The wire is three-dimensional; it’s just its cross-sections which are two-dimensional.
Regardless of what you say, absolute quiet is -infinite decibels, a consequence of the fact that decibels are a logarithmic measure of pressure; if you keep subtracting 20 decibels, you’ll just keep dividing pressure by 10. The way the definition works, to get all the way down to no pressure, you have to go to -infinity decibels.
As you move along a line, your vertical distance and horizontal distance change in a constant ratio; this is called the slope of the line. For example, a line at 45 degrees off of horizontal has slope 1, since movement along the line changes vertical and horizontal distance at the same rate. A line at 60 degrees off of horizontal has slope sqrt(3); moving along this line causes vertical distance to increase sqrt(3) times as fast as horizontal distance. All good and well. But a vertical line? Movement along it doesn’t change horizontal distance at all, while vertical distance zips on by; thus, its slope is considered infinite.
It depends on what framework we’re modelling this within. In the Euclidean plane, will stop touching X once you’ve turned the full 90º. In the projective plane, any two lines intersect, even distinct horizontal lines. But I have no idea where you’re going with this.
Ok. I see. But still, I get the aroma that everything you’re getting at is either approaching infinity (by dividing to ever smaller fractions) only never actually getting there; or approximating infinity, which is another way of saying really really small, or really really big, but not actually infinity. Which means, not infinity.
Aiight. Time to hit the sack. Thanks for the replies all, even the ones I didn’t respond to… it’s been very interesting.
The inability to make a perfect circle is just another way of phrasing the inability to make a length ratio which is exactly equal to π. For any real world length ratio r, the number of digits to which r and π agree cannot be infinite. Fine. Are you willing to say that zero does not exist in the real-world as well, since r - π cannot be 0?
Basically, you’re saying that some particular things can’t be infinite (ratios of non-zero masses, ratios of non-zero lengths, -log_10(r-π), etc.). Fine. But that doesn’t mean the infinite doesn’t correspond perfectly well to other things that can be physically realized or approximated.
There’s nothing intrinsically linking a five meter bar to the number 5; we happen to spot a certain correspondence between a mathematical system and properties of this bar which allows us to think of the bar as representing 5. Under another correspondence, we might think of the bar as representing a different number. Under some useful correspondences, some physical quantities will be thought of as representing, or approximating a representation of, something infinite (as in all the examples I’ve given), even though, under others, this won’t happen.
The only qualms to be found in what I’ve given about approaching infinity but not hitting it right on the nose are the same as would cause us to say “You can only approach exactly 7 meters; you cannot hit it right on the nose”. Infinity is no more troubling than anything else in this regard; see my above question “Are you willing to say that zero does not exist in the real-world as well…?”
Nay, because by saying that r-pi does not equal zero, we don’t refute the existence of zero. We’re refuting the ability of man to construct an r that would be equal to pi, or that we cannot obtain zero through subtracting r and pi. There are other ways of obtaining zero like… having an apple and then taking away an apple. Apples, we can construct, but an r equal to pi we cannot.
yay because:
There are things that we can construct that would yield infinite results if placed in a ratio. Despite not being able to construct a perfect circle, we can certainly construct a perfectly straight line. And if you take that line and orient a coordinate system on top of it with the line going up and down the vertical, we can take the tangent of that line and invariably obtain a real life example of infinity. After that it is even easier to reason that infinity exists on many levels in terms of ratios - all you need is one non-zero number in relation to zero, and then BAM: infinity.
So, does infinity exist? If you’re willing to accept that ratios are real things, than yes. If you’re not choosing to define “exist” because you can’t hold a ratio in your hand, then no.
Let’s look at the golden ratio. We create a vector line inside a computer. It’s one unit long. Finding phi on this line is rather simple, and you can zoom into the line with the phi tangent centered on your screen, theoretically forever, and the phi tangent remains perfectly in place on its infinite ratio.
Now, let’s make a line out of exactly 9000 carbon atoms. Which, I believe we have the technology to do today. (and I argue we can make a duplicate line of the exact length).
Now, let’s find phi on this line. You’d have to put it on the 5,562nd atom. Not quite phi.
If you want to argue for the space between the atoms, we can do that, but we’re not really on the line any more, and then we eventually hit the Planck length… so then what? I’m just not seeing a true infinity there.
