http://www.youtube.com/watch?v=cFods1KSWsQ

Have you thought what that would look like?

(July 29, 2012 at 4:04 am)jonb Wrote: Have you thought what that would look like?

Quite a bit, back in the day. There's a whole bunch of linear algebra lectures/notes (Paul's notes, Khan Academy, etc.) floating around the internet, if you're interested. But if you're talking about the image of those "0x +n" functions, the 'output' is just a single point (every point in the domain/input line ismapped to the same point).

(July 29, 2012 at 4:18 pm)Categories+Sheaves Wrote:

(July 29, 2012 at 4:04 am)jonb Wrote: Have you thought what that would look like?

Quite a bit, back in the day. There's a whole bunch of linear algebra lectures/notes (Paul's notes, Khan Academy, etc.) floating around the internet, if you're interested. But if you're talking about the image of those "0x +n" functions, the 'output' is just a single point (every point in the domain/input line ismapped to the same point).

Can you explain "0x +n" functions again? I'm not clear on what the domain and codomain are, let alone how the points are mapped.

(July 29, 2012 at 8:54 pm)CliveStaples Wrote: Can you explain "0x +n" functions again? I'm not clear on what the domain and codomain are, let alone how the points are mapped.

It's essentially this way of creating a mapping between lines A and B in the plane via some point c (preferably not lying on either line, but it counts as "x0" if c lies on B)--point a on line A is mapped to the intersection of the line ac with B

I'm looking at projective transformations*, (these can be represented as 2x2 matrices) and saying jonb's functions are precisely the ones whose bottom row is [0 1]** (for "xa +n", the top row is [a n]). Technically then, they're just affine transformations, but I'm trying to work through Beltrametti et. al's book on projective varieties and I like the projective lingo

*this acts on some projective coordinates set on the lines in question.

**it's [0 1] as long as the two lines are parallel. If you want to do this for two lines in the euclidean plane that aren't parallel, this will change to [0 m] for some m. You only see action in the bottom-left entry when the projectivity maps some coordinate to the 'point at infinity' (which jonb is not talking about)

(July 29, 2012 at 9:17 pm)Categories+Sheaves Wrote: I'm looking at projective transformations*, (these can be represented as 2x2 matrices) and saying jonb's functions are precisely the ones whose bottom row is [0 1]** (for "xa +n", the top row is [a n]). Technically then, they're just affine transformations, but I'm trying to work through Beltrametti et. al's book on projective varieties and I like the projective lingo

*this acts on some projective coordinates set on the lines in question.

**it's [0 1] as long as the two lines are parallel. If you want to do this for two lines in the euclidean plane that aren't parallel, this will change to [0 m] for some m. You only see action in the bottom-left entry when the projectivity maps some coordinate to the 'point at infinity' (which jonb is not talking about)

But if many points on the line are mapped to the same point, then the mapping isn't invertible...

(July 30, 2012 at 2:39 am)CliveStaples Wrote: But if many points on the line are mapped to the same point, then the mapping isn't invertible...

Darn tootin'

I still have a problem, Looking at the results I am getting. Is there any work about gaps in the series, how little or much of a gap can there be before the series separates, or do you only need the parameters to create the full series?
Is there a mathematical theory about how much of the Argos you need to have an Argos, or how small a hole makes it sink?

(August 1, 2012 at 11:26 am)jonb Wrote: I still have a problem, Looking at the results I am getting. Is there any work about gaps in the series, how little or much of a gap can there be before the series separates, or do you only need the parameters to create the full series?

There aren't any gaps in the real numbers, and the 'gaps' in the rational numbers (if they can be called that ) are infinitely small. The "size" of the gap between 0 and 1 (if we're talking about the natural numbers) doesn't matter, as long as we can tell the difference between one of them, two of them, etc.

(August 1, 2012 at 11:26 am)jonb Wrote: Is there a mathematical theory about how much of the Argos you need to have an Argos, or how small a hole makes it sink?

You would still have to define what it means for the Argo to be the Argo.
That being said, the image of those "x0" maps should not be considered equivalent to the preimage. You seem to realize that doing so would be problematic, but the way you're considering numbers (It's not clear what the necessary/sufficient properties of a "coherent series" are) aren't giving you any leverage on the problem. I'm not going to shove some mathematical definitions down your throat, but you have to nom on something if you want to get ahead on these sorts of questions.

