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Arguments Against Infinite Quantity

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Since neither sound credulous, it would amount to Galileo's observation of comparing infinities to one another for greater than or less than as essentially meaningless.

Food for thought...

If there is in fact a metric expansion of space, entities can, in a certain sense, be "infinitely" far apart, i.e., they can become causally disconnected, unable to interact with each other by any possible means.

Consider a couple of examples of how large distances are measured, e.g., RADAR or laser ranging of the moon. Over very large distances and in a metrically expanding space, there is the possibility that the radio or light wave never "catches up" with the receding entity. In effect, that entity is "infinitely" far away in the sense that no physical thing, even light, can ever get "there" from "here".

Edited by Alfred Centauri
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Expand? Into what? The universe is all that is. If the new area it 'expanded' into existed, it was already part of the universe to begin with by definition. Sounds a little pardoxical, does it not?

I don't believe "into what" is logically required to assert a thing (or collection of them) expands.

Mindy

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I'm don't think that one can meaningfully talk about the extent of the universe unless it is qualified, e.g., the extent of the observable universe. Surely it's true that the distance between any two entities in the universe is finite and surely it's true that the extent of any entity is finite. I think, in this sense, it is correct to say that the universe is not infinite, i.e., that there are no infinite distances, infinite extents, etc.

Nonetheless, there is no logical reason that distances and extents cannot be arbitrarily large, i.e., have no upper bound.

Why must talk of the extent of the universe be limited to the observed universe?

Mindy

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Food for thought...

If there is in fact a metric expansion of space, entities can, in a certain sense, be "infinitely" far apart, i.e., they can become causally disconnected, unable to interact with each other by any possible means.

Consider a couple of examples of how large distances are measured, e.g., RADAR or laser ranging of the moon. Over very large distances and in a metrically expanding space, there is the possibility that the radio or light wave never "catches up" with the receding entity. In effect, that entity is "infinitely" far away in the sense that no physical thing, even light, can ever get "there" from "here".

This seems to confuse ontology with epistemology. It is about measuring distances, not about extension itself. I don't see the logic of resorting to the term "infinite" because it is alone in its position. (If that is just a calculational convenience, I see its use, but in this discussion, I think it may be confusing.)

Mindy

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@Jake, It's clear that the center of a circle is not in the circle, right? And it's clear that it is special, i.e., unique, right? Thus, how can you say that any point in the manifold can be considered a center?

Moreover, AFAIK, distance is a magnitude, i.e., positive (we're not talking about intervals here). Thus, the sum of the distances to all other points cannot be zero unless all "other" points are the chosen point.

I read "special" as referring to the symmetric placement; I would expect a Mathematics site to use the term "unique" if that is an essential feature. In this sense, "special" is as opposed to "arbitrary."

You're right, I should have said that the sum of all vectors starting at the chosen point is zero, since each vector will have a symmetric partner (equal in magnitude and opposite in direction).

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Food for thought...

If there is in fact a metric expansion of space, entities can, in a certain sense, be "infinitely" far apart, i.e., they can become causally disconnected, unable to interact with each other by any possible means.

The article appears to treat 'space' as something that can expand. I think it qualifies under Harriman identificaion that modern science is reifying space. This was implemented by Newton, perhaps inadvertently with his positing 'absolute space', and has been augmented by Einstein with his application of curvature to it.

As he stated in one of his lectures: (paraphrased) Honey, we have this space in the living room. What are we going to do with it? Are we going to leave it in there, or shall we move the space out into the back yard so we can make room for our furniture?

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Just wanted to reiterate that geometry has nothing to do with metaphysics and ontology. All attempts to answer these questions by consulting special

knowledge is a non-sequitor.

If it is true that geometry has nothing to do with metaphysics and ontology, it is still not true that metaphysics and ontology have nothing to do with geometry. Philosophy covers everything. What "special knowledge" cannot do is contradict, replace, or make moot the rest of our knowledge.

Mindy

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Just wanted to reiterate that geometry has nothing to do with metaphysics and ontology. All attempts to answer these questions by consulting special knowledge is a non-sequitor.

While geometry is a branch of mathematics, infinity, as a concept of method, has its epistemological roots firmly planted within the mathematical pot as well. The OP's stance is that infinte quantity is metaphysically applicable.

The geometric reference was an attempt to illustrate that

We percieve life as being temporal: We try to apply temporalness to the universe.

We percieve existents as having size: We try to apply size to the universe.

We percieve existents as having shape: We try to apply shape to the universe.

We percieve existents as having finite quantites: We try to apply infinite quantites to the universe.

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If it is true that geometry has nothing to do with metaphysics and ontology, it is still not true that metaphysics and ontology have nothing to do with geometry. Philosophy covers everything. What "special knowledge" cannot do is contradict, replace, or make moot the rest of our knowledge.

