Simple Question

"Infinite sum."

This gives me a chance to put right and left next to each other in a sentence:

A point is a "dimensionless geometric object" and is, if I have it right, left as undefinied in math.

Well, we could take this in a variety of directions. One is this. Let's say that I ask you to mentally divide down a line of length D into some number of parts. I then tell you that I've just taken one of your parts and divided it down into some more parts. Then I ask you if you can imagine dividing one of those down further. You say, "Sure, no problem." And you proceed to mentally divide one of them still further. At what point would you or I have to admit that one of these divided line lengths can no longer be further divided?

But another might be like this. I can choose to define the variable 'dD' such that it represents the exact and precise width of a point along a specific line of length D, having defined this situation so that D is infinitely divided into 'dD' lengths. In other words, I define the case such that D has an infinite number of 'dD' in it. These 'dD' parts are non-zero, yet they also must be smaller than the smallest finite number other than zero, too. If you try and imagine the smallest possible finite value next to zero, 'dD' will be in between that finite number and zero, still smaller. Now, I ask you this: "If I now add up an infinite number of these 'dD' values, what do I get?" And if you are wise, you answer, "D, of course!"

By failing to find a way to prove it, I suppose. If one can match up each and every point along a line with, say, the infinity of integers and still finds that there are points left over unaccounted for, then you may have shown just that. On the other hand, if no one can seem to achieve this feat, then perhaps you can safely rest on the assertion that their number is infinity, no more, no less.

But a point isn't defined. That's the power of it.

Have you ever paricipated in the "why game?" Where you ask, "Why?" And someone answers. And you ask, "Why?" again. And so on? Where does this end? No matter how you define things or prove things, ultimately there must be an end point in this 'infinitely telescoping' set of whys, where the person answering just says, "Because it is so." And stops there. One must have a foundation, which itself isn't defined. Otherwise, you'd have circular logic. And that isn't so useful.

I wrote the following quite recently into this group. It might help in some odd way:

Interlude:

There are as many points on a short line segment as there are on a long line segment.

Proof: Draw two parallel line segments, with the shorter one above the longer one. Define a vertex by the intersection of a line drawn through the left endpoints of the two segments and a line drawn through the right endpoints of the two segments. Then pick any point at all on the bottom, longer line segment, and draw a line to the vertex. The line will pass through a unique point on the shorter segment. Every point on the long segment can thereby be mapped one-to-one to the short segment.

"Bob Myers" schrieb im Newsbeitrag news:XWSSg.437$ snipped-for-privacy@news.cpqcorp.net...

The point is, you are doing this in your mind. What I mean to say is, infinity has no meaning outside of your mind. There is no physical representation of infinity, so we can't have a direct experience of it through our senses. And that's not at all "irrelevant". That's probably why people have difficulties in imagining the properties of an infinite grid of resistors, and the reason why they asked it in the job interview.

No, you can't; what is infinity + 1? Is infinity a number that has no successor? Then how can you call it a proper number? I don't know whether infinite series exist. The fact that you can imagine something doesn't automatically bring it into existence. I'd even go a step further and say: you can't imagine an infinite series of anything. You can't write down an infinite series. If you write down its formula using the lemniscate, you are not really writing down the series; rather you are writing a generation formula for it, and that's a different concept. By way of abbreviation you may say: "This is the series", but in fact it is not. Mathematics is full of concepts like this. Take Pi for example. Nobody will ever know its exact value. Yet we are using the symbol to calculate "exactly". The important thing here is: These concepts are just tools used by mathematicians. There's no use in postulating their existence in "reality".

Yes, it is interesting and pointless, I agree. I have proved it some time ago for myself, but maybe you won't accept the proof. It requires the use of a head and a wall ;-)

I apologize for the cheap joke. I have found it much saner to assume the existence of something out of my mind, and for all practical purposes I settle with it quietly. I regard it as a self-evident axiom. Besides, I doubt whether one would ever be able to prove the existence of something _out_ of his mind by using logical concepts living only _within_ his mind. The fact that in the entire history of philosophy nobody has ever succeeded in this attempt, does rather seem to indicate the opposite.

(BTW: This is getting _way_ too OT. Somebody is eventually going to flame me over this ;-)

Leo

"Jonathan Kirwan" schrieb im Newsbeitrag news: snipped-for-privacy@4ax.com...

You're not following the grammatical intricacies. Maybe I should have written: "Show me an 'infinite'".

Let me put it into the correct order: "If undefined a dimensionless geometric object you left, right you are!" (a bit Yoda-like ;-)

If my teachers had accepted that as an answer, my math classes would have been much easier!

I'm not disputing whether your mathematical concept of "infinity" or of points or lines is correct or not. They are mathematical tools that work fine for those who know how to deal with them. But in reality you are not likely to ever encounter "infinity", so you can't have an immediate, sensual experience of it.

