As December 21st, 2012 approaches, more and more people seem to want to know about this. So I shall attempt to explain.

The first thing I want to make clear is that the term "Watkins Objection" was something Terence McKenna came up with, not me (that would have been unforgivably arrogant). He responded to my critique of his "Timewave Theory" by posting a webpage which described it with that label. I think his choice of the term reflected his desire to frame his theory within the language of "serious" academia: "Watkins Objection" sounds like something which belongs in the same category as the "Riemann Hypothesis", "Poincaré Conjecture", "Wigner's Surmise" or "Tate's Thesis" (these, unlike my "Objection", all being serious fixtures in higher mathematics).

In 1996, when I was in my mid-20s, December 2012 seemed like some distant future, but here we are (I'm writing this in August 2010) with not much more than two years to go. Back then, I wrote up an account of our meeting and discussion, explaining my "Objection" in great detail. This was written while events were still fresh in my mind and was put up on the Web for the historical record, but I doubt that many people have actually read it. With its various elements of Mayan calendrics, fractals and the I Ching, not to mention its origins in a psilocybin experience, some people will instinctively dismiss McKenna's Timewave Theory as nonsense they don't feel the need to read a serious refutation of it. Other people just want to believe it it has become an item of faith so they're not going to read something which will only serve to undermine that faith. In any case, the arguments look quite technical to anyone who's not involved in the mathematical sciences…even though there's not really any serious mathematics involved (it just looks like that because of the convoluted way in which TM set out his theory).

So, here's a condensed account for people who can't be bothered with all the technical details.

1. I didn't set out with the intention of "debunking" the Timewave Theory. Having been much impressed by a lot of what TM had to say in the early 90s, I was interested to know more about how he'd arrived at this theory. So I contacted him and asked. But the more I found out, the less inclined I was to take it seriously.

2. We met in Palenque in the spring of 1996, where he was leading workshops at a entheogenic plants conference, and had three long discussions where he listened carefully to what I had to say. By the end of this, he conceded that his theory didn't stand up to serious scrutiny and expressed a willingness to admit this to the general public. I remember being impressed by this wish to spread truth, despite it involving the demolition of a major part of his life's work.

3. Once he got home to Hawaii, though, he posted a webpage describing our discussion and giving his version of my critique of the Timewave Theory. I felt this to be entirely unsatisfactory, if not outright misleading (I don't have access to a copy of this now he took it down fairly quickly). Once I'd expressed my disappointment about this to him, he was fully cooperative and let me write up my own detailed version of the events for his website. That piece (with a little introduction by TM) can be found here he entitled it "Autopsy for a Mathematical Hallucination".

4. It seems that TM wasn't able to let go of the theory, though. He found a physicist called John Sheliak who was willing to look at the original theory, and my "Objection", and then put together an incredibly dense and unnecessarily complicated document involving "vector analysis". TM called this (ironically?) the "Sheliak Clarification" and (laughably) presented it as some kind of major scientific breakthrough. I'm quite certain that even fewer people have read this than ever read my original "Objection".

5. Some people have stuck with the Timewave Theory. TM was able to re-brand my critique as an "improvement": by removing the most problematic element of the numerical manipulations involved in the construction of the Timewave (what his software developer Peter Meyer called the mysterious "half-twist" see below), he claimed that an even better Timewave resulted. A software upgrade (involving Sheliak's adjustments) was even brought out! New Improved Timewave 2.0! Now there are people (not many, I suspect) who spend time comparing and contrasting, "McKenna", "Watkins-corrected" and "Sheliak" versions of this thing (there's a fourth, "Kelley" version circulating, I've since discovered all very confusing, but helpfully cleared up in this article by Peter Meyer). There's not much I can do about that, but with 2012 approaching, it feels like it's time to propagate my own "clarification".

So what was the problem with the theory?

To answer that, I'll have to briefly recall what the theory actually was.

TM, having had a particularly intense psilocybin experience with Psilocybe cubensis mushrooms in the Columbian jungle in the mid-70s, claims that the "Logos" (a kind of higher intelligence) told him to look in the I Ching to find a "map of time".

The I Ching is an ancient Chinese oracle, very much concerned with time and change. He would certainly have been aware of it at that time (the West Coast LSD subculture from which he sprang was very much interested in the I Ching see Tom Wolfe's Electric Kool-Aid Acid Test). It involves 64 symbols called hexagrams. Each hexagram involves six lines, each of which is either broken (yin) or unbroken (yang). The 64 hexagrams traditionally appear in a rather random-looking order called 'the King Wen sequence' (there are various systematic ways in which you can order the hexagrams, but this isn't one of them).

So, when TM had come down from the psilocybin, assimiliated the (profoundly weird) experience and got himself home to California, he delved into the I Ching and tried to find this "map of time". What he actually did was to construct a very complicated-looking graph. It's important to emphasise that there are countless ways he could have done this. What he did looks, from a mathematician's perspective, really arbitrary.

