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?
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!)
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).
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.
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?
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.
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
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”
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.
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.
Clayton -
*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.
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.
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.
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)?
in what sense does empty space have a shape at all, let alone FLAT?
it can’t. if you describe something small, i can describe for you something smaller. we can continue this process until we get bored…
…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…
…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!
