Sadly we here observe a retreat within the simple language of mathematics. I am not decrying mathematics nor am I underestimating the great value of that language in extending knowledge of the physical world by bypassing the complexities and irrelevancies common to the natural languages.
It does, however, suffer from two major weaknesses:
Firstly, like all languages, it is capable of generating fictions—entities and scenarios which have no correspondence wit the real world.
Secondly, it is, like all reasoning or computational processes, raw data sensitive. This is expressed in the venerable IT mantra “Garbage In—Garbage Out”
The second of these is, I believe, the main culprit in generating the conclusions that you affirm. For you seem to have fallen into the same trap as Jonathan Huebner who, using a rather arbitrary criterion of “significant advances” concluded that the rate of innovation has actually been decreasing since a maximum that occurred in 1873. A glance around the shelves of a Chemical Abstracts archive, for instance, will quickly tell a denizen of the real world that something is seriously wrong with that analysis.
Now it is very evident that not all aspects of human activities, even some technologies, are subject to exponential growth, as underlined in your very interesting presentation “The Myth of Accelerating Change” Similarly the evolution of life has not exhibited a consistent acceleration in all its bifurcations and ramifications. The “significant steps” used by Huebner (even after allowing for the strong “self selection” effect involved in that analysis) are merely break points, sharp upheavals resulting from the accumulation of innumerable “baby steps” wherein lies the exponentiality.
The input data for your own analysis lies in the conceptual arbitrariness of “important knowledge” and “useful knowledge”. These are the foundation of your argument. If their validity is in question, as I propose it is, then all the arithmetic in the world will not hold it together. Similarly extrapolations of arbitrary data to towers of exponentiality must be considered as pure flights of fancy. GIGO rules!
The great problem with ascribing importance and utility (citation hits certainly won’t do) is in determining the entity to which these are relative. Certainly these value judgments will vary enormously between individual humans and consensus is probably out of the question. We can tie it down a little by noticing that a fairly constant gross exponentiality appears to be tied to technology rather than other human activities. A distinction not easy to make as many aspect of social and individual behavior are themselves driven by technological change. Such features as sex, art and religion are among those with some immunity.
One way to escape from this dilemma, however, if we are to properly interpret the patterns science observes in nature, is to learn the trick of stepping outside our (very natural) anthropocentric shell so that objectivity is not compromised. My book “Unusual Perspectives” (the electronic edition of which can be freely downloaded) uses this approach to arrive at the proposition that the evolution of living systems and the evolution of technology (with which it is contiguous) are components of an ongoing natural process. From this point of reference we can perhaps better determine what aspects are “important” or “useful” (to the life process). This stance certainly strongly suggests the advent of what some call a singularity but which I prefer to regard as something akin to a fairly imminent phase transition. I further speculate that that, considering the history of the process and the apparent direction of its vector, that the internet could as the result of an inevitable self-assembly mechanism be the most likely candidate for the next prime effector of the process. “Unusual Perspectives” can be downloaded from the dedicated website: www.unusual-perspectives.net
Unfortunately this entire discussion is deeply flawed.
Why? GIGO - Garbage in—Garbage out.
However good the logical systems used for processing information they are of no avail without meaningful input data.
Present technologies cannot be used as a basis for prediction because of the unexpected bifurcations and inherent non-linearities in technological developments.
Further problems stem from the use of the very inappropriate buzz-word “Singularity”. Certainly a dramatic change is imminent, but this is better considered as a phase transition—the emergence of a new and dominant non-biological phase of the on-going evolutionary process that can be traced back at least as far as stellar nucleosynthesis.
Indeed, the inevitable self-assembly of this new entity can be clearly observed in what we at present call the Internet.
The broad evolutionary model which supports this proposition is outlined (very informally) in my latest book: “The Goldilocks Effect: What Has Serendipity Ever Done For Us?” It is a free download in e-book formats from the “Unusual Perspectives” website