Or, from another angle: if you're this curious about this stuff, you're doing yourself a disservice by not sitting through some online lectures or reading through a relevant textbook. (Also, this would force you to phrase your questions in a way that I can make better sense of )

(August 1, 2012 at 4:53 pm)Categories+Sheaves Wrote: Or, from another angle: if you're this curious about this stuff, you're doing yourself a disservice by not sitting through some online lectures or reading through a relevant textbook. (Also, this would force you to phrase your questions in a way that I can make better sense of )

This is a problem though, The language needed to understand maths, is not useful in finding out what I want to know until I have spent many years understanding it. When maths or any discipline will only converse in its own terms is it actually useful any more?
It seems to me the logic of maths is- does this work, if it works then it becomes part of the subject. I do not wish to know whether this or that formula works, but I am seeking if there is an understanding of what the relationship of numbers to the series are.
When looking at this I am often given allegories which are not said to be allegories, but are all the same, and very poor ones at that, so much so one wonders if the mathematics themselves have any knowledge of their subject other than a facility to fiddle with formulas, or the other answer, you must learn maths. Which is not a useful answer because it is not the maths that I want. This is the equivalent of saying I will not tell you a story until you have learnt to read. Sure the ability to read is a good thing, but I cannot do everything. I simply want to know is there a mathematical understanding of the nature of number because I want to make a comparison with classical arts understanding of a similar area.
I do not want to have to learn your language, So outside that language is there anything you can say which is useful or is it just 'academic latin', great for priests to talk to each other, about how many angles can sit on the head of a needle, and that ratifies their position as being priests, but that does not contribute any other field of knowledge?
So can a broken series be seen as a single entity? If so in what curcumstances would a series be a unit and not a unit?

(August 8, 2012 at 8:41 am)jonb Wrote: ...When maths or any discipline will only converse in its own terms is it actually useful any more?
It seems to me the logic of maths is- does this work, if it works then it becomes part of the subject. I do not wish to know whether this or that formula works, but I am seeking if there is an understanding of what the relationship of numbers to the series are.

The numbers themselves are usually defined by how they function. e.g. isn't not entirely clear prima facie what it would mean for one infinite set to be 'bigger' than another, (I'm echoing Wittgenstein here) so this business with the cardinalities of sets has to be grounded in the behavior of functions between sets (esp. with bijections). These structures and the ways in which the work co-determine each other, so it's a bit wonky to talk about the 'nature' of numbers sans the way they work.

(August 8, 2012 at 8:41 am)jonb Wrote: I simply want to know is there a mathematical understanding of the nature of number because I want to make a comparison with classical arts understanding of a similar area.

Well, arguably almost all math is investigating the nature of number (from one angle or another).
There is no singular concepts of 'number'. Remember how I dropped that laundry list of "number" systems on your earlier? You can't discuss 'the nature of number' without some provisional statement of what 'number' means. If you pick out a system of numbers (reals, gaussian integers, whatever) I can (in most cases!) talk intelligently about their structure and what motivates their study. If you give me a clear description of what properties you think numbers should have, I can (usually) talk intelligently about the structural implications of that. Right now I'm trying to figure out the thinking behind these pictures (I can extrapolate beyond them, but that's not where you are) and it's hard to get traction here .

(August 8, 2012 at 8:41 am)jonb Wrote: I do not want to have to learn your language, So outside that language is there anything you can say which is useful or is it just 'academic latin', great for priests to talk to each other, about how many angles can sit on the head of a needle, and that ratifies their position as being priests, but that does not contribute any other field of knowledge?

-I don't stand a very good chance of saying anything useful if I can't understand what you're trying to use this stuff for (I currently don't).
-String theorists need the axiom of choice to well-order the real numbers and make sense of their Feynman Integrals. Other folks use this voodoo too

(August 8, 2012 at 8:41 am)jonb Wrote: So can a broken series be seen as a single entity? If so in what curcumstances would a series be a unit and not a unit?

-What do you mean by 'series' again? You called these projections of numbers/lines you made 'coherent series' but that's all the leverage I have on this term.
-What do you mean by 'broken series'? Even in the worst possible interpretation of 'broken series' I'm sure the answer is "yes, you can see it as a single entity" (sets have to get way weirder before they're considered proper classes/not suitable as sets) but since I don't see this as problematic (and you're still hesitant about this) I'm not entirely sure you interpretation of "see as a single entity" is the same as mine. What sort of single entity?
-unit = single entity?