in what sense does empty space have a shape at all, let alone FLAT?
Space has shape in the same way as the surface of Earth has shape (the latter happens to be spherical). In space - when a massive body like a star is present - the three angles of a triangle won’t add up to 180 degrees exactly, but will add up to some other value. The circumference of a circle won’t be pi times the diameter.
By flat I meant using an identity matrix as the metric tensor (in conjunction with imaginary time). An approximation would be in an area of the universe distant from any matter. And for any reference frame - whether it be inertial or non-inertial - one can errect a coordinate system that approximates the space as flat space for a tiny 4-volume.
"How could something be “infinitely small”
>it can’t … i can describe for you something smaller. we can continue this process until we get bored…
Infinity is merely a metaphor for the ultimate results of doing something forever - which obviously can’t be done in practice. It’s up to you if you want to cripple yourself by rejecting this useful metaphor.
we enter the realm of RELIGION. we have come up with a comforting answer that sounds nice…
waylayed into a century-long academic circle jerk rather than trying to answer real questions.
These sound like personal problems. Like everyone in practice, I use calculus, infinitesimals, infinity, etc. all the time.
“pi exists” … it’s a ratio…but what do we mean by ratio? unless we are true-believer platonists, we don’t believe numbers or ratios exist. numbers and ratios are processes. they are ACTS, entailing an observer. hence it is clearer to use verbs and gerunds to talk about them. natural numbers are counting. decimals are multiple interations of counting with decimal places used for accounting purposes.
pi as an “infinite” decimal expansion is, i’ll go ahead and use the word, INCONCEIVABLE. you cannot see it in your mind, or sense it in your mind in any way. but pi to a certain number of decimal places is conceivable.
yet, you say, we can picture a circle and its diameter, the implication being that we can picture the ratio of the circumfrence of the circle to its diameter. actually, we cannot picture it as a static frame, but we can animate it into an internal movie…a process. it is the ACT of imagining the diameter pushed up against the circle once, twice, three times in a continuous line, and then having a little left over. for the partial decimal expansion, you cut the diameter into ten equal pieces, then a hundred equal piece, then a thousand…and attach one of the first pieces onto the continuous line, four of the second piece, one of the third…3,14159…continuing this process until you get bored.
pi is a process, an act by an observer…hence it doesn’t exist - any more than farting exists. it is not sitting there, inherent in the circle itself; it only “exists” as a possible action we can take.
By the way, he also made a correlative “mental” argument for the same thing:
(Though I like his other one much better.)
I don’t see a difference between the images in the outside world and those in my mind’s eye, except that the first ones are more vivid than the second.
(So I’m not sure how I could disregard “the nature of the physical world”.)
Actually a line in what sense?
In the sense that its appearance is actually that of a line, obviously yes.
But, in the sense that it’s actually a 3D object that’s… well, I don’t know where you’re going with this.
What does it mean to “actually become a line”? What exactly is an “actual” line?
Tis universally allow’d, that the capacity of the mind is limited, and can never attain a full and adequate conception of infinity: And tho’ it were not allow’d, 'twou’d be sufficiently evident from the plainest observation and experience. 'Tis also obvious, that whatever is capable of being divided in infinitum, must consist of an infinite number of parts, and that 'tis impossible to set any bounds to the number of parts, without setting bounds at the same time to the division. It requires scarce any induction to conclude from hence, that the idea, which we form of any finite quality, is not infinitely divisible, but that by proper distinctions and separations we may run up this idea to inferior ones, which will be perfectly simple and indivisible. In rejecting the infinite capacity of the mind, we suppose it may arrive at an end in the division of its ideas; nor are there any possible means of evading the evidence of this conclusion.
'Tis therefore certain, that the imagination reaches a minimum, and may raise up to itself an idea, of which it cannot conceive any sub-division, and which cannot be diminished without a total annihilation. When you tell me of the thousandth and ten thousandth part of a grain of sand, I have a distinct idea of these numbers and of their different proportions; but the images, which I form in my mind to represent the things themselves, are nothing different from each other, nor inferior to that image, by which I represent the grain of sand itself, which is suppos’d so vastly to exceed them. What consists of parts is distinguishable into them, and what is distinguishable is separable. But whatever we may imagine of the thing, the idea of a grain of sand is not distinguishable, nor separable into twenty, much less into a thousand, ten thousand, or an infinite number of different ideas
"the capacity of the mind is limited, and can never attain a full and adequate conception of infinity"
This statement is trivial. Nothing a human being can do will be full and adequate when compared to a standard of perfection.
