A simplified argument from motion

[punkforchrist]

You can remove elements from an infinite set. You can remove an infinity of elements from the set. The number of elements in the remaining set depends on how the infinity of elements was removed. It’s not enough to say you “subtracted infinity”; it’s necessary to stipulate how the removal was performed.

Subtraction isn’t allowed in set theory, which is contradictory with what we know about real world quantities.

As for “special pleading”, that’s nonsense. It’s like saying a “reason” needs to be given for discarding Euclidean geometry in Riemannian spaces. The reason is that finite set theory was formulated for finite sets, just like Euclidean geometry was formulated for flat spaces.

A reason is given for why Euclidean geometry doesn’t correspond to the physical world. The axioms are inconsistent with Einstein’s Special Theory of Relativity.

The temperature is really absolutely zero. Absolute zero degrees Kelvin. Can’t get any lower. And the ground state is not the lowest level of expected energy, but the lowest level of possible energy (lowest eigenstate in quantum harmonic oscillator).

If the ground state were literally absolutely zero, then there would be no energy level whatsoever. Here’s how the Brookhaven National Laboratory defines “absolute zero”: “The lowest possible temperature in the universe, at which all atomic activity ceases.” There is no motion at this level, and absolute zero has never been reached.

That’s a circular argument.

It’s only circular if no reason is given for its conclusion. On the other hand, it is circular for Platonists to cite the legitimacy of actual infinites on the basis that actual infinites exist in the physical world. A because B, and B because A.

A paradox is something which seems to violate our common-sense intuition of how things should be.

So identical quantities don’t have to yield similar results?

 

[SeekingCatholic]

Subtraction isn’t allowed in set theory, which is contradictory with what we know about real world quantities.

Please define what you mean by “subtraction”. Removal from sets is certainly allowed in set theory.

A reason is given for why Euclidean geometry doesn’t correspond to the physical world. The axioms are inconsistent with Einstein’s Special Theory of Relativity.

Yes, we know the reason now. But before Einstein, you would have dismissed Riemann as a crackpot and claimed his geometry had nothing to do with the physical world, since before Einstein no reason was given for why Euclidean geometry didn’t correspond to the physical world?

You would have argued, it’s common sense parallel lines never meet. Since parallel lines do meet in hypothetical curved spaces this is contradictory to what we know about the real world. Curved spaces therefore cannot exist in the real world.

If the ground state were literally absolutely zero, then there would be no energy level whatsoever.

This is false. It is not the “ground state” that is absolute zero. It is the temperature which is absolute zero. The ground state need not have zero energy but can have varying energies, depending on the specifics of the system. The ground state is the lowest energy state. A zero temperature is just that, a zero temperature. It is not “no” or a “lack of” temperature.

There is no motion at this level…

Not true. pa.msu.edu/sciencet/ask_st/012992.html

…and absolute zero has never been reached.

That’s true, but we can certainly conceive of systems at absolute zero.

It’s only circular if no reason is given for its conclusion. On the other hand, it is circular for Platonists to cite the legitimacy of actual infinites on the basis that actual infinites exist in the physical world. A because B, and B because A.

I don’t think this is really formulating the argument in the right way, is it? The argument would be something like, all real numbers correspond to ontological realities in the physical world. There are an infinity of real numbers between 0 and 1. Therefore, actual infinities exist. That isn’t circular, although you can dispute the first premise. That is what I am saying you must do (e.g. refute the first premise).

So identical quantities don’t have to yield similar results?

Identical operations on identical quantities should yield the same result. In your examples though you did not perform the identical operation. So there is no contradiction involved in arriving at dissimilar results.

 

[john_doran]

I’ll agree it goes against “common sense”. There are lots of things which go against common sense and are nevertheless true. It’s “common sense” there should be no quantum tunnelling. Yet it exists.

highly speculative interpretations of quantum experimental data are simply not on all fours with mathematical logic and the ontologies such logic are asked to support.

you might as well just state “it’s common sense that there shouldn’t be acausal quantum events. yet they exist”.

says who?

