My statement was not an argument opposed to anyone side of the discussion, I only advanced it because there seemed to be a consensus between the discussants that time was unquestionably infinitely indivisible/continuous/not quantised, and whilst not wanting to argue that it was indeed quantised I wanted to put forth the idea that its a possibility to be considered.
It seemed relevant and worth doing so because several posts by different posters explicitly relied on time's 'continuous' nature.
Where there is no property there is no justice; a proposition as certain as any demonstration in Euclid
Fools! not to see that what they madly desire would be a calamity to them as no hands but their own could bring
nirgrahamUK: infinitely indivisible
infinitely indivisible
Is that supposed to say "infinitely divisible"?
If I wrote it more than a few weeks ago, I probably hate it by now.
good catch!
nirgrahamUK: good catch!
Well, I would argue that "time" dissolves into indivisible moments.
(For vision, I remember a sequence of discrete, 2D appearances that I put into an order.)
Assuming that you were including me in the discussants, where did I imply that I thought the opposite?
>Time is an abstract concept
Not entirely. I can measure space with a ruler. I can measure time with a clock. There's no abstraction there.
>Saying that we're "really" seeing timeless 4-D objects smeared through 3-D space+time, even if true, doesn't help us reach any better understanding of the nature of time.
Who knows what things "really" are? The math and concepts of the 4D "block universe" are convenient. Occam's Razor recommends itself here.
>a four-dimensional complex vectorspace with an actual metric, the Euclidian.
Ya you can use the trick of quaternions here, but special relativity and flat space breaks down as soon as you have a gravitational field - you may have to take into account the metric tensor.
>'plank time' is often referred to as smallest 'meaningful' unit of time, the issue of whether there are smaller 'unmeaningful/unmeasurable' units is then kind of metaphysical...
Planck time is just a bunch of important constants multiplied together to arrive at a specific duration. Into the multiplication you can also throw in dimensionless constants for fun. Hence, the smallest possible duration might be 100x bigger, or 100x smaller. Or there might not be a smallest duration. There's NO experimental evidence. (But in some cases Planck units are useful for facilitating math.)
Just because we humans perceive time as indivisible does not mean it is not quantised...so too for space. i.e. space-time...
True. We cannot distinguish between a sufficiently finely divided substance and a truly continuous substance. But the point remains that we experience the world as a continuum.
Clayton -
Clayton: But the point remains that we experience the world as a continuum.
But the point remains that we experience the world as a continuum.
What's a continuum?
Sorry, but I lost you at "slice the solid block of celluloid at any cut point".
Yeah, that was less than clear.
Consider the equation z = x^2 + y^2, here.
Now, consider an arbitrary plane, z = k, k > 0. The resulting intersection of the plane and parabola will be a circle of radius z^(1/2).
We can think of this shape as something being seen by a Flatworld resident. If we treat the z-axis as "Flatworld time" (t), the Flatworld resident will see a line segment of length 2*t^(1/2). But the time variable is not discrete, it is not a series of "moments", it's a continuous variable.
Even a countably infinite number of discrete circles of the correct radius stacked one on top of another would not be mathematically equivalent to the continuous case described above.
Well, a continuum is something which is "continuous", meaning, it is not discrete.
The number line is the classic example. The discrete number line consists of a series of points... I envision them as having an "air gap" between each one. The continuous number line, however, has no air gaps. Any point along the continuous number line melts into the points on either side of it, so tightly that no air gaps can be found. And no matter how much you compact the discrete number line together, the air gaps never go away (remember, the points on the line are infinitely small), so you can't get a continuous number line by packing in a sufficient number of discrete points (yes, there's a proof for this).
Clayton: remember, the points on the line are infinitely small
remember, the points on the line are infinitely small
How could something be "infinitely small"?
Clayton: Well, a continuum is something which is "continuous", meaning, it is not discrete.
Really I don't even see a reason to think that anything is "continuous".