I like the post you quoted. You nicely described the way mathematical ideas evolved out of the pure mind. In my previous post, that from which this discussion originated, I wrote: 'On the other hand, this is a play on our inability to conceive "infinite". There is no such thing, so our experience is absolutely worthless.'

Maybe you are not unable to conceive 'infinite'. Maybe you know exactly what you are talking about. That's fine.

Anyway, I'm glad I was wrong about the 0 ohms. Imagine the universe was infinite! We'd have short-circuits everywhere! ;-)

Regards,

Leo

Please identify something which does have "meaning" in the absence of a mind to consider it.

Again, simply because you have not learned the mathematical tools used to deal with infinite quantities does not mean that such things do not exist. The concept of infinity doesn't really get brought up in math until you get into calculus, or maybe JUST before that point in terms of infinite series. But you most definitely CAN "calculate properly" with infinity, and in many cases (again, integral calculus and beyond) you can't possibly calculate properly WITHOUT it.

Learn the math, and then you'll know the answers.

Know what a Fourier transform is (just to grab one electronics-relevant example)?

Again, an interesting assertion from a philosophical standpoint, from which it can (and has) been argued either way.

Sure I can. The set of integers is very obviously infinite, and I can't possibly imagine that set without realizing that fact (simply ask the question "what's the biggest integer?" and you immediately realize that there can't possibly BE one). That I can't ennumerate that set (since that would take a literally infinite amount of time) doesn't mean I can't imagine it. You may be unable to imagine such a thing - I can't tell, since there's no way for me to get inside your head and really know what you can and cannot imagine - but I most certainly can.

Different problem. I can't write down even a very, very long finite series (quick, write down the first hundred trillion integers!) simply because I know I have a finite lifespan (whether or not I have better things to do with that finite span...:-)) which does not permit it. So clearly whether or not I can write something down is no indication whether or not it exists.

On the other hand, even if I were to write out each individual number, it could still be claimed that I have not "really" written down the series - it's all just symbols, anyway, and there is no "real" difference between writing, say,

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10

and the "shorthand" expression of that which would involve the use of the "sigma" symbol. Neither has any true inherent meaning, and would be equally gibberish to someone who didn't understand our symbology.

So are you similarly going to claim that pi, like infinity, doesn't "really" exist?

Strictly speaking, that's no proof at all; sure, the wall FEELS "real," and causes you pain when you pound it with your head - but I could argue that it's still "all in your mind," and you'd have no way to show that that claim is wrong.

Of course, as we all do in our everyday lives. But if we look at it critically, it can be very important to realize that a lot of what we believe really IS simply a convenient assumption on our parts. See, I knew this would wind up getting more and more into philosophy....:-)

Bob M.

Oh, I followed the grammatical intricacy just fine. You wanted an adjective and I gave you one.

I probably am too burrowed into the narrow meanings I carry and cannot connect them with an imprecise question, except imprecisely. So I'm still not sure what you want, here.

hehe. Anyway, a point is undefined. It must be, in fact. Just as the situation of 0/0 is also undefined. To define it would be to define it in terms of yet something else, which would need to be defined... in terms of something else... which would itself need to be defined, again in terms of yet something else. Somewhere, this has to end. In the case of formal mathematics, it ends at the point and at

0/0. To define them would be to make math circular or to go find something else that is left undefined, instead. As it is, this is where it is undefined.

hehe. You can never say what the future may yet bring in physics. But in math, I suppose it is possible to prove things. But not all true statements within a coherent, non-trivial system can be proven within that system -- Goedel.

My real point is that unless you define what you mean by "more than" when discussing infinities, you are in trouble. So you need to be precise when asking a question like that. And you weren't. So rather than belabor that, I was flip about it.

Mathematics just _happens_ to sometimes overlap natural reality as we experience it. But there is no reason to imagine that all real concepts in mathematics correspond to sensual experiences. In fact, I'm pretty sure it's the case that they rarely do. It's just that some aspects of math are practical, much to the chagrin of mathematicians. (John Conway once worked on a bizarre 26-dimensional system with the belief that no one would _ever_ be able to find a practical use for it -- he hoped, anyway. Sadly, he was wrong about that.)

It does turn out that nature does behave (on a macro scale) that can usually be treated as though infinitesimals are real. Whether or not they are, is another issue. Einstein's general theory of relativity treats things this way, without quantums, and it corresponds pretty well. At the micro scale (subatomic) things are different. But that's not usually at the experiential level we more directly 'know.'

Thanks.

For higher levels of math, organic experience isn't much of a guide, at all. There is no necessary relationship of natural experience with mathematics. Mathematics is its own universe, so to speak. Where it touches upon reality is a matter of fortunate circumstance, not necessity. It's quite possible that nature would operate in no way similar to mathematics. We are fortunate that is not the case. (Some argue that this is because math is the deeper reality, but I'm not taken with that idea yet.)

