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But what does .9999999999… MEAN?

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Unless the universe is infinite. Nothing contradictory about that.

The Universe is defined as the sum of everything that exists so if this infinite quantity exists it would exist in the Universe but if it did, then it would have to be finite so it must be bigger than the Universe, which is a contradiction.

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The Universe is defined as the sum of everything that exists so if this infinite quantity exists it would exist in the Universe but if it did, then it would have to be finite so it must be bigger than the Universe, which is a contradiction.

As a matter of logic only, that argument can be turned around. The so-called infinite quantity, existing in the Universe, just makes the Universe infinite.

-- Mindy

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As a matter of logic only, that argument can be turned around. The so-called infinite quantity, existing in the Universe, just makes the Universe infinite.

-- Mindy

No, then the Universe would be undefined, which is why that was the first thing I defined.

In general, yes, it is unproductive to talk about self-contradictory statements. But it is one way to illustrate the absurdity to someone who is doing it, and I was sure to point out that a contradiction was contained therein.

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Non-terminating decimals are not quantities. Quantities are determinate.

This may be a good question to ask you and everyone else participating: what is a "quantity?" It seems that different sides of this discussion are answering the question in wildly different ways.

As for real numbers not necessarily being quantities, what do you think about the square root of two? Is it a number, a quantity, or what? Or are the only numbers rational numbers since they're the only ones that can be physically constructed?

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Non-terminating decimals are not quantities. Quantities are determinate.

One third is a quantity whether expressed as a ratio or a decimal. Just as "dog" in English and "perro" in Spanish are two words in different languages for the same set of referents, 1/3 and 0.333... have the same referent. The referent of π and the non-terminating decimal pi is the quantity of radians in a semicircle. The referent is the meaning of a symbol. Using the definition as the meaning of a symbol is an error. The decimal format of one third is a definition within that notation system. Every weighted placeholder notation system (base 2, base 3, base 10, ...) will have certain quantities that will be ill-defined in the sense of being non-terminating. Comparing several notation systems makes clear the distinction between a quantity and its expression, which is a relationship analogous to that between an object and its appearance.

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This may be a good question to ask you and everyone else participating: what is a "quantity?" It seems that different sides of this discussion are answering the question in wildly different ways.

As for real numbers not necessarily being quantities, what do you think about the square root of two? Is it a number, a quantity, or what? Or are the only numbers rational numbers since they're the only ones that can be physically constructed?

The history of mathematics tells us that numbers began by a process of forming one-to-one correspondences between, for example, the sheep in a herd, and a pile of pebbles. One pebble is taken out for each sheep that passes. If the sheep and the pebbles run out at the same time, that is, if a complete one-to-one correspondence can be made, all the sheep are present. (Yes, I'm assuming the pile of pebbles was made by adding one each time a sheep left home.) Notice that there is not any idea of how many sheep there are, just that all of them are present.

Imagine two or three shepards who take their flocks out to graze together, and it happens that two of them have 5 sheep. They are doing their counting in sight of one another, and one notices that their pebbles match the way the pebbles match to the sheep. Once they notice the piles of pebbles match one another, they are attending to the specific quantity (here, 5) that makes certain piles of pebbles match (correspond on a one-to-one basis.)

So there are piles that match one another, and ones that don't. And there are piles, sometimes, that match other piles. Each such matching set of piles is given a name, or symbol. Then, the characteristic that that set of symbols has, is number.

All this took a great deal of time, of course. And, the historical course of development does not dictate to the logical structure. But that is the account given of the discovery of numbers. The idea of quantity is one step of abstraction above that of numbers. Quantity is what all numbers have in common.

You can see that being based on one-to-one correspondence eliminates the idea of infinite quantities. You can also see that the purpose and impulse of each step of development of numbers is specificity. The shepard knows some sheep returned. He wants to be sure none are left behind. When similar piles are noted, the fact that they are the same as one another, and different from other piles--their specificity--leads to their getting symbolized.

-- Mindy

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One third is a quantity whether expressed as a ratio or a decimal. Just as "dog" in English and "perro" in Spanish are two words in different languages for the same set of referents, 1/3 and 0.333... have the same referent. The referent of π and the non-terminating decimal pi is the quantity of radians in a semicircle. The referent is the meaning of a symbol. Using the definition as the meaning of a symbol is an error. The decimal format of one third is a definition within that notation system. Every weighted placeholder notation system (base 2, base 3, base 10, ...) will have certain quantities that will be ill-defined in the sense of being non-terminating. Comparing several notation systems makes clear the distinction between a quantity and its expression, which is a relationship analogous to that between an object and its appearance.

