For Jim Stone: Proof that Unicorns Exist

[Jerry Freeman]

consciousness is a locally confined (in my body, somehow) epiphenomenon of neurophysiology

Yes!now there's an idea!

Yeah. That's how the world looks to most people, so they assume there's no other way to see.

Best wishes,
Jerry

[elendil]

hmmm. i hope this doesn't mark the point where we all start saying what we really think.

[Lorenzo]

The word "idea" has no charge to it, no matter what our opinion, but there are surely such things as positive, negative, or neutral ideas.

[Ridseard]

infinite numbers? ridseard is our resident mathematician, so he should weigh in on this. however, , in the order of is-ing, i think there's only one number = 1. whatever is, is a 1. therefore, there can't be an actually infinite number, since 1+1+1... at any given moment always adds up to a definite number = a finite number, limited. number is a description of limitation, at least in the order of is-ing, so infinity doesn't apply to number in that sense. i know mathematicians speak of infinite numbers, but i think they're using the term in a different manner. perhaps, potentially infinite, but not actually infinite, which is a contradiction in terms.

First of all, the thing designated by a sideways "8," commonly called "infinity," is not a number. It is just a symbol used (mostly) in calculus to indicate that a quantity or process is unbounded. I.e., there is no bound on how large it can get.

A transfinite number, on the other hand, is an actual number, as concrete and definite as the usual finite numbers like 3, 17, 1578943, etc. Essential to the set-theoretic approach to numbers is the concept of cardinality. For example, a set is said to contain 5 elements (or have a cardinality of 5) if it can be put into one-to-one correspondence with the set consisting of the numerals 1,2,3,4,5. I.e., the elements in the set are counted and found to be 5. Transfinite cardinal numbers are also used to "count" how many items are in a set. For example, if a set can be put into one-to-one correspondence with the set of all natural numbers (1,2,3,...,etc....), then it is said to have a cardinality of aleph-nought (designed by the first letter of the Hebrew alphabet with a "0" subscript). Lots of sets contain aleph-noughtelements, such as the set of all positive and negative integers, the set of all even integers, and the set of all rational numbers. The one-to-one correspondences of these sets with the set of all natural numbers are very easy to set up, but a little beyond the scope of a C&F post. At first it seems as if the set of all natural numbers should be twice as large as the set of all even integers, but in fact these two sets are exactly the same size, both having the same cardinality. This is a characteristic of infinite sets: a proper subset may have exactly the same number of elements as the whole set. Another (larger) cardinal number is the number of all real numbers (all possible numbers, including infinite non-repeating decimal numbers), called C (the number of the continuum). It is very easy to see toat C is strictly larger than aleph-nought. However, last time I messed with set theory (many years ago), it was not known whether C is the next higher transfinite number after aleph-nought (i.e., aleph-one). The conjecture that C = aleph-one is called the "continuum hypothesis."

There are a lot of popular books which explain all this a lot better than I can. I mainly just wanted to convey the idea that transfinite numbers are very definite, concrete mathematical objects. There is nothing at all wishy-washy, ambiguous, or mysterious about them.

[jim stone]

Good old Ridseard. The Hebrew aleph here comes from
the mathematician Cantor, who invented the famous diagonal proof,
which showed that some infinite sets are larger than
others. Cantor was the paradigmatic mad genuis
mathematician. There's a story that Cantor was
riding in a horse drawn taxi and began writing a
proof on the inside of the cab. When he got out
he immediately forgot the proof and ran desperately
after the taxi to copy it.

I don't really understand this stuff,
which is why my salary was so pitifully low.
But Ridseard does.

[Ridseard]

Poor, mad Cantor was staggeringly brilliant. He knew he was onto something big and revolutionary. His theory of transfinite numbers is irrefutable. Yet his ideas were ridiculed by other, less brilliant but more prestigious mathematicians, and this totally unwarranted rejection affected his career and probably contributed to his madness. It ain't fair!

[Walden]

There be philosophies, methinks, which hold that the supreme Deity exists, but that we do not.

[Jerry Freeman]

You also get this sort of statement, from across many traditions, "It is no longer I that live, but God who lives in me."

[peeplj]

There are indeed, Walden.

There are also philosophies that we exist and Deity does not.

The are yet also philosophies that nothing exists, that the very concept of existance implies an inherit miscomprehension of the actual.

I figure at the end of the day, all of the philosophies in the world aren't any good to you if they don't help you sleep better.

The question is not really rather or not God exists, or even whether or not we exist. The question is, being here in this world, do we dare to wait for some higher power to give our brief lives meaning, or do we do what we can to find a meaning and a purpose ourselves?

