>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.
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...
"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.
Clayton: To quote the immortal Inigo Montoya... "You keep using that word. I do not think it means what you think it means."
To quote the immortal Inigo Montoya... "You keep using that word. I do not think it means what you think it means."
How could something not mean what I think it means?
Clayton: 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.
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'm pretty sure that you misunderstood me, but I won't try to explain it again until we have some more common ground here.
Clayton: I've never perceived a point of an image in my mind.
I've never perceived a point of an image in my mind.
Do you see anything similar between this image and this one?
Clayton: My sense of sight presents itself to my consciousness as an image that is whole and undivided
My sense of sight presents itself to my consciousness as an image that is whole and undivided
Mine too!
Clayton: not composite.
not composite.
I'm not sure what that means.
Clayton: 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.
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.
Let me quote David Hume, who indirectly answered that objection:
Clayton: The first notion of space and extension is deriv'd solely from the senses of sight and feeling; nor is there any thing, but what is colour'd or tangible, that has parts dispos'd after such a manner, as to convey that idea. When we diminish or encrease a relish, 'tis not after the same manner that we diminish or increase any visible object; and when several sounds strike our hearing at once, custom and reflection alone make us form an idea of the degrees of the distance and contiguity of those bodies, from which they are deriv'd. Whatever marks the place of its existence either must be extended, or must be a mathematical point, without parts or composition. What is extended must have a particular figure, as square, round, triangular; none of which will agree to a desire, or indeed to any impression or idea, except of these two senses above-mention'd. Neither ought a desire, tho' indivisible, to be consider'd as a mathematical point. For in that case 'twou'd be possible, by the addition of others, to make two, three, four desires, and these dispos'd and situated in such a manner, as to have a determinate length, breadth and thickness; which is evidently absurd.
The first notion of space and extension is deriv'd solely from the senses of sight and feeling; nor is there any thing, but what is colour'd or tangible, that has parts dispos'd after such a manner, as to convey that idea. When we diminish or encrease a relish, 'tis not after the same manner that we diminish or increase any visible object; and when several sounds strike our hearing at once, custom and reflection alone make us form an idea of the degrees of the distance and contiguity of those bodies, from which they are deriv'd. Whatever marks the place of its existence either must be extended, or must be a mathematical point, without parts or composition. What is extended must have a particular figure, as square, round, triangular; none of which will agree to a desire, or indeed to any impression or idea, except of these two senses above-mention'd. Neither ought a desire, tho' indivisible, to be consider'd as a mathematical point. For in that case 'twou'd be possible, by the addition of others, to make two, three, four desires, and these dispos'd and situated in such a manner, as to have a determinate length, breadth and thickness; which is evidently absurd.
So let's not reserve the word "point" for anything else but the simplest parts of an image!
Clayton: 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.
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.
David Hume wasn't familiar with that wonderful technology, was he?
Clayton: But that metaphor is hindering you, I think.
But that metaphor is hindering you, I think.
It's not a metaphor.
But, then again, I don't think that you have understood me yet.
(Not your fault!)
Clayton: Try thinking about sound being made up of "points".
Try thinking about sound being made up of "points".
Can't, but I can do something similar, which might be the subject of your next question.
Clayton: 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)?
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)?
Can I divide a sound into a sequence of appearances?
Of course I can do that, but I'm not sure whether that's what you're asking.
If I wrote it more than a few weeks ago, I probably hate it by now.
>"we don't believe numbers or ratios exist. numbers and rations are processes. they are ACTS"
If you can only deal with numbers as high as you can count on your toes, then I feel sorry for you.
Clayton: Hume makes an interesting and important physical argument
Hume makes an interesting and important physical argument
By the way, he also made a correlative "mental" argument for the same thing:
Clayton: 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.
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.
(Though I like his other one much better.)
Clayton: 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.
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 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".)
Clayton: 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.
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.
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?
>"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.
Actually a line in what sense? In the sense that its appearance is actually that of 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's an "actual[...] line"?
In the sense that its appearance is actually that of line, obviously yes.
What does it mean to "actually become a line"? What's an "actual[...] line"?
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.
Clayton -
baxter: 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.
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.
He's talking about single images present to your mind.
(It doesn't allow different sizes, which come in much later, when your mind starts categorizing your 2D appearances into 3D objects.)
baxter:Space has shape in the same way as the surface of Earth has shape (the latter happens to be spherical).
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.
baxter: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.
