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Elliot BenjaminElliot Benjamin is a philosopher, mathematician, musician, counselor, writer, with Ph.Ds in mathematics and psychology and the author of over 150 published articles in the fields of humanistic and transpersonal psychology, pure mathematics, mathematics education, spirituality & the awareness of cult dangers, art & mental disturbance, and progressive politics. He has also written a number of self-published books, such as: The Creative Artist, Mental Disturbance, and Mental Health. See also: www.benjamin-philosopher.com.

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Synchronicity
and Mathematics

A Response to the Lanes

Elliot Benjamin

The probability of my experiences being reasonably explained completely by chance factors is infinitesimally small.

I must say that the Lanes, in their recent Integral World article Desultory Decussation [1], were most gracious in their refutation of the openness to spirituality interpretation that I suggested in my Integral World License Plate Synchronicity article [2]. They apparently enjoyed reading my article, and even referred to it as an “intriguing essay” (though for future reference I will convey to them that there are two “l's” in my first name). However, the crux of the Lanes' article is in the same context that they describe for all their Integral World articles, and this is to set forth rational, scientific, mathematical explanations to explain phenomena that many people consider to be of a psychic or spiritual nature.

The Lanes' extreme scientific perspective reminds me of the heated debate and controversy over how to interpret psi phenomena that has been raging for over a hundred years [3]. The bottom line is that when the usual suspects have been controlled for in psi laboratory experiments, to a tremendously high degree that would more than satisfy the requirements for any standard psychology experiment, one can always point to two remaining factors that can never be mathematically proven does not explain the results: fraud and chance. But in my license plate synchronicity portrayals, the Lanes have assumed that fraud is not a factor; i.e. they assume that I have honestly portrayed my experiences, which is certainly the case.

This leaves “chance” as the remaining scientific factor to explain the synchronicity experiences that I have described, and the Lanes have milked this factor for all it is worth. They invoked “Littlewood's Law of Miracles”; they explain that according to the great mathematician John Littlewood, a miracle can be considered to be an event that has a probability of occurring of roughly 1 in a million, assuming that we are highly conscious and observing of what is around us. They then cite the Littlewood punch line that mathematically we can expect to experience approximately “one miracle a month.”

Putting on my Mathematical Hat

I can see that it is time for me to put on my mathematical hat. I have studied some of Littlewood's most well known mathematical results when I took Complex Analysis as part of my mathematics Ph.D. program. I am a mathematician and my dominant research area is actually number theory (Algebraic Number Theory to be specific). Integral World readers are familiar with my love of numbers from my Integral World Integral Mathematics article [4]. Let me therefore try my hand at examining how relevant Littlewood's Law of Miracles and the Lanes' rational/scientific/mathematical argument is in the case of my portrayal of my license plate synchronicities.

However, there are two points I need to emphasize before doing this. The first is that my mathematical knowledge of probability and statistics is by no means on the level of being an expert in the field. Consequently I will be offering merely what I believe is a modest reasonable interpretation based upon my understanding of probability, and my assumptions about how the Motor Vehicles Bureau assigns license plate numbers. The following mathematical probabilities that I offer are merely suggestive, and I welcome responses of a mathematical nature that are more informed than my own mathematical descriptions.

The second and crucial point is that the license plate synchronicity examples I portrayed in my previous license plate synchronicity article are merely the tip of the iceberg. My license plate synchronicity article was significantly reduced from my original article that I decided to keep private, as many of the synchronicities I originally experienced were of a highly personal nature that could have had very negative personal consequences for me if it went public. My original article was 48 pages with 19 license plate synchronicities over a two month period of time. I actually submitted my article to The Ground of Faith journal, and the editor was kind enough to candidly suggest to me that it would be detrimental for me to make my article public, for the reasons I mentioned above. The upshot is that my shortened license plate synchronicity article now appears on two websites in addition to Integral World: The Ground of Faith journal and the paranthropology website, and my full license plate synchronicity article will permanently remain private. However, I do need to say at least something more about the extent of my license plate synchronicities, in order to calculate what I believe are reasonable mathematical probabilities to explain my experiences.