When you contemplate the existance of God, you stretch your mind into the abstract, and while that's good, it will never compare to the simple feeling of your wife going to sleep with her head on your shoulder.

In the one instance you may seek Heaven; in the other, you have already found it.

Best wishes to all,

--James

[Jerry Freeman]

In the one instance you may seek Heaven; in the other, you have already found it.

There's more than one way to interpret "The Kingdom of Heaven is at hand."

Best wishes,
Jerry

[Jerry Freeman]

You know, James ...

That reminds me of the six systems of Indian philosophy, Nyaya, Vaisheka, Sankhya, Yoga, Karma Mimamsa and Vedanta. Each has its own approach to fathoming reality and its own description of what reality is like. If you compare the various systems, you find them to be in contradiction to each other, and many will argue about which approach is valid and which is not.

However, the six systems of philosophy are sometimes referred to as the six "darshanas," which means the six ways of seeing. The implication is that all six of them are correct and true, but different individuals with different experiences and different states of consciousness will naturally see a different reality.

Reading your list of different philosophical points of view, I would be hard pressed to say that any one of them is "right" or "wrong." Seems to me, they're all looking at the same, multifaceted reality, but from different angles, which reveal different realities.

Best wishes,
Jerry

[herbivore12]

Poor, mad Cantor was staggeringly brilliant.

There's a new book out treating Cantor, written by David Foster Wallace:

Everything and More: A Compact History of Infinity (Great Discoveries)

It's worth a read, for those not afraid to bone up on thier math (I am ashamed to say how much I've forgotten). Ridseard and others comfortable with math will have no problem at all, of course.

There are a couple excerpts on the web, including here:

http://www.conjunctions.com/webcon/wallace.htm

Interesting stuff.

[elendil]

ridseard--say what?

i won't even pretend to understand virtually anything you wrote, but that's me, not you. i never got past 2nd year algebra!

however...

my main point was:

number is a description of limitation

if i understood that small bit not included in "virtually anything" then what you're saying is in agreement with that fundamental concept. i'm not suggesting that that concept is at all useful to a mathematician--rather, that it helps to set the proper bounds for mathematics as well as many other methods of approaching reality. those methods are all useful and good within their proper bounds--and their proper bounds may be very broad indeed--but they are confined to finitude, precisely because the finite is susceptible to conceptualization. when we come to dealing with existing, precisely as act, mathematics won't help (nor did you say that it would), but that's the mistake that people from plato to anselm to descartes to hegel and beyond have made.

as to infinity, the sideways 8 concept was well understood, of course, many centuries ago and is what i called, in medieval fashion, potential infinity. that was a big deal in disputes over the supposed eternity of the universe, and augustine got into the whole issue in some depth. i'm not disputing the validity of such a concept (and i think you understood that). what i am saying is that having a concept of potential infinity--or, what many simply call 'infinity'--cannot bring into existence an actual infinity, even if that concept has useful applications in mathematics and physics. i think that's something like what you were saying about anselm's "proof". and that's because existing (is-ing) is logically prior to essence (whatness, mode of is-ing)--what is sometimes called the "real distinction" between essence and existing. from what i've read, it appears that physicists of a reductionist bent run into these issues, too. they want existing to somehow be a concept, but it's an act...or, at least existing is most analogous to what we might call "being in act."

the only reason i'm posting this is to see whether a mathematician agrees with my non-mathematical description of these issues. HOWEVER...

there is one thing you wrote that i want to take issue with.

ridseard wrote:

There are a lot of popular books which explain all this...

sorry, as a non-math guy i just can't buy that!

[elendil]

herb,

i read your link. i liked what the author said about the danger of treating ideas as if they were "things". that's part of what i was trying to say. i think chesterton, whom the author quotes, is saying much the same thing about logic and method: the danger is not in logic and method per se but in their misapplication beyond their proper bounds or range. and that's what logic and method are, after all: ways of either dealing with ideas or of fitting reality within a conceptual framework that helps us to deal with it. perfectly valid and useful in the case of finite existing (always bearing in mind the possibility of misapplication), but not so useful when we ask: why does anything exist? of course, we still have to use ideas--we remain mere humans--and we have to employ logic in using ideas, but since existing is not a concept we will be led astray if we try to treat it as a concept.

thanks.

[Tak_the_whistler]

If a philosophy or religion is true, then any honest man will still want to believe it even if it does not give him any help at all. If it is not true, then no matter how much help it gives him, if he is honest, will not stick to it.

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The lovers on the left bank of the Seine in Paris, experienced the "mannishness" of humanity when they fall in love, and yet they cry; because they do not believe love exists. Isn't this sad?