I have almost no problem with the concepts, as they are narrowly used in mathematics. What problems I do have are usually because of vague meanderings about such things by those who really don't follow the usage well and imagine questions where there are none.

Probably a better way to say the above is this: Sometimes people imagine that there is an end point of some linear spectrum spanning from, say, complete ignorance to perfect knowledge. Or, let's make this less religious and more mundane -- that there may be a spectrum from "perfectly cold" to "perfectly hot." So people may ignorantly seek answers to questions based upon this primitive conjecture. But it's just not even the right starting point. What is even meant by these ideas? Getting answers is more a matter of understanding the questions better.

What is meant by the question, "What happens when the unstoppable object meets the unmovable object?" What does 'object' mean? What is the behavior of 'objects," in some rigorous meaning? What does 'unstoppable' mean? What does 'unmovable' mean? We are in no position to even ask such questions.

We do not have the perspective. Plain and simple.

However, we can ask more modest questions with some hope. And in the case of infinities and infinitesimals, they have a great deal of meaning because the meaning they do have is limited and prosaic.

Your questions smack of making too much out of them. Though I may be mistaken about that.

hehe. You _do_ see!

Jon

That would be Aleph-null, right? ;-)

Thanks! Rich

Sorry if I didn't express it exactly. Maybe it's clearer if I say that the concept of infinity has no real-world equivalent.

You don't seem overly confident in the power of your arguments ;-)

I find it problematic to claim that numbers "really" exist. Of course I don't doubt that they exist as concepts (or "in your mind"). But consider the following reasonings: "Numbers are concepts. Concepts do exist, it follows that numbers exist." "God is, by definition, everything. Being everything includes being a concept. Concepts do exist, it follows that God exists." Something tells me that this line of thought is not really satisfying..

I think there is trouble with the word "existence". If you allow me - as a non-mathematician - to take an example from maths: Suppose you want to prove that there is no integer greater than 2 and smaller than 1. Most probably you'll prove this by contradicting the assumption that such a number exists. But to prove that there is a contradiction you'll have to think and act as if such a number existed. The statement "There is a number x greater than 2 and smaller than 1" is a perfectly valid mathematical statement. It just happens to be false. But how can you assume the existence of something that you later on prove does not exist? What does that mean for the nature of your proof? When did the concept of x come into existence? When did it cease to exist? What exactly was the reason that it ceased to exist? Now if you say that mathematical concepts "really" exist, what does this say about your understanding of "reality"? Can you create and destroy objects at will just by saying that a statement is true or false?

It's simple to avoid all these questions. In maths, "there is" means "we can think about a certain concept in a consistent and logical way, not contradicting our axioms". For example, "there is a number epsilon such that bla bla" simply means "we can think of a number epsilon without getting into logical trouble, such that bla bla". In this sense you can talk about infinity and it's perfectly acceptable.

But: Confusing the mathematical existence quantor with "real world" existence is a logical fallacy, a category error straight from Philosophy

101.

As I mentioned earlier, I am not a constructivist. There is in fact a very convincing argument against radical constructivism, in Rudolf Steiners "Philosophie der Freiheit" (English title: "The Philosophy of Freedom"). I can't repeat it here at least half as convincingly, so all I can do is refer you to the book if you are interested.

True. It is hard to know what's "real" and what is not. But I'm quite sure that numbers - and especially infinity - aren't :-)

Kind regards, Leo

Nah. Come on. Cheap trick.

You expressed my opinion much better than I did!

I was playing the advocatus diaboli a little. Maybe I should apologize. But at least we had an interesting discussion!

Let me quote something you wrote in another post:

Hmm. I know what you mean. But imagine the moment you reach the end of that "other world". The end of reason, if I may say so. Suppose you suddenly understand the nature of the boundary that separates us from reality. Not by mere conceptual understanding, but by pure insight, by "seeing the fact". Suppose you suddenly realize That Which Can Not Be Expressed In Words and how it relates to you and your conceptual universe. You would know how it comes that we are prisoners of our conceptions, because you would have encountered That Which Is Beyond Concepts. You would understand the nature of conceptual frameworks, that "Truth" and "Falsehood" are only defined within such a framework, and that frameworks are many, and that none of them are real. From that moment on you would lose interest in mere concepts because they would seem to you like a child's play. You would also know the immense "mental space" outside of these frameworks. You wouldn't take the juggling of concepts serious any more, because you would have discovered something that is much more fascinating and "real". I'm not sure whether you understand what I mean. But if you offered me a hundred times the mathematical insight you have in exchange for this experience, I would decline.

