John. 1191993590:::Dave Harrison:::dov12348@yahoo.com::::::I thought Objectivism saw existence as eternal. That's a form of infinitude, isn't it? 1191997971:::Aeon J. Skoble:::askoble@REMOVETHISbridgew.edu::::::You lost me. Surely there's nothing in Rand to suggest that the set of integers is finite. The Einstein joke is funny in either case, of course. 1192005895:::Greg Perkins:::greg@ecosmos.com:::http://ecosmos.com:::Dave, as I understand its use in this context, eternal does not mean "infinite time" but rather "out of time": on the Objectivist account, time is in existence -- existence is not in time. Time simply doesn't apply in the case of existence per-se. (Much like 'location' doesn't apply in the case of existence: location is a relational term and existence stands in relation to nothing. So places are in existence -- existence is not in a place. Existence is elocational. :^) 1192014783:::Jason Stotts:::Jason.stotts@gmail.com:::http://erosophia.blogspot.com:::What is this about Objectivists not believing in "actual infinity"? I most certainly do.

Existence is itself infinite in the literal sense of being without limit. As Greg says, "location" does not apply in the case of existence, because there is nothing outside of existence. While the idea of infinite time may be inapplicable to existence due to time being a judgment of the motion of existents ( I agree here with Aristotle's analysis of time), it is preposterous to say that existence has a limit. 1192020191:::Greg Perkins:::greg@ecosmos.com:::http://ecosmos.com:::Jason, an infinite amount or size is a contradiction in terms: infinity is no particular amount, no particular size -- infinities are abstract potentials, not existing concretes. The notion of some existent (or the sum of all existents) as being actually infinite in any respect would be a violation of the axiom of identity -- no such thing could exist because "existence is identity": no identity means no existence. 1192044412:::John Dailey:::phyrm_1@hotmail.com::::::~ After reading the previous comments...I sympathize with Al. --- And I doubt that he was 'joking.'

~ Indeed, that comment of his, if actually made (how'd I miss it?), shows his 'genius' at least as much as his other accomplishments.

LLAP

J:D 1192104413:::Dave Harrison:::dov12348@yahoo.com::::::Greg, do you think the number of past events is finite or infinite? It seems that it would have to be finite, otherwise we could never get to the present moment. However, if that's the case, there had to be a beginning -- a first event -- but then what came before it? And then we're back to the infinite. 1192105029:::Mike Hardy:::hardy@math.umn.edu::::::Greg Perkins wrote:

> Jason, an infinite amount or size is a contradiction

> in terms: infinity is no particular amount, no particular

> size -- infinities are abstract potentials, not existing

> concretes. The notion of some existent (or the sum of all

> existents) as being actually infinite in any respect would

> be a violation of the axiom of identity -- no such thing

> could exist because "existence is identity": no identity

> means no existence.

What if it is proposed that the statement that space is

infinite in extent _means_ only that there is more space

than any (finite!) volume you might specify? Would you

claim that the amount of space that exists must be no more

than some specific number of cubic light years? (Although

Einstein's remark was facetious, I think the part about his

being uncertain of this point was quite literally true.)

(And do you claim that there really is such an entity as

"the sum of all existents"? It really isn't clear whether

you would think that or not, and for that matter, you're

comments look like something that would be written by

someone who hasn't thought this through.

Once I heard a (taped) lecture by Leonard Peikoff in which

he stated that he did not object to statements such as that

the "number series" [sic] is infinite---by "number series"

he seemed to mean the sequence of positive

integers: 1, 2, 3, ..., and he appeared to be agreeing that

it doesn't come to an end. At that time I was doing graduate

work in mathematics, and one thing that usually happens to

graduate students is that they are startled and disoriented

whenever they find that a word means something to someone

outside their field that is different from its technical

meaning in their field. Peikoff's use of the word "series"

for what I would have called a "sequence" threw me mentally

completely off balance for all of a second or so.

If the topic of infinity is to be raised in this context,

maybe someone here can appreciate Euclid's famous argument

that the number of prime numbers is infinite. Before getting

into that, let us recall that Euclid, who lived in Alexandria

in the 3rd century BC, was the most famous of all writers of

classical antiquity (except maybe the authors of the New

Testament), eclipsing Aristotle by an immense margin, until

maybe around 1960 or so. Geometry was one of the four elements

of the "quadrivium" that all students studied in medieval

universities and at that time geometry was identified with

Euclid's books on that subject. Until the second half of

the 20th century everyone who claimed to be educated had

read at least _some_ of Euclid. Among the major intellectual

advances of the last two centuries are the discovery of

non-Euclidean geometry in the 19th century and its application

to physics in the 20th century by Einstein in a famous work

written in 1915 and many later physicists.

Euclid wrote not only about geometry, but also about number

theory.

Euclid did not use the word "infinite" or "infinity".

Translators render his words by saying "Prime numbers are

more than any proposed multitude of prime numbers", or

"...than any assigned multiplicity of prime numbers" or

the like. In modern language one says "...any finite set

of prime numbers."

