Topics in Current Technical Threads

1) Potential versus actual infinity.

2) Are there mathematical sets?

3) Does mathematics need a foundation in set theory?

4) Is there irreducibly plural reference, predication, and quantification? If yes, does plural quantification allow us to avoid ontological commitment to sets?

5) Discreteness, density, and continuity at the level of number theory, geometry,  and nature (physical space and physical time)? 

6) Phenomenal versus physical space and their relation. Homogeneity and continuity in relation to both. Lycan's puzzle about the location of the homogeneously green after-image. Wilfrid Sellars and the Grain Argument. It was (6) that got us going on the current jag.

The above topics which we have recently discussed naturally  lead to others which I would be interested in discussing:

7) How is it possible for mathematics to apply to the physical world? Does such application require a realist interpretation of mathematics? (See Hilary Putnam, Philosophical Papers, vol. I, 74.)

8) Zeno's Paradoxes. Does the 'calculus solution' dispose of them once and for all?

9) The 'At-At' theory of motion and related topics such as instantaneous velocity.

10) Mathematical existence.

11) The Zermelo-Fraenkel axioms, their epistemic status, and the puzzles to which they give rise.

12) How the actual versus potential infinity debate connects with the eternalism versus presentism debate in the philosophy of time.

Comments

[EG]

Bill, even when I do not understand or agree with everything you write, you always provide something that is worth pondering and meditating on, so thank you for sharing your thoughts, arguments and something of your mind.

Ad 1.

You write in another post (Strange Anti-Epicurean Bedfellows: Josef Pieper, Thomist and David Benatar, Anti-Natalist): "To be mortal is to be potentially dead, and living is the gradual actualization of this potentiality."

I wonder whether this analogizes to how potential and actual infinity are related. That is, as we speak of aleph-null to aleph-n, we are speaking about this actualization, both schematically and as it's only coherent representation to us both as we are, delimited in the sense of Kantian phenomena and noumena, of what "actual infinity" "is."

Also, consider your arguments about existence--I appreciate that your paradigm theory is far more complicated and rich and setting it up here may be to muddy things, but I think your explanation of what existence is, is apt here--and what it "is," as opposed to collapsing or trying to limn in it by speaking about existence as a property like "being" red.

[BV]

You're welcome, EG.

'Actualization' is not the word you want. For it connotes a process of actualizing. If the nat'l numbers, say, form a set, however, they form an actually infinite set, and we can ask about the cardinality of this set. Cardinal numbers answer the How Many? question. Cantor's answer to this question re: the naturals is: aleph-null, aleph-nought, aleph-zero, pick your term. There is no actualizing going on, no counting by man or machine: the numbers are 'all out there.' They all exist whether or not anyone or any thing counts them. Counting them is merely to create names for them in some notation or other: Roman, Arabic, base-10, base-2, etc. We create numerals, not numbers. A numeral is not a number, but the name of a number. Actual numerals are finite in number. Are numbers finite in number?

But you allude to an important question: what exactly do 'potential' and 'actual' mean in the context of a discussion about infinity?

For Aristotle, change is the conversion of potency into act. When an acorn changes into an oak tree it actualizes its inherent potential. But nothing like this is going on with respect to infinity. It is not as if there is a thing called infinity that has a built-in tendency or potential to develop into actual infinity.

So it is not like a living man whose mortality consists in his inherent tendency to die.

To say that the naturals are potentially infinite is to presuppose an agent, an operator, a counter (whether a man or a computing machine) who/that implements the successor function, that is, adds 1 to a given number n to arrive at its successor, n + 1, AND can perform this operation repeatedly. Each concrete operation yields a finite number: no infinite number is ever reached.

Analogy: When I drive down a road, does the road come into existence as I drive? Or is the road out there already and I merely move along it? The former is like pot. infinity; the latter like actual infinity.

Or suppose I ride my mountain bike across open (trackless) desert: I lay down a track where before there was no track. The next day I follow my own track, the one I laid down the day before. Which of these scenarios is like pot. inf and which like act. inf.?

[EG]

Bill,

You have raised more questions than you have answered.

1. What is infinity? (And what of eternity, are they different? If so, how? Why?)
2. What do you mean by “all out there”?
3. We have been using analogy, so what is this really doing? Are we drawing conceptual parallels or are we making a “sameness” claim—and then we quibble in the “degree of sameness,” and then argue about whether one is more sensible/intelligible (I smell an apt Kantian digression here)?

[David Brightly]

Bill, Your latest remarks above prompted the following thought: that our understanding of actual infinity rests on our perception of repetition in space, and our understanding of potential infinity rests on our perception of repetition in time. I have a mental model of the natural numbers as regularly spaced fence posts along a line receding into the distance. An actual infinity of places is ever present. That seems analogous to how we experience space. On the other hand, regularly counting the fence posts is a process in time that at all future times will be incomplete. This is counting as we were taught as children, as opposed to establishing in thought a timeless 1-1 correspondence, as we were taught by Cantor. The two modes of infinity correspond to our distinct experiences of space and time as actual and potential.

[BV]

We should distinguish between numerals, numbers, and numerables. A numerable is just a plurality of concrete objects, the pens on a desk, for example. Suppose there are three pens P, P* and P** on the desk and the child is asked to count them. The child assigns 1 to P, 2 to P*, and 3 to P**. The child then announces that there are 3 pens on the desk, the number of pens being equal to the last number assigned. Is this what counting is?

[BV]

David,

Your analogy is interesting. The fence posts are all out there in space. The naturals, if actually infinite, are also all out there, though not in space. Counting is a temporal process. So if I count the fence posts, I always count up to a merely finite number, no matter how long I keep counting.

If the numbers are generated by counting, then the number of numbers, though infinite/indefinite is merely potentially infinite. But are the numbers generated by counting? Is it not rather the case that the numbers must already be there to be assigned to each fence post counted?