SeekingCatholic:

Which we just did with tristram shandy.

no, we didn’t. what we showed was that there cannot be a being that exists through each moment of time AND that there are an actually infinite number of temporal moments.

there’s nothing absurd about the proposition “X is a being that exists in every point of time”; there is only an absurdity if you stipulate in addition that the set of temporal points is actually infinite.

SeekingCatholic:

Whether or not there are an infinite number of temporal moments, the existence of a being that exists continuously through an infinite number of temporal moments is logically impossible, as demonstrated via tristram shandy, because that gives him the ability to do the logically impossible. However, the tristram shandy only demonstrates that such a being cannot logically exist, not that a infinite temporal series cannot exist, if all beings exist for only finite periods of time.

not necessarily. the logical inconsistency can be eliminated by eliminating either of the inconsistent conjuncts: by rejecting the coherence of the proposition “possibly, there is a being X such that X exists in every moment of time”, **OR **by rejecting the coherence of the proposition “possibly, the set of temporal moments is actually infinite”.

and, as i have stated before, it seems obvious to me that rejecting the latter proposition is far and away the most plausible thing to do (just like rejecting “vacuum fluctuations are uncaused” is patently more reasonable than rejecting “everything that begins to exist has a cause”).

 

[SeekingCatholic]

highly speculative interpretations of quantum experimental data are simply not on all fours with mathematical logic and the ontologies such logic are asked to support.

you might as well just state “it’s common sense that there shouldn’t be acausal quantum events. yet they exist”.

says who?

Well I could say that too. In any event, “common sense” says that quantum electrodynamics, with uncommon-sensical things in it, should fail miserably. Yet it succeeds spectacularly. “Common sense” is not always a reliable guide. Sometimes it’s wrong.

no, we didn’t. what we showed was that there cannot be a being that exists through each moment of time AND that there are an actually infinite number of temporal moments.

there’s nothing absurd about the proposition “X is a being that exists in every point of time”; there is only an absurdity if you stipulate in addition that the set of temporal points is actually infinite.

Right.

not necessarily. the logical inconsistency can be eliminated by eliminating either of the inconsistent conjuncts: by rejecting the coherence of the proposition “possibly, there is a being X such that X exists in every moment of time”, **OR **by rejecting the coherence of the proposition “possibly, the set of temporal moments is actually infinite”.

Right, exactly as I said. But the set of possible universes now includes both a) those with finite duration and b) those with infinite duration but where eternal existence of an object is impossible. Why would eternal existence be impossible? Well, for one, it could have physical laws similar to our own, which mandate every object to have a beginning and an end.

and, as i have stated before, it seems obvious to me that rejecting the latter proposition is far and away the most plausible thing to do (just like rejecting “vacuum fluctuations are uncaused” is patently more reasonable than rejecting “everything that begins to exist has a cause”).

Yeah, but while it’s logically impossible for both to be true in the same universe, the set of possible universes can include universes with one or the other proposition true.

You would succeed if you could show that, in every possible universe of infinite temporal duration, there must be an object of infinite temporal duration as well, or at least that an object of infinite temporal duration must be possible. However there are possible infinite universes where all objects are contingent (having a beginning and an end).

 

[punkforchrist]

Please define what you mean by “subtraction”. Removal from sets is certainly allowed in set theory.

I’m not aware of any mathematician who holds to that view. However, math dictionaries will define subtraction as the removal or dimunition of a set from another set.

Yes, we know the reason now. But before Einstein, you would have dismissed Riemann as a crackpot and claimed his geometry had nothing to do with the physical world, since before Einstein no reason was given for why Euclidean geometry didn’t correspond to the physical world?

Scientifically speaking, we have yet to receive any reason why actual infinites ought to be accepted. They cannot be observed.

You would have argued, it’s common sense parallel lines never meet. Since parallel lines do meet in hypothetical curved spaces this is contradictory to what we know about the real world. Curved spaces therefore cannot exist in the real world.

I wouldn’t have argued that, since the notion of parallel lines is based on an axiom of Euclidean geometry that has since been undermined by Einsteinian physics. What scientific discoveries have made actual infinites preferable?

