There is in fact a great deal of good in asking such questions. ianzin challenges my ideas constantly, forcing me to really think them through and explain clearly, which is great if you are an author, as I am. For example: my next book is on spiders. Ianzin challenged my explanation on how a baby spider knows how to spin an orb web. In attempting to answer (probably not to ianzin’s satisfaction) the book is much better for that challenge. He will get his copy when it is published in a few months, and I will be intrigued to see what he thinks.
I have just started the following book. At the moment ianzin considers that I am possibly deluded. But he will still read my emails and force me to justify my claims. And, should I prove that I am not deluded, the book will be much better for his challenges. If he proves me wrong, the book will never see the light of day and I will not make a total fool of myself. We need the questioners and doubters.
The explanations in this thread have improved greatly for ianzin’s questions. And I still think he’s wrong!
It seems to me that you have a fundamental misunderstanding of what a mathmatical limit is. As in: the construction of a mathmatical limit includes a starting point and a trend as a variable moves toward a point (in this case, upper or lower [positive or negative] infinity). This is a mathmatical definition. When we say limit, this is what we mean.
Therefore, the formula n^2 approaches infinity as n approaches infinity. (I don’t know how to make this form do mathmatical fomulas…)
For any particular value of n, there is a particular x value (x being the result of n^2).
The limit has a perceived boundary. The actual equation itself does not. Suggesting that infinity as a limit does damage to the terms ‘limit’ and ‘infinity’ seems short-sighted. Asymptotes are perfect examples of how a limit can be something other than infinity. It just about understanding the contraints in the use of the term in the mathmatical sense.
Perhaps a good calculus teacher in your area could explain it to you better.
Creating a fractal image takes orders of magnitude longer than a simple JPEG, like 10 minutes versus hundredeths of a second. Also, resulting fractal image files with good quality aren’t much smaller than JPEG files. When we tested 4 or 5 years ago, the fractal image wouldn’t hold text very well either, the text tended to get blotchy.
I’m sure fractal images have their place, but not for a general purpose image format like JPEG or PNG.
Except that even that can be a bit misleading, since the modern mathematical definition of a limit makes no mention of anything moving toward anything else, nor need there be any “starting point.” Instead, it relies on closeness: a limit is something you can get arbitrarily close to. And, for something to “approach infinity” just means that it can be arbitrarily large.
(I’ve seen Calculus described as “the mathematics of change and motion,” and in a sense it is, but in another sense, not really: limits, and hence all of Calculus, can be formally explained without any reference to anything moving.)
ianzin, it’s clear from this and other threads that you are some variety of finitist; there are at least some inductive arguments that you don’t accept. I don’t mean to criticize this; you’re welcome to choose whatever axioms you like in playing with mathematics, and of course in real-world science nothing is likely to rely on the particular choice of transfinite axioms anyway, which gives finitism a certain minimalist appeal.
But mathematicians have generally decided on standard terminology for dealing with infinities. You are using these standard terms in nonstandard ways, which just leads to confusion. It’s not reasonable to expect everyone to understand what you mean by “infinite” or “limit” when you’re not using its standard definition, nor is it reasonable to argue that you are “correct” while other people, using the standard definitions and standard set theory, are wrong, just because they proceed from different axioms than you do. To successfully convince mathematicians that your viewpoint is correct, you will have to point to the axiom(s) you dislike, and give them a really good reason for them to dislike it too. (I join with Indistinguishable in wondering exactly which axioms you don’t accept.) Merely involving the infinite won’t do it; showing an internal inconsistency with some other more-beloved axiom almost certainly would, and showing that the axiom leads to an extremely counterintuitive result would probably convince some but not all mathematicians. (Cf. the Axiom of Choice/Zorn’s Lemma/the Well-ordering Principle.)
This standard terminology is not some conspiracy (at least, as far as I know; but I don’t suppose They would tell me anyway). The consensus seems to have arisen because it leads to interesting results; even if not directly applicable to finite real-world situations, they may allow (some people) better intuition about even finite mathematics, such as with the standard treatments of calculus and limits. For example, any real-world approximation of a Koch snowflake cannot have an “infinite” perimeter (whatever a “perimeter” even means at subatomic scales); but if you build a set of them, they can have perimeters scaling faster than linearly with diameter, which is unusual behavior in itself.
Asking about this scaling leads naturally (for some) to the notion of the limit, with the result of an abstract mathematical object (as described by Indistinguishable) having an infinite perimeter (which in the standard mathematical terminology merely means, as Half Man Half Wit has said, that for any finite value x I can prove that the perimeter exceeds this value). These results seem to be consistent, even if they follow from an induction you’re not prepared to accept axiomatically.
In mathematical contexts, limits are the only way that “infinity” or “infinite” has meaning. It’s possible that you have some other definition of “infinite” to which limits do violence, but none of us here know what definition you’re using. Perhaps you could explain to us what “infinite” means to you?
This has nothing to do with anything here, but looking back over that other thread, I feel like Chronos was unfairly slighted a bit, by me among others; we took him to task over “I’m not even convinced that you can always choose an element from a two element set (Pick one element of the set of square roots of minus one, and tell me which element you picked without resorting to circular arguments)” and claimed this had very little to do with the axiom of choice. But thinking on it now, I would not want to say that. Actually, this is a fine example of what, in topos theory, is considered a failure of the “external” axiom of choice (very roughly, the internal axiom of choice is the supposition that choice functions can be proved to exist, while the external axiom of choice is the supposition that a definable choice function can actually be exhibited (though this formulation hides the fact that they’re actually the same statement, interpreted in different settings)). I even think now Chronos gave the appropriate defense in response to claims to exhibit the choice he denied possible (i.e., that these depend on the internal choice of an isomorphism from the many possible). All the attacks on Chronos’s understanding were focused on the internal axiom of choice, but everything he said makes perfect sense considered as regarding the external axiom of choice.
