A review of the Craig v. Malpass discussion

[Boxing Pythagoras]

There is an actual infinity of points in every finite interval! Thus Achilles can catch the tortoise, in Zeno's paradox, by traversing an actual infinity of intervals.

This would be the mainstream, modern view, to be sure. However, Dr. Craig believes otherwise. He's certainly not alone in this-- there are philosophers like Neo-Aristotelians, Constructivists, and metaphysical finitists who would also argue that actual infinites cannot exist. I simply see no reason to think that such a claim is true and the concept of actual infinites forms a rather important structural backbone for the vast majority of modern mathematics.

[shunyadragon]

There is an actual infinity of points in every finite interval! Thus Achilles can catch the tortoise, in Zeno's paradox, by traversing an actual infinity of intervals. You only travers actual infinites of points in time in the human mind, and not in nature.

Blessings,
Lee

Time and space do not have any intervals attached. Even WLC's reference to moments are a human concept of viewing time. Again . . in nature time is not divided into 'moments.'

The closest thing in science is the smallist theoretical measure of time 'planck time,' and the smallest measure used is the Chronon. Again . . . these are human attempts to measure time.

[JonathanL]

That still doesn't make sense to me.

Imagine, for a moment, that I had not clarified that Benardete believed the precise opposite of Dr. Craig regarding the metaphysics of infinity. Which part of Craig's statement would indicate that Benardete says something with which Craig disagrees? It's certainly not the acknowledgment that actual infinites are not logically inconsistent, as Craig himself believes that. It's not in his characterization of actual infinites as "monstrosities" or in the direct claim that they are metaphysically impossible. It is nowhere in his discussion of the infinite book. At no point does Dr. Craig make any indication that he is citing someone who believes the exact opposite of what he is saying.

Well, even before you even brought up the issue of Dr. Craig supposedly misrepresenting Benardete it didn't even occur to me to interpret Craig's words as him claiming that Benardete thought actual infinities were impossible. My immediate reaction was to understand the part that I bolded as him expressing disagreement with Benardete. I could be wrong in the particular details of my interpretation of Craig's word, but I think my general understanding is correct.

And I do think he gives ample indication using tone of voice, facial expression and head gestures to indicate disagreement with Benardete, so even if he doesn't explicitly state that Benardete thinks actual infinities are possible, but that he himself disagrees with that notion, I think he still implies it with all of the previous mannerism I mentioned. Assuming I'm correct that all of the above indicates disagreement, given that he clearly agrees with Benardete that actual infinities are not logically impossible, his disagreement would be with something not directly stated, which would reasonably be the position that actual infinites are metaphysically possible. If I'm correct that Craig is expressing disagreement in the part that I bolded, then it's also reasonable to infer that Benardete's position is the opposite of Craig's. IOW, Craig doesn't have to explicitly state what Benardete's position is on the issue, because by stating his own opinion in a way that expresses disagreement with Benardete, the implication is that Benardete holds to the opposing position.

[QuantaFille]

I honestly don't notice any such shift in tone; however, Benardete never even refers to actual infinites as "monstrosities," even if the second half of the sentence is meant to be Dr. Craig's own thoughts. It certainly seems like Dr. Craig is attempting to cite Benardete as supporting his position-- otherwise, why reference Benardete at all? There's no need to cite another scholar simply to acknowledge that he's not claiming actual infinites are logically inconsistent.

I'm going to add my 2˘. I take Dr Craig to be saying, "look, here's this guy who agrees with me that there is no logical contradiction here, and I'd like to add that there is a contradiction I see and it's a metaphysical one".
I assume, without having read Benardete's book myself, that Benardete is saying that a physical infinite is possible, and he bases that on the fact that there's no logical contradiction and does not address the metaphysical aspect.

[Boxing Pythagoras]

I'm going to add my 2˘. I take Dr Craig to be saying, "look, here's this guy who agrees with me that there is no logical contradiction here, and I'd like to add that there is a contradiction I see and it's a metaphysical one".
I assume, without having read Benardete's book myself, that Benardete is saying that a physical infinite is possible, and he bases that on the fact that there's no logical contradiction and does not address the metaphysical aspect.

Quite the contrary. The title of the book is Infinity: An Essay In Metaphysics. The entire goal of the book is to defend the notion of infinity against both metaphysical and mathematical finitists.

[Boxing Pythagoras]

Incidentally, I just had my wife listen to the section in question, completely without context, and she agrees with Chrawnus and QuantaFille that it does not seem like Dr. Craig is citing someone who agrees with him; so I'll certainly concede that I'm likely the one misunderstanding his intention here.

That said, what do we think about his actual argument? I'm still not clear on how Dr. Craig moves from "these things are strange" to "these things are metaphysically impossible."