One third is a quantity that cannot be expressed as a decimal. All the languages you wish to translate won't make it be a specific quantity. Non-terminating digits, as quotients, are imprecise, as I explained. 1/3 is a specific quantity, but .333... is not. Decimal notation cannot express 1/3.

The difference between an object and its appearance is NOT analogous to the difference between 2/3 and .666... . They are not the same thing in two different languages, nor two aspects of the same thing. They both claim to be the same aspect, the quantitative one. .666... is gotten from the straight-forward language of arithmetic, except that the process of division can't be completed. We mark that FAILURE with the notation that means "non-terminating."

As for your "language" suggestion, you fail to note that if .666... can be expressed as a non-terminating decimal in any other language, 1/3 will not be capable of expression in it. That's a significant oversight, assuming you meant to clarify the problem.

You can't translate incommensurables out of existence.

-- Mindy

-- Mindy

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All right, Mindy, but your development of "quantity" explicitly leaves out the problem of measurement of the continuum, which is the whole point here. It's true that we can only measure physical objects using rational numbers, but 0.999... = 1 is meant in the sense of real numbers, and so is a concept of method.

Anyway, you seem to be bothered by the fact that whatever quantity is denoted by the symbol "0.999..." cannot be represented as a terminating decimal of the form 0.?????. That's true, but has no bearing on the truth of 0.999... = 1.

If you don't want to accept the existence of numerical expressions beyond terminating decimals I suppose that's your business, but don't go around claiming that it can't be done. I handle such objects all the time in a way that ties right back to reality, in the way that has already been presented in this thread.

Anyway, I think we've both made our points, so I'm gonna bow out-- thanks for the discussion.

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As for your "language" suggestion, you fail to note that if .666... can be expressed as a non-terminating decimal in any other language, 1/3 will not be capable of expression in it. That's a significant oversight, assuming you meant to clarify the problem.

Base 3 notation uses the digits 0, 1 and 2. One third is 0.1 and two thirds is 0.2.

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You are still equivocating between math and reality. Nobody here has denied that the concept "infinity" is a useful methodological tool in mathematics

I'm not so sure about that. Some of the people here seem to think that mathematical entities are real, or intrinsically refer to some real-world objects and therefore mathematics itself must be finite. But if that's not the case, all the better.

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

No, you're evading! And if you say you're not, then you're evading that you're an evader!

...

This tactic is childish. I've asked for an argument, and I've received two responses. First, an attempt at an induction, and I pointed out how this fails. Providing a counter-argument is not evading anything--it's pointing out that the argument provided is insufficient. If you insist on evading this fact, so be it. You evader. The second response was just a re-insistence that nothing can be infinite, but this is obviously not an argument.

We are talking about quantities of existents in reality and you are still talking about natural numbers.

I am talking about mappings between natural numbers and existents. Here is an instance of such a mapping: From the set of numbers {1, 2, 3} I map 1 to me, 2 to you, and 3 to Grames. In fact, this is a one-to-one correspondence with the set {me, you, Grames} because it is surjective and injective, and thus the two sets have the same size, i.e. the quantity of things in each set is the same. Thus when I speak of the the universe being infinite, I speak of the set of, say, electrons being in a one-to-one correspondence with the set of natural numbers. There's nothing that anybody has said that makes the coherence of such a notion even dubious.

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.

Aleph_0, which is the cardinality of the natural numbers, is less than aleph_1 (and aleph_1 < aleph_2 < ...), so problem solved.

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.

Maybe not since I have no idea what "pointing" has to do with anything, but I have a clear idea of what "infinity" means and there is nothing contradictory about the hypothesis that the universe contains infinitely many objects.

The Universe is defined as the sum of everything that exists so if this infinite quantity exists it would exist in the Universe but if it did, then it would have to be finite so it must be bigger than the Universe, which is a contradiction.

You're not just an evader, you're a circular evader!

(Sorry, I just find the Objectivist tendency to yell "evader!" when they can't support their claims to be extremely, extremely hilarious. And only something dirty evaders would do.)

One third is a quantity that cannot be expressed as a decimal. All the languages you wish to translate won't make it be a specific quantity. Non-terminating digits, as quotients, are imprecise, as I explained. 1/3 is a specific quantity, but .333... is not. Decimal notation cannot express 1/3.

Again, this is just not the case, as I've explained with another example of the limit of a sequence of elements. Just because a thing is the limit of a sequence does not mean that the thing is itself somehow infinite. It is not as though we have a piece of clay and we add on infinitely more, infinitely smaller pieces of clay to it. The limit of a sequence is a single thing, and there are just an infinite number of other elements that are arbitrarily close to it. Just because we represent this element by a suggestive notation that contains, in some sense, an infinite string of characters does not mean that this element is any less a number or well-defined element in a mathematical system. Now the real number system is not a system that is appropriate for quantifying things, in any sense other than the fact that the natural numbers "live inside" the real numbers. But that does not mean they're not numbers, and it doesn't mean they're ill-defined.