Clayton: But the time variable is not discrete, it is not a series of "moments", it's a continuous variable.
But the time variable is not discrete, it is not a series of "moments", it's a continuous variable.
Interesting example, but I lost you there.
Really why couldn't it just be a series of moments?
(Telling me that the mathematicians use the word "continuous" to describe your example doesn't help!)
Clayton: Even a countably infinite number of discrete circles of the correct radius stacked one on top of another would not be mathematically equivalent to the continuous case described above.
But why couldn't what they experience be inequivalent to the continuous case described above?
(And, by the way, what does it mean to be "countably infinite"?)
Clayton: Consider the equation z = x^2 + y^2, here. Now, consider an arbitrary plane, z = k, k > 0. The resulting intersection of the plane and parabola will be a circle of radius z^(1/2). We can think of this shape as something being seen by a Flatworld resident. If we treat the z-axis as "Flatworld time" (t), the Flatworld resident will see a line segment of length 2*t^(1/2). But the time variable is not discrete, it is not a series of "moments", it's a continuous variable.
Actually I don't know whether I like this example.
Are you saying that the flatlander would see the whole circumference as a line?
(Also what they are seeing would get farther and farther away, but that would be meaningless.)
Edit:
Actually forget about the last point.
Are you saying that the flatlander would see the whole circumference as a line? (Also what they are seeing would get farther and farther away, but that would be meaningless.) Edit: Actually forget about the last point.
Well, now that you mention it, I guess it depends on whether the Flatlander is inside or outside the circle. If he's inside the circle, nothing will change in his view as he will just see a circle around him in every direction (does he have perspective vision?) Assuming he can't immediately perceive distance, he has no way to tell how far away the circle is apart from moving over to the circle and bumping into it.
I was speaking initially as if the Flatlander was outside the circle.
>How could something be "infinitely small"?
Conceptually, it's the idea of a speck - something of negligible size. The concept is important to understanding limits and calculus, and it's often wrongly derided. In nonstandard analysis, a valid branch of mathematics, infinitesimal numbers are first-class citizens. So if f(x)=xx, then (f(x+epsilon)-f(x)) / epsilon = 2x + epsilon. Taking the "standard part" of this yields 2x, or the derivative of f(x). In nonstandard analysis, there is an integer N so large that it's bigger than any real number and a simple example of an infinitesimal number is simply 1/N. I believe surreal and hyperreal numbers have multiple levels of infinitesimals and infinite numbers.
>And, by the way, what does it mean to be "countably infinite"?
Define the sequence 1, 2, 3, etc. as countable. It's also infinitely long.
Then the even numbers are also countably infinite since they can be arranged in one-to-one correspondence with the above sequence: 2, 4, 6, 8, etc. Want negative even numbers? Fine, then use 0, 2, -2, 4, -4, etc.
Even rational numbers (fractions) are countably infinite. A simple way to see this is to create a grid of numbers where the number at position (x,y) is simply x/y. Then draw a spiral going through all the numbers and you've put them in one-to-one correspondence with 1,2,3, etc. (Note: you get some duplicates like 1/2 = 2/4 = -1/-2 = etc. and nonsense like 0/0 and 1/0 so simply ignore these problems as you encounter them).
Cantor's "diagonalization" argument shows that real numbers are not countably infinite. Consider just the real numbers between 0.0 and 1.0. Suppose you succeded at putting them all into one-to-one correspondence with 1,2,3, etc.:
1 0.141592653...
2 0.292222222...
3 0.317493923...
etc.
Then it's easy to find a real number not in your list that proves you wrong. To find this number, for the first digit after the decimal point choose something other than what #1 has - anything but a 1. For the second digit, choose something different from what #2 has - anythnig but a 9. And so on. Then the number 0.235..... differs from every item in your list. It's not in your list.
It's impossible to "count" the real numbers. While, in some sense, both integers and fractions have the same number of elements - both being countable - there are too many real numbers - they are said to be uncountable.