To begin with, the Freeman Dyson description of Littlewood's Law of Miracles, as portrayed by the Lanes in their aforementioned article, needs to be carefully looked at in the case of my license plate synchronicities. Lets assume for the moment that I am conscious and aware of license plates of cars I can see on the road at the rate of 1 per second for the time period that I am driving in traffic (quite the demanding assumption here but I want to be open to Littlewood's assumptions). Lets also assume that the amount of time I am thus engaged averages to approximately three hours a week over a two month time period, resulting in approximately 27 hours of observations (lets use the number 28 for our estimation, as we can assume that I was driving a bit over 3 hours per week, and the second perfect number 28 is calling to me). So we now have 28 hours multiplied by 60 minutes in an hour multiplied by 60 seconds in a minute, which equals approximately 100,000 observed license plates (100,800 to be exact). Not quite enough license plates yet for a “miracle,” but we're getting there.

As a prototype for the license plate synchronicities I described in my Integral World article, lets use the experience I had of seeing the third perfect number 496 license plate soon after I gave my perfect number lesson to my son and his friends in Oregon. The intersection of these two events is what the Lanes refer to as “Desultory Decussation,” at least according to my understanding of what they described in their article. By a crude probability calculation, if there are a total of 36 choices in each of the license plate slots (10 numbers plus 26 letters), and four possible triplets for the six possible slots (1-2-3, 2-3-4, 3-4-5, 4-5-6), then we would have approximately a 1 in 10,000 chance of seeing the 496 (1 in 11,664 to be exact; I am using the standard probability formula for mutually exclusive events: p(A or B) = p(A) + p(B)). This certainly appears to be far more likely than the Littlewood requirement for a “miracle” of 1 in a million.

However, it is now time for me to include some information that pertains to “all” my license plate synchronicities for this two month period. Within a few days after I experienced my 496 in Oregon, I encountered another 496 in Maine. It is certainly reasonable to assume that these two events were “independent” (i.e. that it was not the same car traveling from Oregon to Maine). Consequently the probability of both these events happening within a few days of each other can be calculated by the formula p(A and B) = p(A) X p(B) (where X signifies multiplication) which equals approximately (1/10,000) X (1/10,000) = 1/100,000,000 = 1 over a hundred million. Do we now have a “miracle”? Well, what if I made this kind of probability calculation applied to the full force of my 19 license plate synchronicities? Roughly speaking it may be reasonable to conclude that the probability of “all” my license plate synchronicities taking place is (1/10,000) to the 19th power, which is approximately 1/(10 to the 76th power). Now keep in mind that 10 to the 6th power is a million, 10 to the 12th power is a trillion, and anyone who enjoys describing outrageously large number phenomena will have a good time explaining how an event with probability of 1/(10 to the 76th power) is incomprehensively unlikely to take place while we are observing our license plates in this two month period of time, to say the least!

My Full License Plate Synchronicities

Let me now give at least a glimpse of what it is that I am talking about with my “full” license plate synchronicities. In my original summary of my 19 license plate synchronicities, I included four 496s, four ACTs, two 2958s (see my Integral World License Plate Synchronicity article for explanations of the meanings of these synchronicities), plus a number of other license plate synchronicities that had a great deal of personal meaning for me. However, as I mentioned above, my mathematical formulas are merely suggestive, as I do not know many details, such as if the same car appears more than one time for some of my synchronicity observations, and how the Motor Vehicles Bureau decides upon their license plate numbers and letters. But the basic mathematical probability formula p(A and B) = p(A) X p(B) astronomically multiplies the extent of how “unlikely” it is for these events to occur by chance.

To indicate the kind of experiential analysis with a mathematical focus that I have done on all my license plate synchronicities, aside from the more personal explanations that I am not comfortable publicly sharing, I am including the same generic evaluation that I utilized in my License Plate Synchronicity article in Table 1 below. However, Table 2 (see below) is significantly more expansive than in my License Plate Synchronicity article, which is what severely reduces the probability of all this being explained by chance and Littlewood's Law of Miracles to something so infinitesimally small that I must conclude it is “unscientific” to believe that chance alone can explain all of this.