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I'm not so sure about that. Some of the people here seem to think that mathematical entities are real, or intrinsically refer to some real-world objects and therefore mathematics itself must be finite. But if that's not the case, all the better.

Mathematical entities are real. They do not intrinsically refer to, rather objectively refer to real-world objects. Infinity arises within the context of mathematics. It is ignoring the hierarchial and contextual roots of infinity, and then applying it as a stolen concept in a manner that contradicts a conceptually grasped, perceptually validated axiom that is being illustrated.

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So are they entities or do they refer to entities? And in either case, how did you come about this information?

Now perhaps if I were claiming that I have some evidence that the universe is infinite, which I don't, you could claim that I'm making something like the error you accuse me of. But I'm not, and nothing you've said has yet indicated the impossibility of infinite quantity.

To illustrate: I observe that all ravens are black. I come to the reasonable judgement that all ravens are black, though I realize that I might one day find a non-black raven. None of this, however, even suggests that a white raven is impossible.

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Mathematical entities are real. They do not intrinsically refer to, rather objectively refer to real-world objects. Infinity arises within the context of mathematics. It is ignoring the hierarchial and contextual roots of infinity, and then applying it as a stolen concept in a manner that contradicts a conceptually grasped, perceptually validated axiom that is being illustrated.

This is no good. The counting numbers are all epistemological abstractions, they are not metaphysical. They are justified and objective, but calling them real or describing them as entities goes too far.

There are two arguments for the finitude of the universe. Deriving finitude from the Law of Identity applied to the universe as a whole is problematic, first because as the universe cannot have a boundary and second because of the difficulty in pinning down what exactly would be contradicted in the case of a universe of infinite extent. The better argument for regarding the universe as finite is simply that it would be to entertain the arbitrary. Unlike the case of crows where we know feathers occur in different colors on other birds so a white crow is plausible, there is no known instance of an actual infinity.

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Base 3 notation uses the digits 0, 1 and 2. One third is 0.1 and two thirds is 0.2.

I have to do some checking. Can you name the system of notation that makes pi a terminating decimal?

-- Mindy

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Just because we represent this element by a suggestive notation that contains, in some sense, an infinite string of characters does not mean that this element is any less a number or well-defined element in a mathematical system.

It is defined. Limits are no big deal. But, can you deny that the purpose of declaring, outside a specifically mathematical context, that .999... = 1 is intended to confuse and provoke, based on what you call "suggestive notation?" You're amusing yourself with a trick question, by choosing a context in which its "suggestive notation" will be misunderstood.

-- Mindy

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This is no good. The counting numbers are all epistemological abstractions, they are not metaphysical. They are justified and objective, but calling them real or describing them as entities goes too far.

There are two arguments for the finitude of the universe. Deriving finitude from the Law of Identity applied to the universe as a whole is problematic, first because as the universe cannot have a boundary and second because of the difficulty in pinning down what exactly would be contradicted in the case of a universe of infinite extent. The better argument for regarding the universe as finite is simply that it would be to entertain the arbitrary. Unlike the case of crows where we know feathers occur in different colors on other birds so a white crow is plausible, there is no known instance of an actual infinity.

Refering to them as real was not to imply they are existents. Yes, they are justified and objective. They are derived from the observation of the relationship of a group of existents to one of its members taken as a unit.

As to a reference, I thought that Pat Corvini did a compelling presentation in her two ARI talks entitled, 'Two, Three, Four and All That' and 'The Sequel' of the same title.

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Grames: "...the universe cannot have a boundary...".

But it can, and does, have a specific extent, even if it is constantly expanding. So that is not a hindrance to deriving the finitude of the universe from the law of identity.

As to "...the difficulty in pinning down what exactly would be contradicted in the case of a universe of infinite extent[,]" the same objection applies as to anything else claimed to be infinite. Infinite is not a quantity. Quantities are specific, determinate. (See the end mentioned in "terminate?" and the no end in "infinite?") So, if you're talking about the size of something, just forget considering if it might be infinite. If you're talking about processes, etc., infinity might be your ticket.

-- Mindy

Edited by Mindy
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All right, Mindy, but your development of "quantity" explicitly leaves out the problem of measurement of the continuum, which is the whole point here.

I didn't know that measurement of a continuum was the point in this thread. We are at this point only able to approximate measures of a continuum. When and if we get to precise measurements, they'll fall under the concept of quantity. Meantime, we approximate with infinite degrees of near-precision. That's not reason to lose the philosophical distinction.