Interesting example, but I lost you there. Really why couldn't it just be a series of moments? (Telling me that the mathematicians use the word "continuous" to describe your example doesn't help!)
baxter took care of the second half of your post... I'll answer this.
Why can't time be a series of moments? Let me point out that you're dipping your toes into a deep philosophical pool with lots of questions and few answers. The "Continuum Hypothesis" is, indirectly, what's behind all this. It's more than a century old but it remains unconfirmed/undenied by a century's worth of the best mathematical and philosophical minds.
Basically, the mathematician Georg Cantor showed that there are different orders or sizes of infinity. He called these the transfinite cardinals. Numbers have two different senses, the cardinal and the ordinal. The cardinal sense of a number answers the questions "how big?" or "how much?" The ordinal sense of a number answers the question "which one?" If I say, "take the third left after you go past Safeway", I am using the number three in its ordinal sense. If I say, "put three pears in the basket" I am using the number three in its cardinal sense. Cantor derived arithmetics for both transfinite cardinals and transfinite ordinals (the arithmetics of each is slightly different from the other).
Cantor is the father of set theory. Imagine a set of three numbers... {4, 5, 6} (the braces are standard notation for a set). What are all the possible combinations of these three numbers?
Well, the first combination is none of the numbers:
{} (the empty set)
The next three combinations would be one of each of the numbers:
{4}
{5}
{6}
The next three combinations would be two of each of the numbers:
{4,5}
{5,6}
{6,4}
The last combination would be all three numbers taken together (the original set):
{4,5,6}
The set of all combinations of the elements of a set (otherwise known as the "set of all subsets") was called the powerset by Cantor. To take the powerset of a set, you just say P(x) where x is the set whose powerset you want. So,
P({4,5,6}) = {{}, {4}, {5}, {6}, {4,5}, {5,6}, {6,4}, {4,5,6}}
It turns out that there is a relationship between the size (cardinality) of a set and the size (cardinality) of its powerset:
|P(x)| = 2^|x|
Where the ||'s mean "size of" or "cardinality of".
Cantor showed that the powerset operation is valid for infinite sets. But this presents an immediate problem. If the powerset operation is valid for infinite sets, what is the meaning of the cardinality relationship between the set and its powerset? What is 2 to the power of infinity??
Cantor used his famous diagonal argument to prove that the cardinality of the powerset is, in fact, well-defined for a powerset of infinite sets and that it is strictly greater than the size of the original infinite set! He then showed that you can take the powerset of the powerset of an infinite set (and so on), giving rise to an entire (infinite) hierarchy of infinities! It's beautiful stuff. I recommend Rudy Rucker's book Infinity and the Mind if you're curious to learn more (much more). It's approachable and fun and he really delves into the philosophical side of things more than the mathematical (but he's a mathematician, so he actually knows what the hell he's talking about).
That is the definition of a geometric point. It has no length, width or depth, yet it has a definite location. If you stick a ruler up to it to measure its size, the result will always be 0. So, it's "infinitely small" or of zero size. Yet, it definitely exists.
Clayton: Why can't time be a series of moments?
Why can't time be a series of moments?
Real fast, for you what does the word "time" refer to?
(Just to make sure that we are on the same page here.)
Clayton: Cantor showed that the powerset operation is valid for infinite sets.
Cantor showed that the powerset operation is valid for infinite sets.
What's an infinite set?
An infinite set is a set which has infinitely many members (no finite number can denote the cardinality of the set).
Clayton: An infinite set is a set which has infinitely many members (no finite number can denote the cardinality of the set).
Is that even conceivable?
I think you are suggesting that, since you obviously cannot hold the contents of an infinite set in your mind, that an infinite set is not conceivable.
Yet, you also cannot hold all of the digits of the real number pi in your head... do you think that pi is inconceivable? You can hold the picture of a circle and its radius in your mind, so surely pi itself exists... given that pi exists and that the real-number value of pi must have infinitely many digits (we can prove this), how is it problematic to say that all the digits of the number pi do exist as a mathematical object, even though you could not possibly hold them all in your mind?