Here is Euclid's proof: Start with any finite set of prime

numbers. Multiply them, getting a number P, then consider

the number P + 1. Either P + 1 is prime (in which case

there is a prime number not in the finite set we started

with) or P + 1 is a product of prime numbers. In the

latter case, none of the prime numbers of which P + 1 is

a product could have been in the finite set we started

with, since the numbers P and P + 1, being consecutive,

cannot share any prime factors in common (for example,

if N is divisible by the prime number 7, then N + 1 cannot

be divisible by 7; the next number after N that is divisible

by 7 doesn't show up until you get to N + 7). Either way,

we have at least one prime number not in the finite set we

started with. Thus any finite set of prime numbers, no

matter how big, can be extended to a larger finite set of

prime numbers.

Greg Perkins, will you say Euclid was wrong? 1192110343:::Adrian Hester:::::::::Mike Hardy wrote: "Greg Perkins, will you say Euclid was wrong?"

The subject was *actual* infinities, not *mathematical* infinities. 1192112545:::Greg Perkins:::greg@ecosmos.com:::http://ecosmos.com:::Mike, I don't understand why you went to all the trouble of rehearsing Euclid's (brilliant) proof for us about how many prime numbers there are. You could have made the same point more easily with natural numbers, noting that obviously we can always add 1! Of course my answer is that Euclid is right that there are no end of primes -- and so is my nephew who learned how to count and saw there are no end of numbers. "Infinite" certainly applies, and this is completely unobjectionable.

So I don't understand why you made the point at all (much less the hard way). This discussion is about concretes, not abstractions -- about existing in reality, not just thoughts.

In this vein, I would recommend Pat Corvini's excellent lecture series "Achilles, the Tortoise, and the Objectivity of Mathematics" <http://www.aynrandbookstore2.com/prodinfo.asp?number=CC22M>. She has a very nice treatment of infinity in light of the Objectivist theory of concepts, and handily cuts through certain perennial tangles thinkers get themselves into regarding infinity (i.e., Zeno and descendants). 1192116096:::Mike Hardy:::hardy@math.umn.edu::::::Why I "rehearsed" Euclid's proof here I already said.

Yes, the point could have been more easily in the way

you suggest, but see my reason stated above.

On the other relevant point: apparently my citing something

fo this kind (e.g. what your nephew thought of) is what it

took to get you to clarify that when you said "actual" you

meant _concrete_.

(BTW, I have a historical question for anyone who might

know the answer. Is this idea that "eternal" means outside

of time originally due to Augustine, or is it older (e.g.

Aristotle, Plato, whoever......?) 1192116320:::Greg Perkins:::greg@ecosmos.com:::http://ecosmos.com:::Dave asks: "Greg, do you think the number of past events is finite or infinite? It seems that it would have to be finite, otherwise we could never get to the present moment. However, if that's the case, there had to be a beginning -- a first event -- but then what came before it? And then we're back to the infinite."

Hi, Dave. Cosmology is more of a question for science, and so far we seem to have more questions than answers in that area. The current musings of the scientists seem to indicate that our physical universe started around 14 billion years ago (including space and time, our concepts of which are bound up in physical relationships and motion). That's a finite amount of time and a finite number of events, which seems fine to me. The unsatisfying part is that it doesn't seem meaningful to talk about anything "before" time and things. (And so I must punt: Maybe mankind's context of knowledge will grow and we'll develop some new concepts in the domain that will not lead to that sort of unsatisfying quasi-question that has no answer.) 1192117152:::Greg Perkins:::greg@ecosmos.com:::http://ecosmos.com:::Mike Humer writes, "apparently my citing something fo this kind (e.g. what your nephew thought of) is what it

took to get you to clarify that when you said "actual" you meant _concrete_."

Sorry for the confusion, I thought that my writing about how "infinities are abstract potentials, not existing concretes" would make it clear. 1192117273:::Diana Hsieh:::diana@dianahsieh.com:::http://www.dianahsieh.com/blog:::Greg, I think you meant Mike Hardy, not Mike Humer. (Mike Huemer is my dissertation advisor. Totally different person!) 1192119094:::Greg Perkins:::greg@ecosmos.com:::http://ecosmos.com:::D'oh! Sorry about that, Mike and Mike! (Should have heeded that vague feeling of something-not-quite fitting when I typed it...) 1192130124:::Mike Hardy:::hardy@math.umn.edu::::::: I thought that my writing about how "infinities

: are abstract potentials, not existing concretes"

: would make it clear.

No. It certainly will not. Maybe some day when I

have some patience I'll explain why not. 1192165699:::Dave Harrison:::dov12348@yahoo.com::::::Ok, Greg. Thanks for your thoughts.

My morning yesterday: Breakfast, work, coffee break and infinity, work, lunch. As I'm sure you know, infinity doesn't do well around the water cooler. ;) 1192187912:::John Dailey:::phyrm_1@hotmail.com::::::~ Would it be incorrect to say that the proper use of the term 'infinity' will be debated...into infinity?

LLAP

J:D