-- Mindy

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"Universe" is the sum of all existents. If even a single existent was unbounded, there could be no other existents whatever.

Multiplicity is fundamental, ontologically, as far as I'm concerned. I see no way around this. To eschew a finite universe, is to contradict identity. To reject the idea of a jar "with no specific amount of marbles in it", is not a rejection simply of the arbitrary, but of invalid concepts. I submit the willingness of some to do otherwise, is to attempt to preserve other premises of the special sciences held to be true.

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"Universe" is the sum of all existents. If even a single existent was unbounded, there could be no other existents whatever.

Okay, here's a question. Do you mean 'objects' when you say that or all existents, ALL of them? You do realize that between any two objects there are numerous relationships, and between those objects and their relationships there are further more relationships, and that the process of relating relations and the objects or other relations that gave rise to them is non-terminating, don't you? Sure some relationships may be arbitrarily constructed but they nonetheless exist.

Or, if I may use numbers here, if the number 1 exists, then the number 2 exists, and so on and so forth. There are an infinite number of finite numbers. Unless you want to claim that numbers don't exist, you have to accept that the universe contains an infinite number of elements. In fact, set-theoretically, you can't even construct a universal set from the axioms, there are paradoxes which make it contradictory to the axioms. So you can't talk about a universe as being the sum of all that exists, unless you're just talking about objects.

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Okay, here's a question. Do you mean 'objects' when you say that or all existents, ALL of them? You do realize that between any two objects there are numerous relationships, and between those objects and their relationships there are further more relationships, and that the process of relating relations and the objects or other relations that gave rise to them is non-terminating, don't you? Sure some relationships may be arbitrarily constructed but they nonetheless exist.

Or, if I may use numbers here, if the number 1 exists, then the number 2 exists, and so on and so forth. There are an infinite number of finite numbers. Unless you want to claim that numbers don't exist, you have to accept that the universe contains an infinite number of elements. In fact, set-theoretically, you can't even construct a universal set from the axioms, there are paradoxes which make it contradictory to the axioms. So you can't talk about a universe as being the sum of all that exists, unless you're just talking about objects.

I said "ontologically"...Relationships exist yes ,as derivitives . I don't see any reason to object to the ability to concieve of endless permutations of them.This is an epistemological phenomenon only.

Edited by Plasmatic
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"Universe" is the sum of all existents. If even a single existent was unbounded, there could be no other existents whatever.

Multiplicity is fundamental, ontologically, as far as I'm concerned. I see no way around this. To eschew a finite universe, is to contradict identity. To reject the idea of a jar "with no specific amount of marbles in it", is not a rejection simply of the arbitrary, but of invalid concepts. I submit the willingness of some to do otherwise, is to attempt to preserve other premises of the special sciences held to be true.

This is arguing against a strawman (at least here it is, within this thread). There are no magic unbounded single existents, there are no jars containing indefinite numbers of marbles. The problem is proving there must be an upper limit to the number of fundamental finite existents. One of the important obstacles is that there is no such thing as a "fundamental existent" discovered yet.

One speculation is that the finitude of the universe is local. At sufficiently large extra-galactic distances the Hubble red-shift accumulates to point of making all radiation zero frequency, effectively choking off interaction beyond the visible universe. And yet, a hypothetical spaceship slowboating off in any direction would always measure the same size for that visible universe no matter how far it got from its origin on Earth, similar to how the speed of light is always measured to be the same regardless of the local frame of reference.

Amateur cosmology is pure fun, fun, fun! :fool:

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This is arguing against a strawman (at least here it is, within this thread). There are no magic unbounded single existents, there are no jars containing indefinite numbers of marbles. The problem is proving there must be an upper limit to the number of fundamental finite existents. One of the important obstacles is that there is no such thing as a "fundamental existent" discovered yet.

One speculation is that the finitude of the universe is local. At sufficiently large extra-galactic distances the Hubble red-shift accumulates to point of making all radiation zero frequency, effectively choking off interaction beyond the visible universe. And yet, a hypothetical spaceship slowboating off in any direction would always measure the same size for that visible universe no matter how far it got from its origin on Earth, similar to how the speed of light is always measured to be the same regardless of the local frame of reference.

Amateur cosmology is pure fun, fun, fun! :fool:

See Grames , I don't consider this a cosmological (special science)distinction. It's an epistemic one based on the axioms. To be clear I was not referring to fundamental constituent of matter. My response is directed at the discussion that began with Alepho on the bounded universe. What I consider fundamental is boundedness. We don't run a test to see what we mean by "entity" or universe.

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