Clayton: Yet, you also cannot hold all of the digits of the real number pi in your head... do you think that pi is inconceivable? You can hold the picture of a circle and its radius in your mind, so surely pi itself exists... given that pi exists and that the real-number value of pi must have infinitely many digits (we can prove this), how is it problematic to say that all the digits of the number pi do exist as a mathematical object, even though you could not possibly hold them all in your mind?
I can see a circle on a piece of paper, and ultimately pi is the ratio between the amount of points making up its circumference and the amount of points making up its diameter.
It mightn't be possible to conceive of pi in one moment, but it's certainly possible to conceive of it over time.
(I'm not against things that have to conceive over time to be able to conceive. In fact, by definition every set is like that!)
Or an even more simple example: We can hold 2 in mind, we can hold 3 in mind. But what about 2/3? It has infinitely many digits, since 2/3 = 0,666666...
Metus: Or an even more simple example: We can hold 2 in mind, we can hold 3 in mind. But what about 2/3? It has infinitely many digits, since 2/3 = 0,666666...
Oh, I get where this is going.
Well, what's it even mean to have an infinite number of digits?
Clayton: Cantor used his famous diagonal argument to prove that the cardinality of the powerset is, in fact, well-defined for a powerset of infinite sets and that it is strictly greater than the size of the original infinite set! He then showed that you can take the powerset of the powerset of an infinite set (and so on), giving rise to an entire (infinite) hierarchy of infinities! It's beautiful stuff. I recommend Rudy Rucker's book Infinity and the Mind if you're curious to learn more (much more). It's approachable and fun and he really delves into the philosophical side of things more than the mathematical (but he's a mathematician, so he actually knows what the hell he's talking about).
I'm also really confused as to how this applies to my question.
The trouble with using a fraction, as Metus did, is that it has an infinite number of digits depending on which number base you use. In base 10, 2/3 has infinitely many digits.
But what it means for 2/3 to have infinitely many digits in base 10 is that each time you perform the division of 3 into 2, there is a remainder of 2 again (3 goes into 20 six times, giving the digit '6' and the remainder 20-18=2 ... ad infinitum, that is, this process provably never terminates).
the amount of points making up its circumference and the amount of points making up its diameter
But what is the amount of points? Let's say we stacked the points one against the other, from left to right. Since they have zero width, you can pack in as many as you like but the length will always remain zero... 0+0+0+0+0...=0. Or, we can take a different approach and try to count how many times you can cut a line and take that as the number of points on the line. But no matter how many pieces you've already divided the line into, you can always zoom in further and divide those pieces yet again, since the result of dividing a line is always two lines, not a line and a point. You never reach a limit where you have finally divided the line into 'points'.
Or, to put it in formula: For every two (real or rational) numbers x_1 and x_2 there is a third (real or rational) number y with x_1 < y < x_2. Though, Clayton, one has to admit that "length" really is a non-trivial property.
Clayton: But what is the amount of points?
But what is the amount of points?
Here's David Hume's clearest definition of a point:
David Hume: 'Tis the same case with the impressions of the senses as with the ideas of the imagination. Put a spot of ink upon paper, fix your eye upon that spot, and retire to such a distance, that at last you lose sight of it; 'tis plain, that the moment before it vanish'd the image or impression was perfectly indivisible. 'Tis not for want of rays of light striking on our eyes, that the minute parts of distant bodies convey not any sensible impression; but because they are remov'd beyond that distance, at which their impressions were reduc'd to a minimum, and were incapable of any farther diminution. A microscope or telescope, which renders them visible, produces not any new rays of light, but only spreads those, which always flow'd from them; and by that means both gives parts to impressions, which to the naked eye appear simple and uncompounded, and advances to a minimum, what was formerly imperceptible.