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?
baxter:Infinity is merely a metaphor for the ultimate results of doing something forever - which obviously can't be done in practice
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.]
Clayton: 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.
In its own perceptual space!
That's what I'm talking about here.
Clayton: 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.
Of course, and I'm just talking about appearances, nothing else.
(Understanding what 3D objects in the outside world are comes after understanding what 2D appearances are.)
Maybe I've been doing actual useful mathematics for too long, but this pedantic hand-wringing over infinities seems inane.
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.
baxter: Consider 1 = 1/2 + 1/4 + 1/8 + 1/16 + ...
Consider 1 = 1/2 + 1/4 + 1/8 + 1/16 + ...
But what do you conceive of in your mind when you think of that series?
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).
baxter:>"we don't believe numbers or ratios exist. numbers and rations are processes. they are ACTS" If you can only deal with numbers as high as you can count on your toes, then I feel sorry for you.
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.
>so your claim
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?
Clayton: 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".
Did you read that in context?
He brought that up to object to something that certainly David Hume was asking the reader to conceive.
(Maybe not object, but at least question.)
>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"
baxter: Or like Clayton said, I can also view it as a formal expression - a bunch of symbols - which produces a true equation when set = 1.
Or like Clayton said, I can also view it as a formal expression - a bunch of symbols - which produces a true equation when set = 1.
But that would be useless for our discussion.
(We are talking about how we actually see things.)
baxter: 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.
Stop thinking about 3D objects!
2D appearances come way before them!
(After all, 3D objects are sets of 2D appearances.)
>Stop thinking about 3D objects!
OK, cool, let me what know concepts/images are acceptable to use in my thought processes.
I guess the concept of infinity, and visualization of 3D objects, are out. What else?
Clayton, here is something important to answer:
I. Ryan: Clayton: I've never perceived a point of an image in my mind. Do you see anything similar between this image and this one?
I think that answering that will get us a lot further in this discussion.
baxter: OK, cool, let me what know concepts/images are acceptable to use in my thought processes. I guess the concept of infinity, and visualization of 3D objects, are out. What else?
C'mon man, try to make more charitable interpretations.
I'm saying that you have to think in terms of 2D appearances to understand David Hume's argument.
(If you try to think of it in terms of 3D objects, you won't get it!)
sure it does. try it! back away from this set of dots ..................................................
until it looks something like ________________ except that it will appear dark grey instead of black.
EDIT: i see you meant not just that they would appear as a line, but actually "be" a line. i think we should eventually ask what "actually be" really means, but i will withdraw this line of argument for the moment in the interest of focusing the thread.
I. Ryan: 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?
a line is a long, thin rectangle. we're not sophists here, so since we are talking about a visual phenomenon we may as well show the object in question.
EXHIBIT A: a white line on a black background
notice that it is several pixels wide. that is intentional. it is to underscore that the line is not infinitely thin. in fact, "infinitely thin" would be nothing more than blather if it meant anything other than "thin enough that the width is negligible for the present purposes".
to see "negligible width" in a picture is easy for some purposes, hard for others. when it is hard, such as when the line needs to be super-thin, you can view it in a movie: imagine zooming in on the line and showing it not getting bigger for a long time, then finally at the end of the movie the line has a definite width, as it is being viewed in extreme close-up.
Clayton: By backing up, you accept that you are not altering the point itself, only your view of the point.
By backing up, you accept that you are not altering the point itself, only your view of the point.
Of course, and that's what I'm talking about.
We don't have anything but our various views of the point.
So what is "the point itself"? Well, just the set of our various views of the point!
(And I'm saying that each of our "views" is nothing but a 2D appearance made up of a finite number of indivisible points.)
This I agree with.
This I don't agree with - or even if I do agree with it, I don't think you've come anywhere near establishing it as a fact.
Clayton: Baxter:Maybe I've been doing actual useful mathematics for too long, but this pedantic hand-wringing over infinities seems inane. 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.
Baxter:Maybe I've been doing actual useful mathematics for too long, but this pedantic hand-wringing over infinities seems inane.
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.
glad we at least agree here. (this discussion might have little if any bearing on the practical applications of mathematics.)
Clayton: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).
this one is answered above: http://mises.org/Community/forums/p/21376/382702.aspx#382702
to re-gloss, infinities are not implicit in the objects or shapes themselves, but rather we can prove that the PROCESS of creating a decimal expansion of pi or the square root of 2 can be continued for as long as we like. and i am fine with using "ad infinitum" as a shorthand for the underlined, as long as we are explicit about it.