Mindy

My comment only pertains to the "If" part of your post. Any non-contradictory statements from the special sciences do not bolster fundamental truths.

Edited by Plasmatic
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Just wanted to reiterate that geometry has nothing to do with metaphysics and ontology. All attempts to answer these questions by consulting special knowledge is a non-sequitor.

It is equally true that any attempt to deduce facts from metaphysics is pure rationalism.

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The article appears to treat 'space' as something that can expand. I think it qualifies under Harriman identificaion that modern science is reifying space. This was implemented by Newton, perhaps inadvertently with his positing 'absolute space', and has been augmented by Einstein with his application of curvature to it.

I just want to clarify that neither Einstein nor his equations reify space. Later interpreters and people trying to dumb it down for laymen (Brian Greene, et al.) have reified space.

Here's a relevant quote:

In this edition I have added, as a fifth appendix, a presentation of my views on the problem of space in general and on the gradual modifications of our ideas on space resulting from the influence of the relativistic view-point. I wished to show that space-time is not necessarily something to which one can ascribe a separate existence, independently of the actual objects of physical reality. Physical objects are not in space, but these objects are spatially extended. In this way the concept "empty space" loses its meaning.

Italics original

I think that's the third time I've posted that quote to this forum, but it's a point I like to make. The math is solid, even if some attempts at physical intrepretation are faulty.

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Thanks Jake. Dr. Harriman may be referencing the 'dumbed down' version in some of the lectures of his I have had opportunity to listen to. Perhaps I should withdraw judgement on that point until I can make time to familiarize myself with the Einsteins works personally. His greater critique was likely down the avenue of stating that Einstein was merely describing appearances rather than identifying what is.

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It is equally true that any attempt to deduce facts from metaphysics is pure rationalism.

In Philosophy, Who Needs It? Rand imagines a spaceship crashed onto an unknown planet. How will, and should the astronauts act? This, She says, is where philosophy is of vital importance. They know nothing about the world they have landed on. But, they know that whatever it is like, it has identity and that they can discover that identity, and make plans accordingly.

Whatever exists has identity; This planet exists; This planet has identity.

The nature of a thing can be discovered. This planet has a nature (identity;) We can discover the nature of this planet.

Isn't that deduction from metaphysics, in a most practical situation?

Mindy

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In Philosophy, Who Needs It? Rand imagines a spaceship crashed onto an unknown planet. How will, and should the astronauts act? This, She says, is where philosophy is of vital importance. They know nothing about the world they have landed on. But, they know that whatever it is like, it has identity and that they can discover that identity, and make plans accordingly.

Whatever exists has identity; This planet exists; This planet has identity.

The nature of a thing can be discovered. This planet has a nature (identity;) We can discover the nature of this planet.

Isn't that deduction from metaphysics, in a most practical situation?

Mindy

Whatever identity the planet has, it can only be found out by beginning with perception. I repeat, any attempt to deduce facts from metaphysics is pure rationalism.

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I guess Grames thinks that rejecting an invallid concept that contradicts identity is "rationalism".

All of the following are errors (stolen from dream_weaver's post above):

We percieve life as being temporal: We try to apply temporalness to the universe.

We percieve existents as having size: We try to apply size to the universe.

We percieve existents as having shape: We try to apply shape to the universe.

We percieve existents as having finite quantities: We try to apply finite quantities to the universe. (fixed)

You cannot demonstrate a contradiction about the universe's size because the concept size does not apply to the universe.

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All of the following are errors (stolen from dream_weaver's post above):

We percieve existents as having finite quantities: We try to apply finite quantities to the universe. (fixed)

You cannot demonstrate a contradiction about the universe's size because the concept size does not apply to the universe.

The OP is trying to apply infinite quantities to the universe.

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Whatever identity the planet has, it can only be found out by beginning with perception. I repeat, any attempt to deduce facts from metaphysics is pure rationalism.

And would you then go outside the ship to try to perceive whether or not the law of identity held there? Or would you go out to collect data because you knew identity held, so that knowledge would be useful?

Mindy

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  • 2 weeks later...
  • 2 weeks later...

We've established (your point 4 in the OP) that any positive claim of a countably infinite set of physical objects is arbitrary, unless you can provide some sort of very indirect evidence. I want to try to pin you down as to exactly how arbitrary you think it is.

While you seem to accept the higher-level hypothesis that there may exist a scientific theory (as in your post #20) that demands the existence of such a collection, I also take it you are not claiming to have produced an example of such a theory.

Not at all. I may produce scenarios in which a theory that makes use of infinite quantities is the most explanatory, but I don't claim that they're actual. I'm asking a metaphysical rather than a factual question.