'Tis the same case with the impressions of the senses as with the ideas of the imagination. Put a spot of ink upon paper, fix your eye upon that spot, and retire to such a distance, that at last you lose sight of it; 'tis plain, that the moment before it vanish'd the image or impression was perfectly indivisible. 'Tis not for want of rays of light striking on our eyes, that the minute parts of distant bodies convey not any sensible impression; but because they are remov'd beyond that distance, at which their impressions were reduc'd to a minimum, and were incapable of any farther diminution. A microscope or telescope, which renders them visible, produces not any new rays of light, but only spreads those, which always flow'd from them; and by that means both gives parts to impressions, which to the naked eye appear simple and uncompounded, and advances to a minimum, what was formerly imperceptible.
Put a spot of ink on the wall, and then step back until it disappears.
It seems obvious to me that, the moment before it disappeared, it was an indivisible point.
(By the way, that wasn't really a definition, but I didn't know what else to call it.)
Clayton: Let's say we stacked the points one against the other, from left to right. Since they have zero width, you can pack in as many as you like but the length will always remain zero... 0+0+0+0+0...=0. Or, we can take a different approach and try to count how many times you can cut a line and take that as the number of points on the line. But no matter how many pieces you've already divided the line into, you can always zoom in further and divide those pieces yet again, since the result of dividing a line is always two lines, not a line and a point. You never reach a limit where you have finally divided the line into 'points'.
Let's say we stacked the points one against the other, from left to right. Since they have zero width, you can pack in as many as you like but the length will always remain zero... 0+0+0+0+0...=0. Or, we can take a different approach and try to count how many times you can cut a line and take that as the number of points on the line. But no matter how many pieces you've already divided the line into, you can always zoom in further and divide those pieces yet again, since the result of dividing a line is always two lines, not a line and a point. You never reach a limit where you have finally divided the line into 'points'.
Then what should we do?
Clayton: But no matter how many pieces you've already divided the line into, you can always zoom in further and divide those pieces yet again, since the result of dividing a line is always two lines, not a line and a point. You never reach a limit where you have finally divided the line into 'points'.
But no matter how many pieces you've already divided the line into, you can always zoom in further and divide those pieces yet again, since the result of dividing a line is always two lines, not a line and a point. You never reach a limit where you have finally divided the line into 'points'.
But why would you zoom in?
Well, what's it even mean to zoom in?
Zooming in 2X is moving from one 2D picture where something is made up of X points to a new 2D picture where it's made up of 2X points. (At least according to my position...)
But aren't we trying to figure out whether it makes sense to say that a finite number of points makes up an image? Why are you bringing up sequences of images?
Clayton: But what it means for 2/3 to have infinitely many digits in base 10 is that each time you perform the division of 3 into 2, there is a remainder of 2 again (3 goes into 20 six times, giving the digit '6' and the remainder 20-18=2 ... ad infinitum, that is, this process provably never terminates).
How can we conceive of that?
0.666... is adding 0.6 to 0.06 to 0.006 and so on.
But, if a single image in our mind is made up of a finite number of indivisible points, like I think, then we would quickly run up against a problem: How could we keep adding something smaller and smaller? Wouldn't we eventually run up against the indivisible points?
So we would need to zoom out of the picture at the exact rate that we are adding the smaller and smaller parts, in order to make sure that we have a constant supply of new points to use.
Does that make any sense?
>An infinite set is a set which has infinitely many members (no finite number can denote the cardinality of the set).
>Is that even conceivable?
Yeah, but perhaps it's more easily understood if the set is defined by a rule (like "all my members are even integers") rather than trying to think of an infinite number of elements all at once.