Clayton: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)?
indeed, large numbers are famously hard to comprehend. the situation is not as bad as it seems. in fact it is exactly as bad as you thought before you read this, because we are using numbers like 1000 and 1000000 every day without very much problem. we also rely on calculating methods and machines, and do so for series of small reasons each of which can be comprehended, or simply for the reason that they don't often let us down.
we can't even subitize beyond 4. that is, the concept of 5 apparently requires memory for some; 4 is a picture, but 5 may need to be a movie. at least, to an untrained child.
there are all sorts of ad hoc tricks we can use to understand what numbers like 1000 mean to us in terms of anticipated experience. they are not perfect, though, and their shortcomings are often revealed to us when our anticipated experience differs from what actually happens. [if we use "infinity" to mean our experience will have an incessant component in it, that would be fine.]
so the short answer is: 1, 2, 3, and 4 are simple pictures that can be conceived in the single frame. some people can conceive of more, or much more in a single frame, using various tricks, and most people seem to be able to conceive of much higher numbers with some semblance of accuracy.
some tricks: counting groups of 4 or 5, using points of reference such as that $20,000 is enough to buy a nice new car, zooming out, using vague notions like "an order of magnitude is a lot!", etc.
BUT we cannot imagine "infinity" other than imagining something like "way big enough to be big enough for the present purpose" or "don't expect this process to terminate".
Clayton:Another approach is to treat infinities as formalisms. Aleph_0 is just a symbol standing for "something" having this, that and the other properties.
this approach only works if it is clear that the actual "something" in question is not a nonsense notion. i can talk about "some physical object O" or "some number x" because we know what a number is and what a physical object is. in short, we cannot talk about the properties of something until we at least know what type of notion it is; infinity is not a number, for reasons already explained...so is it a strawberry? is it treason? is a triangle? a unicorn? i see a problem with listing properties of an object whose class is wholly undefined.
infinity is not a number
Your objections are at least as valid for all numbers other than natural numbers and rational numbers (I think they're valid for those, as well, since I think I'm a mathematical fictionalist). Does 0 exist? What is it? A strawberry? A unicorn? Does -1 exist? What is it? Is it treason? Does i exist (come on, you have to admit i is really freakin' weird).
I'll try to post a section from an essay I've written on this very subject.
Clayton:But what do you conceive of in your mind when you think of that series? 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).
they are just symbols that mean the same thing, yes, but insofar as they mean the same thing, we haven't LEARNED anything. the practical import is that going halfway there, then a quarter of the way, then an eight of the way, etc. will get you closer and closer to wherever "there" is. it is only when we create a movie in our heads that we actually learn why the equation is useful.
i can understand what this equation means for my anticipated experience by seeing a movie of a dot that is one foot from a line moving 1/2 foot closer, then 1/4 foot closer, then 1/8 foot closer, until i am satisfied that the longer i imagine the movie the closer the dot will be to the line.
to do more complex convergences, i can use logic to derive the rules of working with limits1, and then just calculate using those rules, with the knowledge2 that the rules faithfully represent movies that could be imagined. but truly, there is the rub. symbols and formal calculations only make sense as long as we believe they represent something that can, in principle, be imagined.3 either something we have imagined before, or something we trust that the experts have imagined for us and enough people have checked it and used it so that we can be reasonably confident it is useful.
1which require no concept of infinity as such
2if i have done my logical derivations correctly
3or if the formalism "just works" and we don't know why...but this is equivalent to magic, religion, superstition, mysticism then. that doesn't mean it's not useful: praying to the skiing spaghetti monster every night could help you achieve your goals faster than otherwise. being right for the wrong reasons is still being right, but it doesn't bode well for continuing to be right under different circumstances, because you are focused on the wrong underlying mechanism for your success so far.
baxter:>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.
so it will no longer be triangular, right? and what direction does a sphere near a star "bend"? can you point to that direction?
baxter:>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.
epsilon-delta is hard to understand? you seem to be using the word infinitessimals to mean "a formal representation of something we have previously understood by the logic of the epsilon-delta process", so i do not think we are disagreeing.