So, uh… basically, I’d agree with Chronos on everything now. Just wanted to get that off my chest.
(I’m sure if I read more old threads, I’ll find more instances of “Wait; I shouldn’t have said it like that”, but rarely do you get the opportunity to actually make the amendments)
Hm. I hadn’t even realized that there was a distinction between the internal and external axioms of choice. But I don’t suppose I can argue with someone saying they agree with me.
This sounds like a semantic issue. For instance, I could define a mathematical function as f(x) = x/0 which looks to be completely useless. But that contradicts (or does violence to?) the definition that things function when they work, when they produce a useful output. But the mathematical use of “function” has its own definition, and it’s named “function” simply because of its similarity to the common usage of the word.
In the same way, “limit” has a strict mathematical definition. Visualizing its operation with the common usage of the word can be useful, but it’s wrong to apply the common usage of “limit” to the mathematical phrase “the limit as n approaches infinity” to say that this somehow limits infinity.
Yeah, I didn’t expect that you were overtly aware of these nuances (roughly speaking, it’s something that gets ignored in traditional expositions of set theory but discussed in category-theoretic foundations where a wider variety of models are considered, though it isn’t really intrinsically tied to this particular distinction), but just pointing out that they do give formal substance to the point you were making at a more intuitive level, so that we were off-base in being so dismissive before.
The nice thing about math is that everyone is required to spell out the meaning of their terms in unambiguous symbolic language. In particular, a word like “limit” or a phrase like “the limit is infinity” have been given a standard meaning, which you can find in innumerable mathematics textbooks (and which is essentially the one that I and others have given above).
Your objection seems to be, roughly, “Well, they should have chosen another word for it besides ‘limit’.” Maybe so. There are lots of technical terms which have been given names which might be misleading to lay people not familiar with the term’s technical meaning.
But I don’t think anyone here is saying “‘Limit’ is the best possible word that could have been chosen for referring to this mathematical concept.” They’re just making a statement about limits and infinity using the standard mathematical definition of what it means for a limit to be infinite.
Suppose someone were to say, “Prairie dogs are burrowing rodents,” and I were to reply, “That’s wrong, because no dogs are rodents.” My objection would be incorrect, because in fact “prairie dog” is a term that refers to a species of burrowing rodent. What I’m really trying to say is “I don’t think they should call them prairie dogs.” Which is a fine opinion to have, but that doesn’t invalidate the person’s original statement. Prairie dogs are burrowing rodents, whether the name “prairie dog” bugs me or not.
Likewise, it is true that the limit (as n goes to infinity) of (4/3)[sup]n[/sup] is infinity. Arguing it’s not true “because that’s not what ‘the limit goes to infinity’ means to me” is like arguing that “'Prairie dogs aren’t rodents, because to me prairie dogs are canines who live on the prairie.” Instead of arguing with what people are actually saying, you’re arguing with what you think their words should mean. As I say, the nice thing about math is that the words are assigned a standard meaning in advance. And at this point, changing the definition from the one every college-level math student already knows wouldn’t serve much purpose other than to create confusion.
Forget all the fancy explanations about fractals. Most are a bad reading of Sagan’s books and kind of a fascination with the pretty pictures you can get in a PC using a few simple equations.
Fractals are a subset of chaos theory.
Chaos theory was developed trying to solve the three-body problem, by Poincaré.
If you want a dummie approach, just check ergodic theory at Wikipedia.
Or read the classic, James Gleick’s “Chaos: making a new science” - wonderfully readable. That book, along with the free software, Fractint, changed my life many years ago. Got me so obsessed with chaos and fractals that I did my Masters on them (in education). I have read Gleick a number of times now, and get high on it every time.
I find fractals really interesting and pretty, but as ciroa indicates, they are the mathematics underpinning chaos theory. See the implications in terms of the real effects of chaos - it is then that they really gain their full glory. The less spectacular fractals, such as the bifurcations or the Lorenz Attractor, start to take over as the most intriguing of all.
As a real-world example of fractals, chaos theory, and the relationship between them: Suppose you have two large masses, and put them at, say, the points (0,1) and (0,-1) on a coordinate grid. The large masses are fixed in place somehow, so they won’t move. Now, take a mass to some other point in the coordinate grid, and drop it from rest. Eventually, it’ll hit one of the two big masses. If it hits the mass on top, then you color the point you started from red; if it hits the mass on bottom, you color the point blue.
Now, if you do this using ordinary Newtonian gravity, the picture you get is boring: It’s just all red on the top half, and all blue on the bottom half. But if your two big masses are black holes, and you use general relativity, a very different picture emerges: The top half is mostly red, and the bottom is mostly blue, but there are large loops of blue in the red part, and those blue loops have red loops in them, and the red loops in the blue loops have more blue loops in them, and so on. There are regions near the boundaries of the loops where, if you move your initial particle a very small amount in any direction, you might end up falling into a differernt hole. This is the type of behaviour that chaos theory talks about. And if you draw a picture of the different outcomes you get from the different initial conditions, that gets you a fractal.
I spent a few minutes looking for pictures, myself, before posting that. Unfortunately the only hits I found were in journals that most folks here won’t be able to access. Fortunately the author of several of those papers is one of the professors here, so I might be able to get something from him.
