fdrake

Yeah, some of it really isn't your fault. If you go back through the forum over the years, we get about a few Randians per year. They usually come, ha...

Maybe because you've not let yourself get to know them better. Acting from such a position of condescension confines others to fit your already establ...

I'm always surprised by how many people read Ayn Rand and imagine themselves as similar to her hero, rather than the societal dregs that depend upon h...

I'm finding the same thing, what I'm benefitting most from I think is trying to integrate the imaginative background from the first section with the m...

It's really hard going! Spending hours on paragraphs means you know it's hard.

I wanna highlight something in this post because it's cool, and I just grokked a connection. Jon suggested writing the general quadratic equation (wit...

But yes, I think this gives the correct expression, and you can force the matrix to be either a lower or upper triangle at your whim. Engineer's proof...

Edit: For some reason I thought we were discussing the matrix A above rather than the curvature, apologies for confusions. I've changed this comment t...

Yeah. I screwed up the formulas a few times and have been editing them since. That post's very much a work in progress. It should read something like:...

Finishing §1 in section 2, Riemann starts to introduce the idea of changing coordinate systems - more specifically, using different coordinate systems...

A bad OP which has already generated a decent conversation is more likely to remain. If we catch a bad OP before any discussion starts on it, we're mo...

I don't think there's much extra philosophy in that bit from what we've already covered, it's just detail on how to construct localised distance measu...

Starting §1 in section 2. It's very likely that I have some misconceptions and falsehoods in my presentation since it's outside of my comfort zone. So...

I'm thinking of making a post that describes the flow of the argument without going into the mathematical detail. Similar to @"John Doe"'s post earlie...

Will take some time to digest the argument in §2.

Considering you've just made one post, you could probably ask to be deleted then make a new account. Summoning admins to see if this is possible @"Bad...

I'm going to move onto section 2 tonight, but just the summary of section 1 and the preparatory remarks for the remainder of section 2. Hopefully this...

But yes, the 'ply' corrections below the shapes in your drawing look right. :D

Made a thing with more examples. /uploads/resized/files/33/e5gv0padlmhqvt1p.png Empty areas in the 1D case because we're just considering the lines. B...

That sounds about right Moliere. You extended it correctly (by my reckoning) to the surface of the sphere, so I think we see eye to eye now. Typically...

Switching into the language of degrees of freedom can help. A degree of freedom is a unique direction of variation. Straight lines have 1 degree of fr...

@"StreetlightX"@"John Doe"@"Moliere"@"Wallows" Does anyone have anything they want to discuss before we move onto the meatier/more technical bits?

Another thing which is worthwhile to highlight is that coordinate systems are a way of ascribing positions based on quantities - so we have length mag...

Here's another demonstration of 'innate coordinates', using the simply extended manifoldnesses (1D manifolds) of the original blue curve and the red a...

Riemann has given us a method for constructing higher dimensional manifolds from lower dimensional manifolds. In §3 he does the reverse, giving us a m...

If you want to consider the volume of the vase, then the whole circle is a 2 dimensional object we're 'passing over' with the curve which is a cross s...

Quick corrective note: Riemann equates simply extended magnitudes with 1 dimensional objects, but the points which constitute them are 0 dimensional. ...

So I think §2 is reasonably straight forward, it's an exercise of the imagination which sets out what Riemann means when he thinks of a multiply exten...

@"John Doe",@"Moliere",@"StreetlightX" - already a lot to discuss. Hurrah. Let's hope it continues going this well.

Finishing my exegesis of §1 definite portions could be elements, like A in {A,B} portions of discrete manifoldness, distinguished by a mark (A). Or th...

I don't put much stock in the Ship of Theseus. At most it shows that concepts can have fuzzy boundaries. All I wanted to do in the thread was to highl...

No idea what you're talking about. See here. Usually however we are not talking about empty domains or inexistent objects.

So long as you're not dealing with an empty domain, and you know this a priori. The universal quantifier is equivalent to 'not for some x not (rest of...

So the specialisations bit, this is now discussing §1 in the paper: We can only have magnitudes expressed mathematically if there exists a general con...

I think this is right. A manifold is essentially an object with a coordinate system associated with it. So like a sphere, or the boundary of a circle,...

Do you see any distinction between the issue in the OP and the Ship of Theseus?

I forgot to say, I'll move onto the next section once you've had chance to comment. I might be attuned to the math but I don't think I am to the broad...

I'm happy to put up maths notes explaining what I can. I'm also having to learn more differential geometry to write the exegesis, since I've never stu...

I imagine that even without understanding the math, as I won't at points without lots of work (I have a first principles/pedagogical differential geom...

Third post in a row, sorry fellow mods, I got too excited and hit the 'Post Comment' button too quickly. I think this is about right, wondering how Eu...

The modern notions of coordinate systems and vectors weren't well established at the time of Riemann's writing. Moreover, as will become more clear (I...

Another note I have is that notions like coordinate system, manifold, set and so on are not precisely established in mathematical discourse at the min...

Thank you!

If you're that passionate about the subject, try again, tackle a specific issue the book deals with in some amount of depth. Don't just quote from it ...

Though I wasn't the mod that removed it, I'd allow a link to a blog post or a personal book if the OP was very well written, sufficiently in depth, an...

Nothing in the spam filter. Was it restored?

So that's ludicrously dense argumentation. I'd be interested to see how other people took it, as I likely missed things by focussing on trying to disp...

Still in "Plan of the Investigation", now the second paragraph. Hoo boy. So Riemann's saying that the mathematical accounts in history, while possibly...

Guess I'll start today rather than tomorrow since tomorrow is busy. Here is a copy of the paper that's in plain text (so it allows quotation through c...

One of the reasons why it's so dense is because he's inventing lots of, what are now distinct, mathematical concepts at once. But he's not using the u...