Hmm. It had been a long time since I read that. I forgot that EY specifically compares the Vinge singularity to an event horizon. It’s an apt comparison, although the event horizon is not a singularity (well, it’s a co-ordinate singularity in some co-ordinate systems, but that doesn’t count*). I took a quick look on the internet to see if I could find Vinge actually making that connection, but I didn’t see it. The 1993 document, for example, doesn’t refer to space-time or black holes at all (but it did remind me that, contra Wikipedia [and me above (ETA: actually I didn’t make the erroneous claim, but I think I believed it)], Von Neumann deserves at least part of the credit for being first to say ‘singularity’, since Ulam was paraphrasing him).
There is possibly a tension here between those who instinctively perceive the word ‘singularity’ as a mathematical term and those who perceive it as a (speculative) historical one. Many of us, presumably, can go either way depending on the context, treating the two things basically as [the referents of] homonyms. But with my bromide immortalized in that image, I was attempting to point out what they all have in common: they are by their very nature holes in a structure, not elements of a structure. When the structure in question is a map, they are points where the map fails. When the structure in question is the object of investigation, as it is in mathematics, they are merely discontinuities; but they are not things, they do not exist. For me, the very word ‘singularity’ connotes the tenuousness of our maps.
Of course, it is also important to make a literal/figurative distinction here. Singularities in math and physical theories are literal singularities; historical ones are figurative. And some are more figurative than others; as I’m sure has been pointed out, Kurzweil’s singularity is pretty hard to see in any kind of superexponential growth curve, but perhaps running into a vertical asymptote can be charitably interpreted as hyperbolic.
* That’s actually a good example of what I mean, though. Co-ordinate singularities are in some maps but not others (that we know how to make); ‘real’ singularities (the word is here quoted for bogosity) are in every map we know how to make.
Yes, I was actually thinking of the intelligent explosion type singularity as being the one that resided the most in the map. And the point about the difference between a mathematical singularity and a non-mathematical singularity is a very good one (and the analogy about an event horizon is also interesting. Although even then, there’s a strong territory aspect there because once one is inside the event horizon one cannot send a signal out by any means.)
And some are more figurative than others; as I’m sure has been pointed out, Kurzweil’s singularity is pretty hard to see in any kind of superexponential growth curve, but perhaps a vertical asymptote can be charitably interpreted as hyperbolic.
Actually, if one starts with a differential equation that has a large rate of growth with respect to the function itself it isn’t very hard to force a singularity. But this is a nitpick, such functions don’t exist in real life generally, and on the rare occasions when a model has one it is generally an indication that there’s a problem with the model, not that there’s anything like that in reality.
There’s nothing special going on locally at the event horizon. And on the other hand, seed AI FOOM is just about the most singular kind of historical Singularity I can think of. The entire Hubble volume could undergo some kind of phase transition at the speed of light. (Maybe even faster.)
Not actually literally singular, mind you. Because it’s ‘real’, not ‘abstract’. But not even literally a ‘co-ordinate singularity’, figuratively speaking. (Unless Penrose is right about noncomputable physical action being involved in mental activity. [That is a joke. Perhaps I should label them. The ‘hyperbolic’ thing was a rather clever pun, by the way.])
So I’m not sure what you’re getting at in your first sentence, which is possibly due to lack of sleep. Also:
if one starts with a differential equation that has a large rate of growth with respect to the function itself it isn’t very hard to force a singularity.
I somehow have no idea what this means. But I would like to. Maybe it’ll be clearer in the morning.
Now that they’re all visible, I guess it’s okay to post this: http://img25.imageshack.us/img25/6645/singularitiesareinthema.png
[edited to replace link since I noticed the left edge of the picture was cut off]
That claim depends on what type of Singularity you are talking about.
Hmm. It had been a long time since I read that. I forgot that EY specifically compares the Vinge singularity to an event horizon. It’s an apt comparison, although the event horizon is not a singularity (well, it’s a co-ordinate singularity in some co-ordinate systems, but that doesn’t count*). I took a quick look on the internet to see if I could find Vinge actually making that connection, but I didn’t see it. The 1993 document, for example, doesn’t refer to space-time or black holes at all (but it did remind me that, contra Wikipedia [and me above (ETA: actually I didn’t make the erroneous claim, but I think I believed it)], Von Neumann deserves at least part of the credit for being first to say ‘singularity’, since Ulam was paraphrasing him).
There is possibly a tension here between those who instinctively perceive the word ‘singularity’ as a mathematical term and those who perceive it as a (speculative) historical one. Many of us, presumably, can go either way depending on the context, treating the two things basically as [the referents of] homonyms. But with my bromide immortalized in that image, I was attempting to point out what they all have in common: they are by their very nature holes in a structure, not elements of a structure. When the structure in question is a map, they are points where the map fails. When the structure in question is the object of investigation, as it is in mathematics, they are merely discontinuities; but they are not things, they do not exist. For me, the very word ‘singularity’ connotes the tenuousness of our maps.
Of course, it is also important to make a literal/figurative distinction here. Singularities in math and physical theories are literal singularities; historical ones are figurative. And some are more figurative than others; as I’m sure has been pointed out, Kurzweil’s singularity is pretty hard to see in any kind of superexponential growth curve, but perhaps running into a vertical asymptote can be charitably interpreted as hyperbolic.
* That’s actually a good example of what I mean, though. Co-ordinate singularities are in some maps but not others (that we know how to make); ‘real’ singularities (the word is here quoted for bogosity) are in every map we know how to make.
Yes, I was actually thinking of the intelligent explosion type singularity as being the one that resided the most in the map. And the point about the difference between a mathematical singularity and a non-mathematical singularity is a very good one (and the analogy about an event horizon is also interesting. Although even then, there’s a strong territory aspect there because once one is inside the event horizon one cannot send a signal out by any means.)
Actually, if one starts with a differential equation that has a large rate of growth with respect to the function itself it isn’t very hard to force a singularity. But this is a nitpick, such functions don’t exist in real life generally, and on the rare occasions when a model has one it is generally an indication that there’s a problem with the model, not that there’s anything like that in reality.
There’s nothing special going on locally at the event horizon. And on the other hand, seed AI FOOM is just about the most singular kind of historical Singularity I can think of. The entire Hubble volume could undergo some kind of phase transition at the speed of light. (Maybe even faster.)
Not actually literally singular, mind you. Because it’s ‘real’, not ‘abstract’. But not even literally a ‘co-ordinate singularity’, figuratively speaking. (Unless Penrose is right about noncomputable physical action being involved in mental activity. [That is a joke. Perhaps I should label them. The ‘hyperbolic’ thing was a rather clever pun, by the way.])
So I’m not sure what you’re getting at in your first sentence, which is possibly due to lack of sleep. Also:
I somehow have no idea what this means. But I would like to. Maybe it’ll be clearer in the morning.