Kevin Drum has a graph with one of the weirdest y-axes I've ever seen. "Time till next event" -- does that mean the time until the next dot that gets plotted on the graph? Then the arbitrary decision to assign dots for some things and not for others makes a big difference. If it so happened that humans controlled fire immediately before learning agriculture, and you decided to put in a dot for fire, you'd get an big downspike in the graph. Really, there are tons of other dots you could think up -- "domestication of animals", "man on the moon", "discovery of alcohol"...

From the look of the graph, though, "next event" seems to be whatever happens (roughly) at the present. That would explain why 10^9 years before the present, you get an event that's 10^9 years until the "next event". So it's no wonder that the graph is pretty much a straight line -- the number of years before the present varies directly with, um, the number of years before the present.

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The x-axis, or independent variable, is time, in powers of 10. The dependent (y) variable is time from advancement_sub_(x-1) to advancement_sub_x. In other words, at time 10^10 years ago, it would take 10^9 years to see the next significant human invention. At time 10^9 years ago, it would take 10^8 years. That's one f'ed up graph to look at, but if you simply log the X values and log the Y values, it's a line. Hence, it's a log-log linear relationship.

Now, it's NOT linear with respect to time, and that's the mistake I think most people are making when they read this. They think that Kurzweil just arbitrarily stuck his pin pricks at the moments that best supported his theory. That's far from true, and the discontinuities in the graph are proof enough of that.

It's good to remember that 10^10 minus 10^9 equals 9 * 10^9. That's why you can have these events spaced out in a manner that suggests question-begging; the first actual ordered pair is something like (5*10^9, 2*10^9). That is, life began 5 billion years ago, and the next groovy invention of multicellular life came down 2 billion years later. Leaves an awful lot of time for the next one, and doesn't even knock off an order of magnitude in the process.

Pretty cool graph. Too bad bloggers suck at math.

In fact, the relationship is linear with respect to time; if you take a linear function (with y-intercept zero for simplicity; small complication happens if you don't, but it's just a linear change of variable, so nothing geometrically interesting is involved) y=mx and log both sides, you end up with log y = log m + log x, and thus a linear relationship between log x and log y.

What did happen is that the line you were graphing changed from having slope 10 to having slope 1, but that doesn't somehow make it not a line. This graph is in fact weird, if for no other reason than the definition of "event" seems loose enough to not distinguish between the relative importance of the invention of radio and the beginning of life on Earth, suggesting that maybe we should include many many other points too (as Neil says, fire, alcohol, etc.) The fact that the graph isn't precisely a line (the graph has no discontinuities, it's just not everywhere-differentiable) says nothing interesting about the quality of the arbitrary choices of points.

linear-graphs are common when charting population growth and other things that have exponential growth/decay characteristics.

Dennis, I bow to your actual advanced math skills. I jammed one too many "logs" into my definition: it's log-linear. (of course you have to log both sides of an equation, else it's not an equation anymore!) And I said "discontinuity" because I thought that a point at which the slope of a line changes was by definition a discontinuity. But the curve still only has one value at that point. Thanks for refining my initial pass at this.

Neil, you satisfied yet?

PS -- if you ever want to see the relationship between a log graph and a linear graph, go to finance.yahoo.com and grab a chart for your favorite stock or mutual fund. For some reason the default chart display is logarithmic.

Yeah. Well, I guess I should say that what weirded me out about the y-axis wasn't that it was logarithmic -- hell, I've invested for long enough that most of the graphs I've looked at in my life are probably logarithmic. It was the "Time until next event." I was wondering what of significance could possibly depend on the time until the next plotted event, especially given the arbitrariness of the point selection.

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