"any finite quality, is not infinitely divisible"
I’m not sure what Hume means by division here. Is he talking about quotients? Or about pieces/segments? If the pieces are allowed to have different sizes, then things can be divided infinitely. Consider 1 = 1/2 + 1/4 + 1/8 + 1/16 + …
Zeno’s paradox is not really a paradox. The runner does overtake the tortoise. Maybe I’ve been doing actual useful mathematics for too long, but this pedantic hand-wringing over infinities seems inane.
In the same sense that Hume is speaking of an actual point. He notes that if you used a device you would be able to separate the dot again, even after you have stepped back from it. By backing up, you accept that you are not altering the point itself, only your view of the point. Hume is simply noting that our perception of things is what it is and when you step back from the point, a condition arises where the point itself is indivisible by your mind in its own perceptual space. This is not a statement that the physical world itself has changed simply because you backed up, which is what he means when he says that you could use a telescope to separate the light beams and see the dot in more detail even after having backed away from it.
If you back away from a series of closely spaced (yet separated) dots, they will start to look like a solid line. But if you got the telescope out, you would again be able to see that they are, in fact, separated points. The points do not physically merge together by virtue of your stepping back any more than the point on the paper becomes physically indivisible by virtue of your stepping back.
a planet is an object, not empty space. is empty space an object? not a trick question - rather, i’m asking you what you mean by “space” so we stay on the same page.
so your claim, rather than that space is actually curved near a star, is that when a very long wire, which was straight in an area of the universe distant from any massive bodies, comes near a star, it ceases to be straight?
for practical purposes, words like infinite and forever mean variously, “as long as necessary,” “as long as is feasible,” “as long as i can imagine”. we enter religion, or really just incoherent babbling if we venture beyond those uses of the word. [the definition of a limit in calculus does not require infinity anyway, as i’m sure you’re aware. instead it describes a useful process that one may continue for as long as is needed for the application in question.]
I agree with your frustration. However, a lot of people are hung up by it and I think the hang-up is the idea of containing “an infinity” inside one’s mind which is obviously impossible. There are two ways to deal with this. The first is to point out infinities that are implicit in very ordinary things (like the irrationality of the length of a hypotenuse of a right triangle with sides of length 1… or pi, entailing an infinitude of digits). The second is to note that very large finite finite numbers also do not fit within the mind, yet they are not infinite, so what’s wrong with them? Consider the number 1,000! Can that number fit within your mind? If you find that easy to contemplate, how about the number pi^googleplex? Or how about B(100) (with B() the “busy beaver” function)?
Another approach is to treat infinities as formalisms. Aleph_0 is just a symbol standing for “something” having this, that and the other properties. That “something” can be called “infinity”, if you like, or call it something else if you’re uncomfortable with that word.
But, yeah, there’s no good reason to associate the infinite with the mystical in a post-Cantor world.
That’s the point of formalism… you don’t need to conceive it in your mind, you simply show that the first object has the same formal properties as the second object, differently written, and you’re done. They are “equivalent”.
1/2+1/4+1/2^n is just another way of writing the number 1 because both objects have the same properties (dividing both by x yields 1/x).
erm? i have a calculator, and i have learned the rules of arithmetic. even if for no other reason, i trust what they tell me because they have served me well so far. i trust that the results outputted represent the results of any counting process(es) i would have undertaken in the absence of these tools.
Not really mine. More like Einstein’s and every non-crackpot physicist for the last 80 years.
a very long wire… comes near a star, it ceases to be straight?
Yes. The same way a straight wire wrapped around a sphere ceases to be straight. A gigantic, triangular, rigid object dragged near a star will cease to have angles adding up to 180 degrees.
the definition of a limit in calculus does not require infinity anyway, as i’m sure you’re aware. instead it describes a useful process
Yeah, great, too bad the epsilon-delta definition is useless in practice. Infinitesimals are easier to understand, are more useful, were used by Leibniz a founder of calculus, and are logically valid per the branch of mathematics known as nonstandard analysis.
we enter religion, or really just incoherent babbling
Yeah, that’s pretty much what this arguing is. If you need help doing actual math, or actually calculating something, let me know.
I assume you are using the word “religion” (or superstition) to refer to speculation about things that do not actually exist. Well, I hate to break it to you, but finite numbers do not exist any more than infinite numbers do.
Incoherence is the result of ignoring contradictions in one’s system. What contradictions must be ignored to operate with infinities?
But what do you conceive of in your mind when you think of [1/2 + 1/4 + 1/8 + 1/16 + …]?
A bar of unit length, cut down the middle. In the right half, there is another cut down the middle. In the right half of that, there is another cut. I see more and more cuts as I look to the right.
Or like Clayton said, I can also view it as a formal expression - a bunch of symbols - which yields truth when prepended to “=1”