He counted the number of lines which changed from one hexagram to the next in the King Wen sequence, generating a list of 64 numbers between 1 and 6. He then replicated this list six times, stretched it out horizontally and vertically by various factors, overlayed the various stretches, etc. You can find all the details here, if you're sufficiently interested. What you end up with is a jagged-looking graph, the sort of thing you might see in a physics textbook. But someone else with the same psychedelically-inspired motivation could have started with the sequence of hexagrams in the I Ching and come out up with an entirely different graph.

TM interpreted his graph as showing the rise and fall of what he called "novelty" (you can find numerous writings and recordings online wherein he explains this concept in great detail, so I won't bother). The way he constructed the graph meant that, at some point in history, it would drop to zero, supposedly indicating some kind of "infinitely novel event", beyond which nothing could be said or described. This is where the December 21, 2012 date comes in.

A lot of people, having casually looked into this, read or heard TM talking about it (or someone else's second-hand account) might be under the impression that he had arrived at that date via some kind of calculation. He certainly gave that impression. I remember being somewhat disappointed to find out that he hadn't calculated it he'd merely slid his graph back and forth along the timeline of history (as he understood it) until he got the best possible fit with the rise and fall of novelty (as he saw it).

One account [L.E. Joseph, Apocalypse 2012: A Scientific Investigation into Civilization's End (Morgan Road Books/Random House, 2007), p.204] indicates that TM originally decided that the "best fit" would occur if he placed his "zero point" in mid-November 2012…but then he learned about the significance of December 21, 2012 for the Mayans, and decided to shift the graph along by a few weeks.

When I spoke to TM myself, I got a variation of this account: one of the key periods of time he was working with was 67.28 years: this equals both 6 x 64 x 64 days (6 x 64 = 384 days is quite close to 13 lunar periods, he noted, so 67.28 solar years could arguably be understood as 64 "lunar years") and, approximately, six 11.2-year minor sunspot cycles. He took this to be deeply relevant, as it related the cycles of Sun, Moon and Earth via the numbers 6 and 64, which are central to the I Ching (TM's account of all this can be found here). For some reason, he decided (reasoning backwards?) that the first use of an atomic bomb on a civilian population (Hiroshima, August 6, 1945) was especially "novel", then added on his 67.28 years to arrive at November 16, 2012 (which would have been his 67th birthday, as well as being very close, in historical terms, to the winter solstice of 2012).

On some occasions, TM claimed (contrary to the above) that he had, working with his Timewave, arrived at the December 21st, 2012 date prior to learning about the Mayan calendar (see 2:20 into this video). Unfortunately, he was never entirely transparent about the detailed origins of his theory. For example, he gave the impression that the original formulation of the Timewave involved "fractal mathematics", something that grabbed some people's attention: there was a lot of excitement about fractals in the 1990s (particularly in the drug culture), but in the mid-70s they were barely known about, so TM seemed really ahead of his time in this respect. The early-90s republication of his 1975 book The Invisible Landscape certainly reinforced this impression, yet when I was finally able to get access to the original (very obscure) 1975 version, I found there to be no fractal mathematics present he had simply "fractalised" it at some point later, presumably to make it seem more relevant to the latest (sub)cultural trends.

Peter Meyer has clarified that it was he who introduced the fractality to the Timewave when coding the Apple //eversion of the TWZ software in 1986 but he insists that "the fractal nature of the timewave was implied (or was implicit) in the original formulation even though its author was not aware of this, and he only became aware of it after Mandelbrot had popularized the concept." This "implicit fractality" is debatable, in my opinion. The original formulation involved an infinite sum of a basic 'shape' at ever-larger scales. PM later introduced another infinite sum going in the other direction, that is, to ever-smaller scales (he has told me that this is based on something which appears in the original The Invisible Landscape, but neither have a copy of this to hand this should be cleared up). I still feel TM was being somewhat disingenuous (or at best bandwagon-jumping) when hyping the 'fractal' aspect of his Timewave, although this feeling has nothing to do with mathematics, and everything to do with interpretation and presentation.

Of course, if you were determined to believe that there's really something in all this, you could convince yourself that he was being guided by some kind of higher intelligence as he put his theory together, revised it, etc….but then you might as well become a religious fundamentalist. TM himself, to his credit, would have expected you to evalutate it on its merits via the framework of experimental science how effectively does it describe the rise and fall of "novelty" throughout history? And how effectively does it predict it?