Wait, what's that? Someone at the door? Oh no! It's the newsgroup police! I'm being arrested for being off topic! Help! I want to talk to my lawyer! Help......

Well, perhaps. Not worth worrying about, one way or another.

I did?

:)

Hehe. The nice thing about these "eyes" is that they can be developed by anyone. One of the truly important aspects of mathematics is that when a Greek describes something 3000 years ago, I can put in my head

3000 years later almost exactly what they had in mind. Developing a language that speaks across centuries, across culture, across fad, across place, ... and to do that in such a precise way that the rigorous deductions to specifics of any of it would also be the same rigorous deductions to specifics that someone else thousands of years later in an entirely different place and culture.... well, you have to admit that is pretty important, I think.

This isn't religious experience. Anyone can share the concepts and do so with near perfect replication. The world I see is the same world another sees on their own, no matter their culture, place, or time. In that sense, it is as objective and real as the world around us. Not some individual experience that cannot be proved, so to speak.

You are already going too far, though. We have no idea that there is such an "end," what it may look like, or even be able to ask questions about the concept. It's not objective to suggest it and it is otherwise beyond our ability to speculate about, should there be such a thing. What you are asking is almost like, "What would a perfect human do in this case?" We have no definition of "perfect," certainly no idea of what a "perfect human" is, and the very question itself carries no rigorous meaning. There is no way to make an answer we can share.

That doesn't mean the question is meaningless to a person. It just means that it cannot be objective. I can say that "I am happy right now" and, though it may certainly be true, it is not objective. You cannot test this, you cannot weight it, you cannot arrive at an independent opinion based on facts. You can only accept my word. This is because it is an internal state of mind.

So I really don't like questions like this, because they advance nothing.

Yup, you've gone way off the deep end! ;)

How can you say? I've no idea what I'd do or not do, or even if the suggestion is possible. Or even what it means, for gosh sake.

Neither of us can say what would happen.

Well, you have selected what amounts to a religious experience, plain and simple. More particularly, and even more alarming in a sense to me, is that you are posulating a capital-T Truth as opposed to a more mundane, scientific fact or lower-case-t "truth," which remains tentative. I have a healthy (nay, strong) reaction to folks looking for Truths, instead of truths. We cannot ever know Ultimates. If for no other reason, then because we cannot possibly say what the future will bring us.

Look at it this way. If we were able to say something that is True, no matter what the future may yet say about it, then we have found something that doesn't need to be examined and, in fact, is immune to evidence. It must be immune, since by definition it is True and not just true. Since it is True, no matter what the future may yet say, and is thus immune to evidence, it must be the case that evidence isn't about this Truth. But evidence is nature. And therefore, this Truth isn't about nature itself. Since evidence and nature cannot affect it, it is therefore NOT about "reality."

Frankly, I'm not interested in those things. Worse, it worries me when others are -- because they almost always then try and force the meaning of their perceived Truth onto my life. And since nothing I can say, no argument I can make, no evidence I can present will have any impact on their committment to control my life, we cannot find a way to negotiate a compromise. Not a real one. And so, they are very dangerous people.

That worries me.

;)

Jon

"Homer J Simpson" wrote in news:%uVQg.32558$cz3.4043 @edtnps82:

Incorrectly I would say

--
Posted via a free Usenet account from http://www.teranews.com

:-)

Well, there's one natural constant: human curiosity. Remember the moment you left your Lego's or whatever when you went to school and discovered an entirely new, exciting world? It doesn't mean that you leave your toys for ever, but the feeling just isn't the same any more.

Plain and simple, no. I am not a religious person, and I have definitely not _selected_ anything. "religio" means "duty", and I think that an experience has nothing to do with duty. Seen by daylight, it's just an experience that our minds enable us to have (excuse the generalization), and an experience that offers some deeper understanding of how our mind works. Maybe I have expressed myself a little too poetical, so it sounds like mysticism ;-)

Have a closer look and you'll see that I put the "Truth" in quotes to paraphrase exactly what you describe here. I don't think there are Truths at all; I believe that the truth of all statements is determined relatively to the conceptual frameworks we live in. This is just a belief, nothing I can prove (how would you prove that, btw?), though I think there are some quite convincing arguments. I also don't deny that these truths can come very close to what we perceive as reality (as you said, maths sometimes just happens to overlap perceived reality) - but then there's the strange effect that our concepts can sometimes influence our perceptions. Personally, I find this tricky to sort out. Meanwhile, I hold it with Karl Popper who said that statements can never be proved true, only false.

I'm sorry if it seemed that I wanted to impose something on you. That's definitely not the case. No need to be worried ;-)

They let me go when I pointed out that others were even more OT. Have you read the thread "Smoothing Capacitor Values"? What a steaming pile of B.S.!

Kind regards,

Leo