## What is the difference between actual infinity and potential infinity?

• 1.2k
In that vein, do you recognize that there's a conceptual distinction between an "actual infinity" and a "potential infinity"?
— Relativist
Yes, it corresponds to the difference between metaphysical actuality and logical possibility. Again, mathematical existence refers to the latter, not the former.

I wonder if I understand that. Potential infinity is often taken to be the collection (but not set) of the natural numbers 0, 1, 2, 3, ... It's potential in the sense that given n you automatically have n+1; but you never have all of them taken together at once in a set.

But how is that metaphysical actuality? There's nothing in the physical world that corresponds to the endless sequence of natural numbers. It seems to me that 0, 1, 2, 3, ... and {0, 1, 2, 3, ...} are equally abstract. One can accept or reject the axiom of infinity; but either way you end up with a structure whose "actually" is certainly in question. Yes? No? I have no idea.
• 1.2k
But how is that metaphysical actuality?
It is not; as I said, mathematical existence--including the potential infinity of the natural numbers--is not metaphysical actuality, it is logical possibility. From your other comments, I think that we agree on this; perhaps you misread my previous post.
• 1.2k
perhaps you misread my previous post.

I think I still don't know what actuality means.

But you said that potential infinity has metaphysical actuality. Don't the natural numbers (as modeled by the Peano axioms) contradict that?
• 1.2k
But you said that potential infinity has metaphysical actuality.
No, I said exactly the opposite of that.
In that vein, do you recognize that there's a conceptual distinction between an "actual infinity" and a "potential infinity"?
Yes, it corresponds to the difference between metaphysical actuality and logical possibility. Again, mathematical existence refers to the latter, not the former.
Actual infinity corresponds to metaphysical actuality, while potential infinity and mathematical existence correspond to logical possibility.

Since I have found you to be a normally clear-headed and insightful participant here, I fear that your persistence in dealing with @Metaphysician Undercover lately may be producing some unfortunate side effects. :grin:
• 325
It is not; as I said, mathematical existence--including the potential infinity of the natural numbers--is not metaphysical actuality, it is logical possibility

Metaphysical Actuality: The philosophical position that thought becomes actual by becoming concrete. Subjectivity, the "I" has constitutive validity, having sole omnipotence.

This seems to me an extreme position. How then does it interact with historical actuality? :chin:
• 1.2k

That is not what I mean by "metaphysical actuality." I just mean the modal property of being actual, rather than merely possible or strictly necessary, such that something possessing it acts on and reacts with other things in the environment.
• 1.2k
Since I have found you to be a normally clear-headed and insightful participant here, I fear that your persistence in dealing with Metaphysician Undercover lately may be producing some unfortunate side effects.

LOLOL.
• 325
That is not what I mean by "metaphysical actuality." I just mean the modal property of being actual, rather than merely possible or strictly necessary

So, combining "metaphysical" with "actual" means someone is thinking a metaphysical thought? Or does the expression imply an interaction with physical reality? I am going on a classical definition of the expression. What do you really mean? Please clarify with examples.

I am not a philosopher.

Thanks. :chin:
• 1.2k
So, combining "metaphysical" with "actual" means someone is thinking a metaphysical thought? Or does the expression imply an interaction with physical reality? I am going on a classical definition of the expression. What do you really mean? Please clarify with examples. Thanks.

I didn't understand a word @aletheist wrote but I was embarrassed to admit it.

I do not understand what metaphysical actuality is. Are the natural numbers metaphysically actual? The completed set of natural numbers? The square root of 2? Chaitin's constant, which is known to be noncomputable? Is a brick metaphysically actual? How about an electron? A quark? A string?
• 1.2k
So, combining "metaphysical" with "actual" means someone is thinking a metaphysical thought?
What does thinking have to do with anything? Metaphysics is a branch of philosophy, within which "actual" has a technical meaning that distinguishes it from "possible" and "necessary."

Or does the expression imply an interaction with physical reality?
That is closer, since whatever is physical is actual in the relevant sense. However, just to be clear, I hold that reality is not coextensive with actuality; there are also real possibilities and real necessities.

If I say that I have an apple, what I usually mean is that I have an actual apple. If I posit a set of apples in the strictly mathematical sense, then I am talking about something that is logically possible, but not (necessarily) actual. :smile:
• 325
If I say that I have an apple, what I usually mean is that I have an actual apple. If I posit a set of apples in the strictly mathematical sense, then I am talking about something that is logically possible, but not (necessarily) actual

"Metaphysics is a branch of philosophy, within which "actual" has a technical meaning that distinguishes it"

Speak to metaphysics, please. Define "actual" in that context. :nerd:
• 1.2k
Are the natural numbers metaphysically actual? The complete set of natural numbers? The square root of 2? Chaitin's constant, which is known to be noncomputable?
No, these are all numbers; and again, existence in mathematics entails only logical possibility, not actuality in metaphysics.

Is a brick metaphysically actual?
Yes, in accordance with how I was using that term.

How about an electron? A quark? A string?
That depends on whether one is a scientific realist about each of these entities. I am currently inclined to say yes, probably, and maybe.
• 325
Speak to metaphysics, please. Define "actual" in that context.

Metaphysics is a branch of philosophy, within which "actual" has a technical meaning that distinguishes it from "possible" and "necessary."
• 1.2k
Speak to metaphysics, please. Define "actual" in that context.
The actual is that which acts on and reacts with other things.
• 325
I don't understand your definition re: metaphysics. What are things? We seem to be in a downward spiral here.

On the other hand, for a little clarity:

Des Bosses to Leibniz (1700s): "Monads are metaphysical actualities."

Now, that makes sense. :cool:
• 6.6k
Do abstractions exist at all? I suggest they don't. A number line "exists" only as an abstraction, but this is not true existence. It's just a concept, in which a set of logical/mathematical properties are considered abstractly. The same is true of numbers, whether rational or irrational. "3" doesn't exist, but collections of 3 objects exist - so we can think abstractly about 3-ness. Neither does Pi exist; nevertheless we can abstractly consider the fact that all "circles" (another abstraction) have Pi as the ratio between their circumference and diameter.

That's right, we covered this earlier in the thread. We need to distinguish between three things, the symbol, what the symbol means (abstract concept, what you called "3-ness"), and what the symbol is being used to refer to (physical object, and groups of physical objects). There appears to be an inclination in this thread, to conflate the latter two things, and claim that what the symbol means, and what it refers to, are one and the same thing, i.e. the symbol simply refers to a concept. This is an ontological error, meaning is not a thing.

LOLOL.

Alethiest and I go way back. We're not too far apart metaphysically, only disagreeing on some finer points. But in relating ontology to mathematics, altheist employs intentional vaguery and ambiguity in terms, as well as outright contradiction to support unreasonable mathematical principles. So we part here.
• 1.2k
But in relating ontology to mathematics, aletheist employs intentional vaguery and ambiguity in terms, as well as outright contradiction to support unreasonable mathematical principles.
LOLOL.
• 6.6k

It's a good thing we all have a sense of humour here. LOLOL!
• 227
Anecdote: my Father was a great believer in these 'after-death experience' stories with their shining tunnels and meeting people again. He 'died' for some time in an ambulance and they revived him in the hospital. He was glad to re-join us, but depressed by the experience. There was Nothing there. No sort of argument, but emotionally strong with me!
bold
italic
underline
strike
code
quote
ulist
image
url
mention
reveal