Clayton:I assume you are using the word "religion" (or superstition) to refer to speculation about things that do not actually exist.
things that do not actually exist, OR actions that are not actually possible. (or any other word uttered by someone that does not correspond to any consistent thought they have had [or often, corresponds to some other thought entirely that does not work for their theory])
Clayton:Well, I hate to break it to you, but finite numbers do not exist any more than infinite numbers do.
numbers are symbols that represent, at bottom, the act of counting. they aren't subject to questions of existence or non-existence, any more than any other act is. counting 7 watermelons is a possible action one may take, for instance.
the key difference is that finite numbers can be imagined by visualizing the movie of counting for small numbers, and by various other more-or-less reliable tricks for larger numbers; but "infinite numbers" cannot be imagined, other than as a movie that one doesn't expect to end. there are no tricks to get us to understand "infinity", other than the tricks to get us to understand very large numbers, where "very large" means large enough for their diminuitiveness to be negligible in the given context.
again this may not even matter for mathematics in practice, but there is a good reason for talking about it regardless: it certainly may matter for when the formalisms of mathematics are carried over to other fields based on the belief that the mathematicians are super-rigorous. if mathematics is not rigorous in certain aspects because it doesn't need to be1, that's fine for mathematics; the problem is when that lack of rigor is transferred elsewhere on the mistaken trust that such careful people as mathematicians have already proved it is valid for us.
SUMMARY: mathematics is only rigorous in the aspects where it needs to be; "infinity" is not one of those aspects (although numbers are treated more carefully by some people and in some textbooks). this may not be a problem for math, but it can be a problem for other fields that appeal to the authority of math, mistakenly believing it to be rigorous in ALL aspects, even those aspects that don't matter for math done in practice.
1baxter, you asked me to call you when i need to calculate something. i would happily do so. the problem here is not (necessarily) for the field of mathematics, but for those who point to mathematical formalisms that may only be useful in pure math or in certain delimited applications and try to invoke them in other contexts where the formalism would be entirely divorced from reality or utility. that would be an example of a lost purpose.
Clayton:Does 0 exist? What is it? A strawberry? A unicorn?
0 is a bookkeeping symbol, which doesn't represent anything all by itself. "0 apples" simply means "no apples". "0 velocity" means "motionless". "1 - 1 = 0" is a praxeological statement-form that can be applied to create a whole class of valid statements, one of them being "if you pick up one apple, then reverse that action (by putting it back down), you will end up with no net change in your apple holdings".1
Clayton:Does -1 exist? What is it? Is it treason?
the minus-sign is just a bookkeeping symbol, so -1 means nothing until it has content: putting an apple back down, for instance. processes don't exist or non-exist, they are instead possible to perform or impossible to perform. at bottom, -1 just means doing the opposite of whatever 1 represents, and of course it can only have that meaning when it is readily apparent what "opposite" means in the context.
if we take the symbol "1" to simply represent a movie of someone counting, then the symbol "-1" would be meaningless because such an act is impossible to reverse or do the opposite of, unless we define it in a special way, like "i owe him 5 apples and have no apples; in other words i have -5 apples" or "i count here 5 apple IOUs, that is, -5 apples".
finally, even the number 1 has no intrinsic meaning, necessarily, until we say "1 football" or "1 kick". counting is a natural match because we can understand the idea of counting some fuzzy, unknown objects, but really i should note that counting is always about counting something. (we also call reciting the natural numbers "counting" but that is a different use of the word.)
1is praxeology contained in mathematics or is mathematics contained in praxeology? it is a question of how one defines mathematics, but i would argue that at least most of mathematics is best demarcated as a sub-branch of praxeology; as i have been trying to show, it all is based on human action and human observation (or the action/observation of any intelligent being).
whoops, i missed this:
Clayton:Incoherence is the result of ignoring contradictions in one's system. What contradictions must be ignored to operate with infinities?
incoherence is also the result when someone utters a word that doesn't match any cognitive action or object in their mind. like "blicksthlezorp". or even "apple" if the person has no perception or memory of an apple in their mind.
Phew, I see some major differences in views of mathematics. Anyway, Zang, what about if I say that a "blicksthlezorp" is a object that has certain properties? Let's say that such an object has the same properties as "1", or in your words, the act of counting one. Can we write "blicksthlezorp" = 1? Can we say this "blicksthlezorp" is the same as 1, or the act of counting one?
Even more in "practical mathematics" than in theoretical we have to say those tho objects are the same since we have no possibility to distuingish them.
Thoughts on this?
1D creatures would only see points
2D creatures would only see lines
3D creatures would only see planes
At every level the creatures would infer a higher dimension. For example a line has no depth, and thus, in a sense, it you could ask how can it see a line of infinitely small depth? Yet, the 2D would creature would be seeing a cross section of a 3D world. They would be inferring depth. But, do we see in planes and infer volume with our memory of the collection of planes?