But to have it said: If we are no longer disputing the impossibility of infinite quantities, and it is recognized that no satisfying argument has been provided to that effect, then I am happily unconcerned with other issues. However, I'll still respond to the questions below.

(1a) Suppose it is shown that no such scientific theory is possible. Is there then any other way in which evidence can be proffered for the existence of an infinite number of physical objects?

You mean, suppose that it is impossible for there to be infinite quantities, and then you ask if there is any other way in which evidence can be proffered for the existence of infinite quantities? I doubt it, but if it can be shown that infinite quantities are impossible, then I will be satisfied in the first place.

(1b) If, for the sake of argument, such a scientific theory was proposed, do you think it would be sufficient to justify the literal existence of such a set of objects?

Yes. Why do I accept the existence of atoms? Because the atomic theory is that which best explains and predicts behavior which I can observe. I have never observed an atom, but the atomic theory has too many theoretical virtues to justify rejecting it. It is (relatively) simple, lends to calculation, predicts events accurately, accurately suggests ways to manipulate the world around us, and so on. In fact, simplicity was the original reason for the popular adoption of heliocentrism. Geocentrism was never disproved, probably until we launched a person into space. While Ptolemaic geocentrism could accurately predict the location of celestial bodies, for an observer at a fixed parallel, when shifting parallels the calculations were no longer accurate. Trying to come up with Ptolemaic models of celestial positions and orbits became computationally nightmarish, when Copernicus's incredibly simple model lent to easy computation with just as much accuracy, and so it was adopted. I don't believe either of these theoretical changes are philosophically suspect, in spite of not having been supported by direct observation.

Likewise, in general, if a theory has such theoretical virtues, then I come to believe the theory without insisting on direct observation. Thus if a theory has these virtues and makes essential claim to an actual infinity of objects, then I will believe in the theory and thereby believe in an infinity of objects.

Next, regarding the acceptance of arbitrary claims:

(2a) If you seriously accept arbitrary claims such as infinite numbers of objects as a possibility due to a lack of self-contradiction, do you regard as equally admissible physical accounts appealing to the existence of Zeus, Ra, etc., as there is equal evidence for both?

So long as the theories are not self-contradictory, yes, I take them to be possible in that sense. I take the claim that there is a person on the roof of my building to be equally arbitrary in that sense, and equally possible in that sense. Some of the arbitrary claims will sound fantastic because we have never seen or heard anything like it before, some of them will sound mundane and more plausible because we have seen similar things, but that doesn't affect the fact that they're equally unsupported by observational evidence and so--in the sense you're using the term "arbitrary" here--equally arbitrary. And fittingly, I don't believe in the existence of Zeus, and I don't believe in the existence of an infinite quantity.

Now, if somebody were to claim to have evidence for the existence of Zeus, I would be more suspicious since there seems to be so much evidence against it: What's he been doing for the last couple thousand years at the least? Where's he been? Moreover, the story of Zeus seems to be too easily explained by child-like fantasies and desperate attempts by ignorant people to explain the world. That is to say, I would be more reluctant to accept the Zeus hypothesis because I already have another one that is working quite well. But if, some how--say by direct observation, or by theories which make essential use of the Zeus hypothesis to predict all events better than current theories, etc.--there were better evidence for Zeus than against him, then I would believe that Zeus exists. Infinity, on the other hand, has none of these unpleasant properties. Current theories, as far as I can tell, are completely agnostic on this account and so there is no evidence against the existence of infinite quantities. Moreover, it doesn't seem like some infantile attempt at explaining anything. So far, it's no explanation at all in the first place--there is no attempt at explaining any fact by reference to infinite quantities. The hypothesis of an actual infinity does not claim to actually solve anything, just yet. But if it did, and if there were good evidence and theoretical virtues, then it would have a case worth listening to. And as we seem to have established, there is no reason why this theory would violate the Principle of Identity, and so there would no reason to reject it out of hand.

Along comes Cantor. Not satisfied with the limitation imposed by the relation established between the concept and the percieved, sweeps this aside insisting that 'countable' suggest that it can be counted, dismissing that it requires a completion of the task to be considered counted, and states that in an infinite series of counting numbers, they are 'countable', even though it can never be completed. As TheEgoist points out in his post in dealing with infinity, there is no way to determine how far you are from the end at any point.

Why is that problematic? There wouldn't be an end-point. So what?

Galileo made some similar observations about infinity in his observation that dealing with multiple instances of infinity, there is no way to compare equal to, greater than, or less than. He concluded that such comparisons where meaningless.

Reference? And why can you not talk about "greater than"? Sure, the natural numbers are infinite, but you can still pick some two natural numbers and compare whether they're greater, less, or equal to each other. Comparing some natural number to the number infinity itself? Well, here you'd have to somewhat expand your notion of "greater" and "equinumerous", or you'd be right, there would be no sense in comparing them. But if your notion of "greater" is redefined by the notion of mappings, as we started with, then infinite sets will be greater in quantity than any finite set.