[shunyadragon]

I'm going to add my 2˘. I take Dr Craig to be saying, "look, here's this guy who agrees with me that there is no logical contradiction here, and I'd like to add that there is a contradiction I see and it's a metaphysical one".
I assume, without having read Benardete's book myself, that Benardete is saying that a physical infinite is possible, and he bases that on the fact that there's no logical contradiction and does not address the metaphysical aspect.

First, I am not satisfied with Benardete's argument.

Personally my approach to the problem from a matter of fact of the limits of descriptive math when applied to metaphysical (?) assertions. Further, clarification is needed what is considered to be the metaphysical aspect as opposed the math of actual versus potential infinites as applied to our physical existence. Actual infinities are defined as 'closed' sets of infinities, and potential infinities are' open' sets without any defined limit. The usefulness of Math is defined as the ability to be descriptive of our physical existence. The only way an actual infinity can be applied to the past is to define a closed infinity with a beginning point, which cannot be done, nor can it be a closed set in the present.

Again . . . please explain What is the 'metaphysical aspect' where the math of infinities can be applied when they are asserted to be descriptive of the nature of our physical existence. This is as slippery as trying to nail a ten pound block of Jello to the wall.

[Boxing Pythagoras]

First, I am not satisfied with Benardete's argument.

Do you mean Dr. Craig's argument? We haven't actually discussed any of Benardete's arguments, other than to say that he thinks actual infinites ARE metaphysically possible.

Actual infinities are defined as 'closed' sets of infinities, and potential infinities are' open' sets without any defined limit.

This isn't quite correct. An actual infinite is any set which can form a bijection with a proper subset of itself. The classic example is the Natural numbers which can be placed in one-to-one correspondence with, for example, the Even numbers. Since the Evens are a proper subset of the Naturals, this implies that the Naturals form an actually infinite set.

A potential infinite, on the other hand, describes a process and not a set. A potential infinite refers to a process which can be iterated indefinitely. Here, the classic example is simple counting. This is a step-by-step process which can (at least theoretically) continue indefinitely, since every number counted has a successor which can be counted in the next step.

The usefulness of Math is defined as the ability to be descriptive of our physical existence.

Quite a number of mathematicians and philosophers of mathematics would disagree with you, there.

The only way an actual infinity can be applied to the past is to define a closed infinity with a beginning point, which cannot be done

Neither of these points is actually correct. It's fairly trivial to imagine an infinite past without a beginning point. And even if we suppose that time is finitely bounded in the past direction, it's not difficult to define a scenario in which the number of past moments is still actually infinite.

nor can it be a closed set in the present.

Not sure what you mean, here.

Again . . . please explain What is the 'metaphysical aspect' where the math of infinities can be applied when they are asserted to be descriptive of the nature of our physical existence. This is as slippery as trying to nail a ten pound block of Jello to the wall.

I'm not sure what you are getting at, here, either. There's quite a number of metaphysical questions as regards mathematics, in general, and actual infinites, in particular.

[shunyadragon]

Do you mean Dr. Craig's argument? We haven't actually discussed any of Benardete's arguments, other than to say that he thinks actual infinites ARE metaphysically possible.

Both arguments are flawed. I do not believe actual infinities are metaphysically meaningfully possible.

This isn't quite correct. An actual infinite is any set which can form a bijection with a proper subset of itself. The classic example is the Natural numbers which can be placed in one-to-one correspondence with, for example, the Even numbers. Since the Evens are a proper subset of the Naturals, this implies that the Naturals form an actually infinite set.

I disagree, as far as an infinite set of either even or odd numbers represent potential infinities as you describe below.

A potential infinite, on the other hand, describes a process and not a set. A potential infinite refers to a process which can be iterated indefinitely. Here, the classic example is simple counting. This is a step-by-step process which can (at least theoretically) continue indefinitely, since every number counted has a successor which can be counted in the next step.

Agree with your description of a potential infinity, I disagree. In math infinities are also defined in terms of sets.

Neither of these points is actually correct. It's fairly trivial to imagine an infinite past without a beginning point. And even if we suppose that time is finitely bounded in the past direction, it's not difficult to define a scenario in which the number of past moments is still actually infinite.

Not sure what you mean, here.


I'm not sure what you are getting at, here, either. There's quite a number of metaphysical questions as regards mathematics, in general, and actual infinities, in particular.

I am not sure what you are getting at. Philosophers try to apply math to metaphysical question, but I consider it fruitless effort. I have not problem with disagreeing. Applying math to this apologetic argument needs too many metaphysical assumptions. Examples please. I disagree. No explanation is forthcoming how these assertions of metaphysical claims can be justified by math. The question still stands what are the metaphysical aspects involved here?

[shunyadragon]

Incidentally, I just had my wife listen to the section in question, completely without context, and she agrees with Chrawnus and QuantaFille that it does not seem like Dr. Craig is citing someone who agrees with him; so I'll certainly concede that I'm likely the one misunderstanding his intention here.

That said, what do we think about his actual argument? I'm still not clear on how Dr. Craig moves from "these things are strange" to "these things are metaphysically impossible."