Read until you have something to write...Write until you have nothing to write...when you have nothing to write, read...read until you have something to write...Jeremiah
>And I'm saying that each of our "views" is nothing but a 2D appearance made up of a finite number of indivisible points.)
I cannot see this as being true except to the extent that our vision depends on a finite number of biological cells in our eyes. But tricks such as optical illusions demonstrate that one's subjective visual image is further "post-processed" by the brain/optic nerve. I know of no way for one's subjective visual image to be shared by another person or converted into pixels.
BTW, I guess I don't really understand Hume or why it's important to deal with 2D images. The mind has direct access to 3D geometry through the senses of touch and proprioception.
>and what direction does a sphere near a star "bend"? can you point to that direction?
The curvature is a tensor, which can be collapsed into a scalar (i.e. a curvature value): http://en.wikipedia.org/wiki/Ricci_curvature. Physics theory and experiment have no answers as to whether the universe is really 4D or simply a 4D manifold embedded in higher space. In any case, if there were a "curvature direction", it would point outside of the manifold of our universe, and couldn't be "pointed to" by an element of the universe.
>you seem to be using the word infinitessimals to mean "a formal representation of something we have previously understood by the logic of the epsilon-delta process"
Not at all; infinitesimals have nothing to do with the epsion/delta definition. Infinitesimals are elements that stand on their own both conceptually and formally in the branch known as nonstandard analysis. Furthermore the epsilon/delta definition of limits is NOT a process, but rather contains words like "for each" and "there exists". Not "as it gets smaller", etc. It is non-intuitive, awkward, useless in practice, hampers students' education, and was contrived by people like you who for some reason can't abide the simple and useful concept of infinitesimals.
0 is a bookkeeping symbol,
Yes, that is, in fact, how the use of zero originated but I don't think it helps your point in any way. I asked whether zero, in fact, exists and your response is "it's a book-keeping symbol." Fine, then so is infinity (or, to be more specific so are the well-defined notions of infinity which are part of transfinite mathematics). Note also that you conveniently ignored the imaginary unit, i. Is it a book-keeping symbol? If so, to what does i correspond?
Mathematical formalism is any discipline of symbol manipulation. What the symbols represent (your primary concern) is irrelevant to the study of the rules of manipulation. The symbols could represent nothing that is part of our experience or even anything that we can envision or imagine in our minds, even in principle. This is praxeologically possible for the same reason that any aesthetic endeavor is... for the sheer pleasure of watching the symbols be manipulated. I have a great example to illustrate this, it's called the See-and-Say sequence invented by John Conway. Basically, you start with any number. Then, you repeat the first digit in each run of digits in the number with the length of that run prepended to it. For example:
1, 11 (the first element is "one one"), 21 (there are "two ones" in 11), 1211 (there is "one two and one one" in 21), 111221, 312211, 13112221, 1113213211, ...
I'm pretty damn sure there is nothing in the real world which corresponds to the See-and-Say sequence (maybe run-length data compression but that's artificial, too).
>"0 is a bookkeeping symbol, which doesn't represent anything all by itself."
The view of it as a formal symbol on a piece of paper is only one viewpoint. Zero is also a concept grasped by the human mind. It's the number of items left when you start with one item and take one away (number-as-a-construction). It's the number of steps you have to take to stay in the same place (number-as a-motion metaphor). It's where you're standing right now (number-as-a-place metaphor)
>"1 - 1 = 0" is a praxeological statement-form
I don't see what it has to do with praxeology. It is simply a datum. Even birds understand 1 - 1 = 0. When one threat comes near its nest, and then one threat is seen to be leaving, the bird will relax.
>most of mathematics is best demarcated as a sub-branch of praxeology
Are you saying that Earth, Venus, and Mercury weren't 3 planets, and didn't travel in ellipses, in the aeons when Earth was devoid of people?
baxter: I cannot see this as being true except to the extent that our vision depends on a finite number of biological cells in our eyes.
I cannot see this as being true except to the extent that our vision depends on a finite number of biological cells in our eyes.
I want to give you a better answer, but for now let's work with this:
We can't even come up with the idea of "biological cells" or "eyes" without thinking about 3D objects, but we need sets of 2D appearances before we can get to 3D objects, so it's clear that we need to answer what 2D appearances are before we get to talking about 3D objects.