Table 1 - Sample Experiential License Plate Synchronicity Scale

Characteristic Rating
1) The experience cannot reasonably be explained by chance factors.  
2) The experience cannot reasonably be explained by beliefs, expectations false memories, or illusions.  
3) The experience is directly related to a significant life event.  
4) The experience is directly related to an intensive emotional need.  
5) The experience is directly related to the resolution of an intensive emotional need.  
Average Rating:  

Note. The rating scale from 1 to 5 is from low to high, where 1 is the lowest and 5 is the highest, and the average rating is rounded off to one decimal place.

In Table 2 below, I have included my experiential ratings on 19 license plate synchronicities over a two month period of time, that I consider to be unusual enough to warrant inclusion in the table [6]. The ratings correspond to the five numbered statements in Table 1; the four ACTs, three 496's, four ABJs, and two 2958's are labeled as ACT(#1), ACT(#2), ACT(#3), ACT (#4), 496(#1), 496(#2), 496(#3), ABJ(#1), ABJ(#2), ABJ(#3), ABJ(#4), 2958(#1), and 2958(#2). A rating will be listed in parentheses next to a numbered statement followed by a colon; to illustrate, a rating of 4 for Statement #2 will be denoted in the format 2: (4).

Table 2 - Experiential License Plate Synchronicity Scale

License Plate Symbol Statement Ratings Average Rating
ACT(#1) 1: (3), 2: (4), 3: (5), 4: (5), 5: (4) 4.2
ACT(#2) 1: (4), 2: (4), 3: (5), 4: (5), 5: (5) 4.6
ACT(#3) 1: (3), 2: (4), 3: (3), 4: (3), 5: (2) 3.0
ACT(#4) 1: (3), 2: (4), 3: (4), 4: (4), 5: (4) 3.8
496(#1) 1: (5), 2: (5), 3: (5), 4: (2), 5: (1) 3.6
496(#2) 1: (4), 2: (5), 3: (4), 4: (2), 5: (1) 3.2
496(#3) 1: (4), 2: (5), 3: (4), 4: (5), 5: (5) 4.6
ABJ(#1) 1: (2), 2: (2), 3: (3), 4: (2), 5: (1) 2.0
ABJ(#2) 1: (2), 2: (2), 3: (3), 4: (2), 5: (1) 2.0
ABJ(#3) 1: (2), 2: (2), 3: (3), 4: (3), 5: (3) 2.6
ABJ(#4) 1: (2), 2: (2), 3: (3), 4: (3), 5: (3) 2.6
2958(#1) 1: (4), 2: (4), 3: (3), 4: (2), 5: (1) 2.8
2958(#2) 1: (3), 2: (4), 3: (3), 4: (5). 5: (3) 3.8
ACM 1: (3), 2: (3), 3: (5), 4: (5), 5: (5) 4.2
GRAM 1: (3), 2: (3), 3: (5), 4: (5), 5: (5) 4.2
469 1: (3), 2: (2), 3: (3), 4: (1), 5: (1) 2.0
6496 1: (4), 2: (3), 3: (4), 4: (1), 5: (1) 2.6
7496 1: (4), 2: (4), 3: (4), 4: (5), 5: (5) 4.4
497 1: (3), 2: (3), 3: (4), 4: (3), 5: (4) 3.4
Average License Plate Synchronicity Rating: 3.3

Now none of this “proves” that something resembling Littlewood's Law of Miracles cannot account for my license plate synchronicity experiences. But according to my mathematical understanding, the probability of my experiences being reasonably explained completely by chance factors is infinitesimally small, and consequently it is astronomically unlikely that the events occurred completely by chance. From a “scientific” perspective it therefore strikes me that the most logical explanation is something “other than chance.” At this point we can now bring in all kinds of alternative explanations, ranging from quantum physics theories to subtle energy fields to God. The Lanes themselves, in their previous Integral World article Mysterium Tremendum [7], described quantum physics interpretations of various phenomena that I believe could certainly include synchronicity phenomena.