As Plasmatic points out, he does not prove axiomatic things. The axiomatic is outside the provence of proof. It can be validated, if an individual chooses to validate it for themselves, but an explaination alone does not suffice. The validation, for lack of perhaps a better way of putting it, also has to be 'experiential'.

Well then, what's the axiom? You can't just say it's an axiom without naming what axiom you're appealing to. And if you say "identity", then you have to say how identity precludes infinity. Because as far as I can tell, there is no connection whatsoever between the axiom of infinity and identity. If you bring up the point about boundaries, then you again have to say why it is that lacking a boundary on quantity is impossible.

I.e., you have not said anything new here, and only suggest a rehearsal of a long conversation that has already played out, and ended with no good argument against the possibility of an infinite quantity. This is intellectually irresponsible, since it attempts to suggest that the case has been made to satisfaction, for the impossibility of an infinite quantity, when no such thing is even remotely true.

Are we not more aptly stated, a part of the universe, rather than inside of something, suggesting that an outside be inferred? Are we going to pick and choose the attributes we wish to anthropomorphise in the case of having a beginning or end? Are we going to induce size and shape and extrapolate it to that which has no 'outside'? And then, inducing from the evidence that all that we have encountered to date are indeed finite in quantity as well as their other attributes, choose to disregard that induction in an attempt to apply infinite to attributes that do not appear to fit without contradictions?

My apologies if this is not stated in the clearest, most precise terms.

This point has been made before, and refuted before, unless you can come up with a counter-counter-argument.

Your injective map definition assumes the natural numbers have infinite quantity, so it is circular.

To say that the natural numbers are infinite is to say:

- there is no largest natural number, or

- the natural numbers are unbounded, or

- for any natural number n, n+1 is a natural number, or

- the series of natural numbers is open-ended.

It does not mean that there are an infinite number of natural numbers.

By saying that the natural numbers are infinite, I am merely taking this to be a piece of our mathematical rules. It's not circularity, but statement of definition. To have the size of the set of natural numbers is just to stand in an bijection relation to the natural numbers, which has the properties you mention above. But sets do not grow by thinking up new members--sets are determinate collections of objects. So in sets, there is no such thing as the possibly infinite. There is only what is actually in the set and what is not.

However, as stated a few posts into the discussion, even if we amend our mathematics to accommodate a notion of possible infinity, I may simply re-phrase my question: Is it possible or impossible for there to be, for every set of natural numbers, an injective map from that set to some set of disjoint physical objects? If so, then I would take this to be my definition for there being an infinite quantity.

The concept of "quantity" refers to the amount of something, which is determined by counting the units which comprise it. This is not a "special notion." The idea of "infinite quantity" is invalid because the Law of Identity demands that whatever the final count of units is, it is that count. If the counting process is open-ended, the counting is never complete: there is no final count; there is no quantity.

I've responded to all this before.

How does all this gainsay William Lane Craig's Kalam argument? He argues against an actual infinity.

He argues that the universe was caused, from the assumption that it has a beginning time. I don't know if he argues, or how he argues, the acceptance of the assumption, but it's not the same as arguing against an actual infinity. If you know of some such argument that he does employ, please reproduce it here because it is not an article of common knowledge, or even of widely known philosophical knowledge.

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  • 3 weeks later...

aleph_o, WLC maintains, with his discsussions of Hilbert hotels, infinite books in a library and other red herrings, that non-mathematical infinity is impossible. He maintains that by successive addition, no one can attain infinity,but that is the essence of ininity! Everyday arrives on time as Kyle Williams and Aquinas argue, so that an infinity of days- real stuff, not a mathematical series: there is indeed no final count!

He fails to realize that infinity minus infinity, the series of odd and even numbers + the series of just odd or just even numbers and such matters, whilst different from finite numbers, do not invalidate the notion of infinity for material stuff.

He fails to realize that the Big Bang was no creation but a transformation. Morriston, Grünbaum and Oppy,amongst others, have presented the case against the Kalam.

Edited by skeptic griggsy
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Aleph_0:

What I was trying to get at is this: right now you're holding out hope that a future scientific theory may provide indirect evidence for an infinite quantity. If it can be shown that scientific theories by their nature cannot justify the existence of infinite quantity, you have no grounds to suppose them except to reify them from advanced mathematics.

However, scientific theories are created in a finite context, by observing finite relationships between measurable phenomena in a specific range justified by the data (one of the points made by Harriman's new book). I can't see how, from this, you can ever extrapolate to justifying an infinite number of objects. The analogy to atoms is inappropriate, since you claim that infinite quantities are justified by scientific theories, not just inperceptibly large (or small) quantities.

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