This is false. It is not the “ground state” that is absolute zero. It is the temperature which is absolute zero. The ground state need not have zero energy but can have varying energies, depending on the specifics of the system. The ground state is the lowest energy state. A zero temperature is just that, a zero temperature. It is not “no” or a “lack of” temperature.

If you’re using the phrase “absolute zero” in the sense that there is still motion, albeit very little motion, then we’re not barking up the same tree. I’m referring to a state in which there is absolutely no motion (let’s call it absolute absolute zero). In your definition, there is still something in motion, whereas if there is no motion whatsoever, then it is a lack of motion.

Not true. pa.msu.edu/sciencet/ask_st/012992.html

The definiton on the above site is using “absolute zero” in a different sense, as explained above.

I don’t think this is really formulating the argument in the right way, is it? The argument would be something like, all real numbers correspond to ontological realities in the physical world. There are an infinity of real numbers between 0 and 1. Therefore, actual infinities exist. That isn’t circular, although you can dispute the first premise.

I believe it is circular. Let’s take each of your above statements one by one. Let “all real numbers correspond to ontological realities in the physical world” = A. “There are an infinity of real numbers between 0 and 1” = B. In order to arrive at the conclusion C, that actual infinites exist, A must be true, but A is only true if B is true. However, B’s truth is contingent on the instantiation of A. The syllogism is viciously circular, logically speaking.

Identical operations on identical quantities should yield the same result. In your examples though you did not perform the identical operation. So there is no contradiction involved in arriving at dissimilar results.

The operations are identical if both sets contain the same measurable quantity.

 

[SeekingCatholic]

I’m not aware of any mathematician who holds to that view. However, math dictionaries will define subtraction as the removal or dimunition of a set from another set.

In other words, the inverse of set union. It is, in fact, possible, to define inverse operations on transfinite objects.

Scientifically speaking, we have yet to receive any reason why actual infinites ought to be accepted. They cannot be observed.

I would argue that the knowledge that many systems show fractal properties provides a reason to accept actual infinities. Of course, the fractal progression can’t be observed to infinity, due to our observational limitations. However, the fractal property certainly can be, and is.

I wouldn’t have argued that, since the notion of parallel lines is based on an axiom of Euclidean geometry that has since been undermined by Einsteinian physics. What scientific discoveries have made actual infinites preferable?

I’m talking about what you would have argued before the discovery of Einsteinian physics. The point is, as just as you cannot use the axioms of Euclidean geometry to argue against the possibility that space is Riemannian (and if you would have you would have ben wrong), you cannot use the axioms of finite mathematics to argue against the possibility of actual infinities. The axioms of finite mathematics don’t apply to transfinite entities, just as the the axioms of Euclidean geometry don’t apply to Riemannian space.

If you’re using the phrase “absolute zero” in the sense that there is still motion, albeit very little motion, then we’re not barking up the same tree…

I’m using the phrase “absolute zero” in the sense T = 0, which is what it means in physics. There is not an “absence” of temperature, but a zero temperature. Here zero is ontologically real, not just an absence of something.

I believe it is circular.

You are wrong.

Let’s take each of your above statements one by one. Let “all real numbers correspond to ontological realities in the physical world” = A.

This is the Platonist premise.

“There are an infinity of real numbers between 0 and 1” = B.

This is a mathematical fact.

In order to arrive at the conclusion C, that actual infinites exist, A must be true, but A is only true if B is true.

Not so. A’s truth is not contingent on B. A could be true and B false, or vice versa. A could be completely false, and B nevertheless true. Or, A could be true, and B nevertheless false.

However, B’s truth is contingent on the instantiation of A.

B’s truth is a simple mathematical fact not dependent on A.

The syllogism is viciously circular, logically speaking.

No. To defeat the syllogism you need to deny the premise A.

The operations are identical if both sets contain the same measurable quantity.

No, they aren’t. This is only true in the case of finite sets.

I recommend you read the linked article. Your arguments seem to be based on the writings of William Lane Craig. The following article refutes Craig:

webspace.utexas.edu/deverj/personal/papers/worlds.pdf

 

[punkforchrist]

Thanks for the link. Craig has also written a number of responses, which you may be familiar with. leaderu.com/offices/billcraig/menus/existence.html