I’m not arguing you can’t approximate infinities… I’m arguing that once you do, it’s not really infinity anymore, just as much as it’s not really phi anymore. Pretty damn close to our eyes – and all practicality, but just not the same thing.
I think this is too simplistic. Just because I can construct an infinite series that sums to 2 doesn’t mean that my experiences of 2 contain infinity. Math’s power at describing certain facets of reality is, I think, undeniable. I have argued on these boards that math is just the language of certainty. But this is not the same thing as saying infinity has a real analog. The use of 2 in regards to real objects is something we’re all familiar with. And, after a time, imaginary numbers. It might be that after a time we’ll be familiar with some use of infinity in everyday life. At such a time, I will agree with you.
That’s what I really like about the GR view of gravity, as just an aspect of space. Because time behaves in space the same way for acceleration and for gravity. The closer a clock is to the speed of light, the slower it moves from the perspective of an observer at rest. Likewise, the closer a clock is to a body of mass, the slower it will run. At the speed of light, and at a point of infinite gravity, it will stop altogether.
eta: As I think of it, I wonder whether the bounds of the universe could be described in terms of acceleration and gravity, rather than in terms of more traditional dimensions like length and width.
I’ve heard that the shape of the universe can be described as a chronosynclastic infundibulum, which also pretty well describes the shape of a black hole’s space.
[scribbles down “chronosynclastic infundibulum.” Googles it. Adds to personal lexicon, in hopes of whipping it out in conversation tomorrow, during Thanksgiving, to repel disliked family members.]
Yes, but that’s using zero for a quite different purpose. If we’re going to switch our methods of corresponding numbers with physical quantities, I can easily construct quantities corresponding to π or infinity as well. Which is my point; just because under some correspondences, we can’t physically realize certain concepts, it doesn’t follow that they aren’t realized under other natural correspondences.
Yes, indeed. The only reason to bother with using “slope” explicitly is to point out an example where such a correspondence between ratios and the physical world is already in natural use.
(Though, let me ask, why do you believe we can construct a perfectly straight line but not a perfect circle?)
Well, you can’t hold any abstract concepts in your hand, whether ratios or discrete cardinalities or whatever else, yet people seem to have no qualms with saying 2 exists.
Fine. Under this particular correspondence of a mathematical system with physical reality, you can’t realize φ. So what? Under this particular correspondence, you can’t realize -5 or 3/4 either. Under some other correspondence, you can.
I guess my point is, it’s meaningless to ask “Does mathematical object X exist in the physical world?” without specifying a particular manner for corresponding the mathematical framework in which X exists with entities in the physical world. And, even if there are some applications for which no such natural correspondence is forthcoming, there may be (and, indeed, often are) others for which such can be found, or even is already in common use.
So your objection is only that the layman is not already familiar with these particular uses of mathematics to correspond to physical reality? That doesn’t quite seem like it addresses the same skepticism the OP is getting at.
I would argue that saying infinity has a real analog means only that there is some natural analogy between mathematical systems containing infinity and the physical world; regardless of whether the layman is explicitly aware of them, exceeding simple such analogies are already in ubiquitous use, at least among those with such baseline scientific sophistication as to be worth looking at for this. The infinite is no more problematic to correspond with the physical world than -5 or 3/4; the trick, always, is to choose an appropriate system of correspondence.
That’s what I like best about the way you think, Indistinguishable. You take a Bobby Fischer approach, by which I mean you never lose sight of the basics. I’ve heard that Fischer, from time to time, watched the beginner’s chess lessons on public television so as never to lose touch with the elements of the game.
Sorry, I was a bit unfair, cmyk; the correspondence you had chosen does, of course, realize 3/4, as the 6750th atom. But my point stands; you can’t realize an imaginary number, a 3-dimensional vector, an angle, or a sine wave under this correspondence with atom positions within a line, either. It demonstrates nothing. All those concepts, like infinity, continue to have their physical correspondences elsewhere.
Thanks. It is indeed unfortunately easy to get so swept up in the popular heuristics taught over a lifetime of abstraction (e.g., representing so many physical quantities as either reals or integers) that one elevates them to dogma, forgetting to stop and look around at what else there is.
That’s okay. You have a better grip on this than I do, it seems. I am, after all a layman.
I’m trying to visualize your examples as best I can, but somehow I still remain unconvinced you can point them and say there’s a pure infinity in there.