Here we run into problems. "Novelty" is a term which he used a lot, and which works quite well intuitively, but when you try to put together a precise definition or, worse, try to quantify it, things start to get rather vague. Beyond this, if you want to overlay his novelty graph onto our experience of time and history, you must come up with some kind of scale. TM's basic unit based on his observations about lunar cycles, solar years and sunspots being linked by the numbers 6 and 64 is one Earth day. And that's one Earth day as it is now bear in mind that Earth days haven't always been the same throughout the history of the Earth. And yet, this graph is supposed to show the fluctuations of novelty not just in human history or Earth history, but the history of the entire Universe. He liked to claim that he could correlate his Timewave graph with key events in the formation of new structures of matter, stars, planets, etc. If so, why would the time scale generated by the rotation of (to quote Douglas Adams) a "an utterly insignificant little blue-green planet far out in the uncharted backwaters of the unfashionable end of the western spiral arm of [an equally unremarkable] galaxy" (which wasn't going to exist for billions of years) be particularly relevant?

But the final blow to the theory (as far as I was concerned) was my discovery of a footnote buried in an appendix of the user manual which accompanied TM's Timewave software. In fact, this footnote even appeared on the wrong page, so I wouldn't be surprised if no one had even noticed it previously Timewave enthusiasts seeking to explore the intricacies of historical novelty with this software were unlikely to comb through the appendix of the accompanying documentation, after all. In this footnote, a puzzling reversal of half of the 384 "data points" which are involved in the construction of the first stage of the Timewave graph is described (by Peter Meyer, who wrote the software and the accompanying manual) as follows:

This is the mysterious "half twist". The reason for this is not well understood at present and is a question which awaits further research.

I challenged TM on this he had, for no obvious reason, reversed half of the "data points" in his construction, and now it was being claimed that "this is not well understood" and "awaits further research". I asked him why he did it, in this case. He simply claimed not to remember (it was a long time ago, and a lot of psychoactive substances were involved, after all). It is not known who formulated the algorithm which generated the 384 numbers given in the original (1975) edition of The Invisible Landscape, since it may have been one of Terence's FORTRAN programmer friends. Peter Meyer informed me that he discovered the half-twist only when he reconstructed the algorithm in 1994, and said: "We have no information regarding what explanation Terence gave to his programmers, and in fact cannot rule out the possibility that they made some original contribution. So who it was who reversed half of the data points is not known.") The most likely explanation I could see at the time was that he had been playing with the "data points", trying to produce a wave that would produce a better fit with historical novelty. He initially accepted my point that if he couldn't justify this step in the construction, then he couldn't expect anyone to take his "theory" seriously.

There were a couple of remarks he made in regard to all this which are worth recalling here:

(1) He later claimed (when I challenged him on why he'd done a U-turn and attempted to shore up the theory with obfuscating "clarifications", after initially admitting that it didn't stand up) that even if it had no basis in truth, it was still worth propagating. His reasoning was thus: if enough people expect some kind of major global transformation in 2012, this will shape their actions in such a way that makes such a transformation more likely! At the time, I found this incredibly irresponsible, but he may have had a point. It's impossible to quantify, but I reckon that TM, with his Timewave Theory, has had a greater role in the current proliferation of 2012-related hype than any other individual. And now we see things emerging like the documentary 2012: Time for Change, wherein viewers are urged (by Sting, among others!) to work towards this much-need "change", regardless of the significance of the Mayan calendar or anything else.

(2) Bizarrely (but not surprisingly, given his "alogical" approach to reality), he pointed out that there was a very large spike in "novely" in fact, the largest to date at that time in the spring of 1996, around the time we met in Palenque, and suggested that this could be interpreted as the Timewave predicting its own downfall. It's hard to know how to respond to this!

Peter Meyer has recently issued a CD-ROM containing (a) the latest (and last) version (7.10) and (b) a German version of a slightly earlier version.of the software, accompanying documentation and many articles on the subject. An introduction is available here and an overview is available here.

* * *

I will always be grateful to Terence McKenna, though, because it was he who first got me thinking about prime numbers.

In my initial email to him, sent while I was working on my PhD thesis, I had offered my mathematical services, should he require them to further clarify or elaborate his theories. In his response, he told me that he had "an idea involving prime numbers" which he wanted me to look at but he didn't tell me what it was. I didn't find out for months (and when I did, it was easy to dismiss it he was hoping that his Timewave might be able to predict new, large prime numbers, thereby bringing it fully to the world's attention…anyone who's studied a bit of higher mathematics and who looks at the way that he generated his graph from the 384 "data points" will immediately be able to see that this is not going to happen!).

But, in the intervening months (this was '94 or '95), I had excitedly tried to imagine what Terence McKenna would be thinking about prime numbers. I tried to "get inside his worldview" and then think about prime numbers. And this led to some very strange thoughts! These strange thoughts were the beginning of my interest in number theory (a branch of mathematics I had previously been largely unfamiliar with), culminating in the recent (June 2010) publication of The Mystery of the Prime Numbers (Volume 1 of the Secrets of Creation trilogy).

*Image by Beverly & Pack on Flickr courtesy of Creative Commons Licensing. *

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