Note that not not all mathematicians accept the idea as valid. "Most modern constructive mathematicians accept the reality of countably infinite sets (however, see Alexander Esenin-Volpin for a counter-example)." - from http://en.wikipedia.org/wiki/Intuitionism
Even more extreme: http://en.wikipedia.org/wiki/Ultraintuitionism "some ultrafinitists will deny the existence of... exp(exp(exp(79)))"
Not everyone in mathematics accepts the same axioms. I guess when you get too abstract, there isn't a lot of reality to help guide you. Sometimes either choosing an axiom or choosing its negation can lead to uncomfortable conclusions: http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox "a solid ball... can be split into a finite number of non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball"
baxter: Define the sequence 1, 2, 3, etc. as countable. It's also infinitely long. Then the even numbers are also countably infinite since they can be arranged in one-to-one correspondence with the above sequence: 2, 4, 6, 8, etc. Want negative even numbers? Fine, then use 0, 2, -2, 4, -4, etc. Even rational numbers (fractions) are countably infinite. A simple way to see this is to create a grid of numbers where the number at position (x,y) is simply x/y. Then draw a spiral going through all the numbers and you've put them in one-to-one correspondence with 1,2,3, etc. (Note: you get some duplicates like 1/2 = 2/4 = -1/-2 = etc. and nonsense like 0/0 and 1/0 so simply ignore these problems as you encounter them). Cantor's "diagonalization" argument shows that real numbers are not countably infinite. Consider just the real numbers between 0.0 and 1.0. Suppose you succeded at putting them all into one-to-one correspondence with 1,2,3, etc.: 1 0.141592653... 2 0.292222222... 3 0.317493923... etc. Then it's easy to find a real number not in your list that proves you wrong. To find this number, for the first digit after the decimal point choose something other than what #1 has - anything but a 1. For the second digit, choose something different from what #2 has - anythnig but a 9. And so on. Then the number 0.235..... differs from every item in your list. It's not in your list. It's impossible to "count" the real numbers. While, in some sense, both integers and fractions have the same number of elements - both being countable - there are too many real numbers - they are said to be uncountable.
Thanks.
Clayton, one has to admit that "length" really is a non-trivial property.
For a mathematician, yes. However, neither myself nor IRyan are mathematicians and our interest is more general than the specialized interest of a mathematician in precisely defining the objective attributes of mathematical objects, such as the length of a line.
Here's David Hume's clearest definition of a point
Hume makes an interesting and important physical argument but we can define a 'point' analytically to have certain attributes that we desire it to have, regardless* of the nature of the physical world. I don't think Hume's discussion directly bears on the issue at hand, either - can you mark dots on a piece of paper, separated by whitespace, walk backwards and thereby they (actually) become a line? I think not.
*The "regardless" is not as "regardless" as those who enjoy analytic study more than synthetic study tend to think... we usually choose to think about certain analytic objects exactly because of their resemblance to the physical world.
Clayton: But what it means for 2/3 to have infinitely many digits in base 10 is that each time you perform the division of 3 into 2, there is a remainder of 2 again (3 goes into 20 six times, giving the digit '6' and the remainder 20-18=2 ... ad infinitum, that is, this process provably never terminates). How can we conceive of that?
To quote the immortal Inigo Montoya... "You keep using that word. I do not think it means what you think it means."
The "and so on" is uber-important. :-)
What you're really saying is that:
2/3 = Sum_n=1-oo [ 6/10^n ]
Read: "Two-thirds is exactly equal to the sum taken from n equal to 1 and onward, ad infinitum, of six divided by ten raised to the power of n."
2/3 = 6/10 + 6/100 + 6/1000 + 6/10000 ...
Note that, since this could be formally proven (I can't remember how to do proofs with infinite sums right at the moment, but they're fairly straightforward), it's not a matter of saying that the sum 'approaches" 2/3 eventually. It's just saying that you can rewrite 2/3 as the above sum.
But, if a single image in our mind
I've never perceived a point of an image in my mind. My sense of sight presents itself to my consciousness as an image that is whole and undivided, not composite.
is made up of a finite number of indivisible points, like I think, then we would quickly run up against a problem: How could we keep adding something smaller and smaller? Wouldn't we eventually run up against the indivisible points?