(How could anybody really disagree with the idea that we see a sequence of 2D appearances? How could anybody really argue that we see depth immediately, like we see the other 2 dimensions, and not through experience?)
baxter: But tricks such as optical illusions demonstrate that one's subjective visual image is further "post-processed" by the brain/optic nerve.
But tricks such as optical illusions demonstrate that one's subjective visual image is further "post-processed" by the brain/optic nerve.
Same for this.
Let's forget about this stuff and get back to what we actually see!
(Until what we see are those things!)
baxter: I know of no way for one's subjective visual image to be shared by another person or converted into pixels.
I know of no way for one's subjective visual image to be shared by another person or converted into pixels.
The pixel thing was just a metaphor.
It didn't really work, so we should probably forget about it.
baxter: The mind has direct access to 3D geometry through the senses of touch and proprioception.
The mind has direct access to 3D geometry through the senses of touch and proprioception.
That pushes the subject radically deeper than my constant one-line responses would suggest.
I will try to get to that later.
baxter: BTW, I guess I don't really understand Hume or why it's important to deal with 2D images.
BTW, I guess I don't really understand Hume or why it's important to deal with 2D images.
I will try to expand on this when I get some time.
Metus:Anyway, Zang, what about if I say that a "blicksthlezorp" is a object that has certain properties? Let's say that such an object has the same properties as "1", or in your words, the act of counting one. ... Can we say this "blicksthlezorp" is the same as 1, or the act of counting one?
mostly yes.1 once you define the word in a way the audience understands, you can use it.
1only thing i would point out is that "1" (interpreted as counting) is a process or act, not an object. [1 watermelon is an object.]
baxter:>and what direction does a sphere near a star "bend"? can you point to that direction? The curvature is a tensor, which can be collapsed into a scalar (i.e. a curvature value): http://en.wikipedia.org/wiki/Ricci_curvature. Physics theory and experiment have no answers as to whether the universe is really 4D or simply a 4D manifold embedded in higher space. In any case, if there were a "curvature direction", it would point outside of the manifold of our universe, and couldn't be "pointed to" by an element of the universe.
if we are not talking about bending in any direction, in what sense are we saying that a long, thin pole "bends"?
baxter:>you seem to be using the word infinitessimals to mean "a formal representation of something we have previously understood by the logic of the epsilon-delta process" Not at all; infinitesimals have nothing to do with the epsion/delta definition. Infinitesimals are elements that stand on their own both conceptually and formally in the branch known as nonstandard analysis.
Not at all; infinitesimals have nothing to do with the epsion/delta definition. Infinitesimals are elements that stand on their own both conceptually and formally in the branch known as nonstandard analysis.
if so, provide a definition. i found the following definition in Nonstandard Analysis, by Dr. J. Ponstein:
Ponstein in Nonstandard Analysis:An infinitesimal is a number that is smaller than every positive real number and is larger than every negative real number, or, equivalently, in absolute value it is smaller than 1/m for all m ∈ I = {1, 2, 3, . . .}.
this definition smuggles in the idea that it is even possible or meaningful for there to be a number smaller than every positive real number, and (AFAICT) he chugs right along without ever dealing with that issue of possibility. had he later gone on to prove the existence of such numbers, that would be one thing; but in failing to do so, he tacitly attempts to use his definition as an existence proof.
it would be as if i defined "plizzle" as "a real number that is neither positive, negative, nor zero"...and then just kept writing. or if i defined "god" as "god is an omnipotent and omniscient being" and acted as if i had proved God exists......now is this MATH or is it RELIGION?1
1i realize this is not your definition. my exuberance is directed squarely at Ponstein.
baxter:Furthermore the epsilon/delta definition of limits is NOT a process, but rather contains words like "for each" and "there exists". Not "as it gets smaller", etc.
it is essentially a set of instructions for how to perform the process, although it may not be phrased exactly that way.
baxter:It is non-intuitive, awkward, useless in practice, hampers students' education, and was contrived by people like you who for some reason can't abide the simple and useful concept of infinitesimals.
you're admitting that it's a pedagogical gimmick and a practical handwave. handwaving is often useful; my criticism is that it is not rigorous, and this is not necessarily a criticism that applies to mathematics proper2, but to those who believe all of accepted mathematics is rigorous, and would attempt to carry over certain handwavy concepts to other fields to "prove" things there.
2it is fine for math to be only as rigorous as it needs to be for the given application...but that last part is key