I do not believe it can be determined if 'these things' are metaphysically possible nor possible with this argument,

[Boxing Pythagoras]

Both arguments are flawed. I do not believe actual infinities are metaphysically meaningfully possible.

Why not? They seem to me to be just as meaningful for metaphysics as any other aspect of mathematics.

I disagree, as far as an infinite set of either even or odd numbers represent potential infinities as you describe below.

No, they don't. We can meaningfully talk about the whole set of Even numbers without having to enumerate them in a step-wise fashion. And it's rather simple to show that the Even numbers form a bijection with proper subsets of the Even numbers (for example, the multiples of 4). This is precisely what mathematicians have meant by an actually infinite set for centuries, now.

Agree with your description of a potential infinity, I disagree. In math infinities are also defined in terms of sets.

Actual infinites refer to sets, in math. Potential infinites refer to processes.

I am not sure what you are getting at. Philosophers try to apply math to metaphysical question, but I consider it fruitless effort.

Why do you consider it fruitless? Quite the contrary, mathematics seems very fruitful for metaphysics. After all, if something is mathematically impossible most people would agree that such a thing is metaphysically impossible.

Applying math to this apologetic argument needs too many metaphysical assumptions.

What metaphysical assumptions would those be?

No explanation is forthcoming how these assertions of metaphysical claims can be justified by math.

What assertions are you talking about?

The question still stands what are the metaphysical aspects involved here?

The concepts of number, magnitude, and quantification.

[shunyadragon]

Why not? They seem to me to be just as meaningful for metaphysics as any other aspect of mathematics.

I do not consider metaphysics to be an aspect of math. Math is independent of metaphysics and it is essentially a neutral tool box to be used and applied to problem solving, or in the case of metaphysics misused.

No, they don't. We can meaningfully talk about the whole set of Even numbers without having to enumerate them in a step-wise fashion. And it's rather simple to show that the Even numbers form a bijection with proper subsets of the Even numbers (for example, the multiples of 4). This is precisely what mathematicians have meant by an actually infinite set for centuries, now.

Yes if defined as infinite they are potential infinite sets. A simple text that explains this Infinity and the Mind by Rudy Rucker

Actual infinities refer to sets, in math. Potentially infinite refer to processes.

No, both may be expressed and defined by sets.

Why do you consider it fruitless? Quite the contrary, mathematics seems very fruitful for metaphysics. After all, if something is mathematically impossible most people would agree that such a thing is metaphysically impossible.

To be blunt no, unless you can provide some coherent examples. Please give examples, and explain what was meaningful referred to specifically as 'metaphysical aspects.'

What metaphysical assumptions would those be?
What assertions are you talking about?

The assertions that are the presuppositions of apologist arguments, such as 'The universe began to exist.'

The concepts of number, magnitude, and quantification.

Unless you are counting Gods one believes in, I believe metaphysics, such as the apologist arguments, are too subjective to use math to support their arguments.

Still waiting for what are the metaphysical aspects I asked about. There is no explanation forthcoming.

[lee_merrill]

Time and space do not have any intervals attached. Even WLC's reference to moments are a human concept of viewing time. Again . . in nature time is not divided into 'moments.'

No, but we can divide time and space into intervals.

The closest thing in science is the smallist theoretical measure of time 'planck time,' and the smallest measure used is the Chronon. Again . . . these are human attempts to measure time.

And just because there is a smallest measure of time that we can theoretically measure does not mean we cannot consider intervals of time (and space) that are arbitrarily small.

Blessings,
Lee

[shunyadragon]

No, but we can divide time and space into intervals.

This an artificial construct of using math for science, and does not reflect the nature of time nor space it self, nor whether it is finite nor infinite, or temporal nor eternal.

And just because there is a smallest measure of time that we can theoretically measure does not mean we cannot consider intervals of time (and space) that are arbitrarily small.

The point was that system of measurement and math are developed for understanding our physical existence is a tool of science, and time and space has no increments of time naturally and cannot be used to artificially demonstrate whether space nor time is infinite or finite. .

Actually science is not all that concerned whether our physical existence is fine nor infinite, because it cannot be falsified either way. In fact in math has been demonstrated in physics to work very well even without the symbol of infinity.

Metaphysics makes claims based philosophy and beliefs, which are premises in their arguments to justify their beliefs. Science does not make claims that cannot be falsified by scientific methods.

[QuantaFille]

Quite the contrary. The title of the book is Infinity: An Essay In Metaphysics. The entire goal of the book is to defend the notion of infinity against both metaphysical and mathematical finitists.

I think what I was thinking was more along the lines of, Dr Craig doesn't address what Benardete thinks about the metaphysics of it, only what Benardete thinks about the logical aspect. I do definitely hear the shift in tone that Chrawnus mentioned. He starts speaking with a sort of half laugh in his voice right around the word "monstrosities".