Let me help you get out of the mental trap you're stuck in. What is a 'point' of sound? What is a 'point' of smell? What is a 'point' of taste? What is a 'point' of proprioception (the sense of taking up space and having mass)? If you're going to start your analysis from sense perception, then don't just look only at one sense. Look at your other senses. As a result of the wonderful modern technology of electronic raster imaging devices (televisions, computer monitors, etc.) we have this digital/composite "metaphor" that we can rely on when thinking about the visual sense. But that metaphor is hindering you, I think. Try thinking about sound being made up of "points". Can you divide sound into instantaneous pieces which, when pieced together in time, form the complete sounds you hear in your head (warning: this is a trick question)?
>>2/3 = 6/10 + 6/100 + 6/1000 + 6/10000 ...
>Note that, since this could be formally proven
Multiply each side by (1 - 1/10):
2/3 - 2/30 = 6/10 - 6/100 + 6/100 - 6/1000 + 6/1000 = etc.
This alternating, telescopic series collapses to yield an obviously true statement:
2/3 - 2/30 = 6/10
You can similarly prove 1+2+4+8+16+... = -1 by multiplying by (1-2). This is another thing that not all mathematicians would accept. (Euler would.)
Gotta love p-adic numbers. :-D
>Gotta love p-adic numbers. :-D
There's also
1+2+3+4+...=-1/12
1x1+2x2+3x3+4x4+... = 0
1x1x1+2x2x2+3x3x3+4x4x4+... = 1/120
1x2x3x4x5x... = sqrt(2pi)
2x3x5x7x11x... (i.e. prime numbers) = 4pi*pi
1-1x2 + 1x2x3 - 1x2x3x4 + ... = 0.403... = 1 - gompertz' constant
baxter:...special relativity and flat space breaks down as soon as you have a gravitational field
in what sense does empty space have a shape at all, let alone FLAT?
I. Ryan:How could something be "infinitely small"?
it can't. if you describe something small, i can describe for you something smaller. we can continue this process until we get bored...
Clayton: I. Ryan:0.666... is adding 0.6 to 0.06 to 0.006 and so on. The "and so on" is uber-important. :-) What you're really saying is that: 2/3 = Sum_n=1-oo [ 6/10^n ] Read: "Two-thirds is exactly equal to the sum taken from n equal to 1 and onward, ad infinitum, of six divided by ten raised to the power of n."
I. Ryan:0.666... is adding 0.6 to 0.06 to 0.006 and so on.
...but what we can never do is continue it "to infinity". the moment we posit an idea which we ourselves admit that we cannot ratiocinate at all, we enter the realm of RELIGION. we have come up with a comforting answer that sounds nice...
Wikipedia:"Most modern constructive mathematicians accept the reality of countably infinite sets (however, see Alexander Esenin-Volpin for a counter-example)."
Clayton:The "Continuum Hypothesis" is, indirectly, what's behind all this. It's more than a century old but it remains unconfirmed/undenied by a century's worth of the best mathematical and philosophical minds.
...and we have heard reassuring noises from the AUTHORITIES. but we have lost our original purpose, waylayed into a century-long academic circle jerk rather than trying to answer real questions.
...
if we want to understand what we are trying to get at with the word "infinity", we must first understand what we mean by "number". what is 7? it is not a thing but an act, a process: it is counting. arbitrarily, it is a MOVIE of a someone moving their index finger up and down over and over, then stopping.
and with that, the fount of nonsense runs dry. there is something we can actually see in our mind's eye, something someone can actually tell us how to think of.
so what is "infinity" - nay, the real question is "what do we mean by infinity, if anything?" INCESSANT COUNTING! a movie of someone moving their index finger up and down...but the movie never seems to end...so far. what we cannot do is imagine a movie playing on for "eternity."
if you can say a word, but you cannot actually think of or imagine the concept behind it, the word has no correspondent in your thoughts: in common parlance, the word is nonsense. infinity is a plizzlevatch, a foogalop, a smeegram!