quantity, at least so long as we specify infinity of size aleph_0 or some such. It's perfectly well-defined and specific, just infinite. Why is it that unlimited should imply undefined? This doesn't constitute a proof, you're just asserting in slightly different words that infinite quantities don't exist. But that's not any axiom that I have. I have A = A, and consciousness, and the value of my life, but I don't have that infinite quantities is an inherently self-contradictory idea.

[Trebor]

A quantity without limit is still a specific quantity, at least so long as we specify infinity of size aleph_0 or some such. It's perfectly well-defined and specific, just infinite. Why is it that unlimited should imply undefined? This doesn't constitute a proof, you're just asserting in slightly different words that infinite quantities don't exist. But that's not any axiom that I have. I have A = A, and consciousness, and the value of my life, but I don't have that infinite quantities is an inherently self-contradictory idea.

Surely you are joking!

What is the specific, finite quantity of an infinite quantity?

Why is it that unlimited should imply undefined?

You have to equivocate on the meaning of "defined" to ask this question. I've already addressed it.

This doesn't constitute a proof, you're just asserting in slightly different words that infinite quantities don't exist. But that's not any axiom that I have. I have A = A, and consciousness, and the value of my life, but I don't have that infinite quantities is an inherently self-contradictory idea.

There's no proof of Identity. Proof presupposes identity. Existence is Identity. To be is to be something as opposed to something else, limited as opposed to unlimited. There's no actual infinite or unlimited existent or unlimited quantity of existents.

For the sake of absurdity, let's suppose that there are an infinite quantity (number or count) of marbles (the regular type of marbles that kids play marbles with). Even if you held that there's an infinite universe (which isn't), the quantity of if marbles is limited by the existence of other non-marble existents. Replace marbles with anything else.

[aleph_0]

Surely you must be joking! I just said that a quantity can be specific and yet infinite, so I obviously can't--and don't need to--tell you any quantity that is equal to an infinity and which is finite and specific. That there is at least a quantity aleph_0 of electrons in the universe would just mean that there is an injection from the natural numbers to electrons, which is perfectly well-defined, just not finite.

And I'm not asking for a proof of identity, I've already told you that I accept A equals A. I'm asking for a proof that this in any way has anything at all to do with anything about infinity.

And why would the quantity of marbles be limited by the existence of other things? What is contradictory about supposing that, for every ten thousand light-years away from me in (something like) a straight line, there is a marble, regardless of how many ten thousand light-years you actually travel? Though you will never actually travel even one of those light-years, what is contradictory about the idea of there being at least one marble on such an endless path?

Edited July 31, 2010 by aleph_0

[Trebor]

Surely you must be joking! I just said that a quantity can be specific and yet infinite, so I obviously can't--and don't need to--tell you any quantity that is equal to an infinity and which is finite and specific.

If you can make such a statement, I can't be of any help. Such a statement is simply too absurd for me to give any of my limited, finite time to.

Much good luck!

[aleph_0]

Then you can't prove your case, and it's not an axiom. I was quite confident of that in the beginning, to be honest.

[Grames]

One third is 1/3 = 0.333...

Two thirds 2/3 = 0.666...

Three thirds 3/3 = 0.999... = 1.0 = 1

[Mindy]

Then you can't prove your case, and it's not an axiom. I was quite confident of that in the beginning, to be honest.

An infinite quantity is an oxymoron. "Quantity" means a determinate size or amount.

-- Mindy

[aleph_0]

No, quantity just means "how many", or to be more mathematical about it, to be in an equivalence class defined by standing in a bijective relationship. There's nothing contradictory about the idea that I could put the natural numbers in a one-to-one correspondence with some collection of physical objects.

[Marc K.]

Yes. To say you have an infinite amount of something is self contradictory. Such a quantity wouldn't fit in the Universe.

[Mindy]

One third is 1/3 = 0.333...

Two thirds 2/3 = 0.666...

Three thirds 3/3 = 0.999... = 1.0 = 1

Non-terminating decimals are not quantities. Quantities are determinate. When truncated, they have information as to an approximate quantity. Since they aren't quantities, they can't be treated as such through arithmetical operations, etc. In fact, we cannot express what one divided by three equals as a decimal, though we have a symbolism for the partial process.

So, .33333333 times 3 = .99999999, but .333... times 3 does not equal .999... .

It is not true that 1/3 = .3, and it is not true that 1/3 = .33, and it is not true that 1/3 = .333, etc. Notice that no number of threes would suffice, but the symbolism contains nothing but threes. It is a variation on the meaning of the equal sign, that we express 1/3 = .333... . The non-terminating symbolism explicitly says that the series is insufficient.

-- Mindy

Edited July 31, 2010 by Mindy

[aleph_0]

Yes. To say you have an infinite amount of something is self contradictory. Such a quantity wouldn't fit in the Universe.

Unless the universe is infinite. Nothing contradictory about that.

[Mindy]

No, quantity just means "how many", or to be more mathematical about it, to be in an equivalence class defined by standing in a bijective relationship. There's nothing contradictory about the idea that I could put the natural numbers in a one-to-one correspondence with some collection of physical objects.

Setting the members of two sets in a one-to-one correspondence means nothing unless you exhaust at least one of those sets, remember?

-- Mindy

Edited July 31, 2010 by Mindy

[Marc K.]

No, quantity just means "how many", or to be more mathematical about it, to be in an equivalence class defined by standing in a bijective relationship. There's nothing contradictory about the idea that I could put the natural numbers in a one-to-one correspondence with some collection of physical objects.

You are still equivocating between math and reality. Nobody here has denied that the concept "infinity" is a useful methodological tool in mathematics but my first reply to you was in response to this:

I've seen no argument, from the premise that A equals A, to the conclusion that all quantities of things in reality are finite. It is not enough to assume that all things which exist, exist as finite beings. [emphasis added]

That argument has now been made but you are evading it.

We are talking about quantities of existents in reality and you are still talking about natural numbers. You could name a specific natural number and we might be able to find a quantity of that number in reality. But if you named the "number" "infinity" we could find no quantity in reality to match it with because it is not a specific number -- it is bigger than any number I can name.

Infinity points to no existent in reality, it is not derived directly from perception of reality, it is an abstraction from abstractions, it is a method of the mind.