Two Concluding Examples

I would like to conclude this article by giving two more concrete examples of the result of applying probability theory to determine how likely it is to explain certain unusual events completely by chance factors. For my first example, I have recently been corresponding with someone who responded to my License Plate Synchronicity article by sharing with me some of his own synchronicities. His name is Samuel Hunter, and Samuel shared with me the following series of synchronicity events he experienced related to the number “23” (for more information about synchronicities related to “23” see Wikipedia). I will number Samuel's sequence of synchronicities and subsequently give some informal probability estimates. To be conservative, I'll eliminate Samuel's Synchronicity #5 in my probability calculations, since this is closely related to his Synchronicity #4, both signifying the date of 9/23.

Here is Samuel Hunter's “23” synchronicity series:

“About 9 days ago I went to the grocery store with Gisela, who works in my home.

  1. when I parked, the odometer said 117770 which when added up equals 23--I showed this to Gisela because I could feel this coming...
  2. as she went to get groceries, I went to Western Union where I had to take a number which was J23.
  3. I went back to the store and joined her in the checkout line where, when the check-out lady tried to scan the apples and did not know the code...she was told 5558 which when added equals 23.
  4. when I got home I asked “spirit” why 23??? and as I did, my eyes all on their own glanced down at my PC where I saw the date--Sept. 23rd hahaha.”
  5. that night my wife (who was in Mendellin) video called me and while we were talking she started to type (and she can barely type). Then her typed message appeared and it said: 9/23/2010 8:47:16PM Cristina Chester recuerda hoy es cumpleaños de Aleja, which means “remember today is Alejandra's (her child) birthday!”

The probabilities I assigned to these events that I consider to be reasonable estimates are as follows: 1) 1/10; 2) 1/100; 3) 1/30; 4) 1/30. Multiplying the probabilities together yields approximately .00000111, which is approximately one in a million—a miracle! Now put this together with the full range of Samuel's experiences with these kind of “23” synchronicities (which he has not yet shared with me), and we quite likely may be back to a description of the kind of astronomically unlikely events to occur by chance that I have described above for the full range of my license plate synchronicities.

Charles Tart (2009) [7] has used the term “scientism” to describe the “fundamentalist” allegiance to rigid dogmatic narrow-minded beliefs that are dressed up in a scientific language but in actuality are completely the opposite of the true nature of open-minded scientific investigation. From what I understand of the mathematical unlikelihood of chance factors being able to reasonably explain the mysterious events that I have described in this article, I cannot help but wonder if the phenomenon of people clinging to chance factors to explain synchronicity events that are astronomically unlikely to occur by chance, represents science or “scientism.” .

To give one final example of how “unlikely” chance factors may be to completely explain some mysterious events, I will briefly describe the most recent license plate synchronicity that I experienced.

About five weeks ago I was with my MR client and I parked my car in a community college library parking lot and noticed that the license plate of the car opposite me said “Jenny.” Now Jenny happens to be the original name of my mother, who died over 35 years ago. But the intensive phase of my license plate synchronicities was over, and I decided to attribute this to be nothing more than an interesting coincidence. However, I must admit that I felt the goose pimples and I looked further at the remaining letters on the license plate, and they were “Be.” My goose pimples suddenly became quite large, as “Be” are the first two letters of my last name and my mother's last name, and there in front of me was spelled out the first seven letters of my mother's name: “JennyBe”! This deeply struck me in an emotional way as “beyond reasonable explanation” by chance factors.

In my license plate synchronicity evaluation framework from Table 1 above, I would assign the highest ratings of “5” in each of the five categories to this most unusual event.

Lets play a bit with the probability mathematics involved. As I have no reason to think otherwise, I'll make the assumption that each of the seven slots in the license plate could have been assigned any number or letter, which yields a total of 36 to the 7th power possible license plates, which is approximately 78 billion possible license plates. I think we have now entered territory where Littlewood's Law of Miracles no longer applies. In other words, I think there is something going on here that needs to be explained beyond mathematical probability and chance. What exactly this explanation is, I do not know. But I believe that calling it “synchronicity” is both warranted and appropriate.

NOTES

1) See David Lane and Andrea Diem Lane (2010), Desultory Decussation: Where Littlewood's Law of Miracles Meets Jung's Synchronicity; Integral World website: www.integralworld.net

2) See Elliot Benjamin (2010), License Plate Synchronicity: An Experiential Account and Analysis; Integral World website: www.integralworld.net

3) Sse for example Harry Irwin & Carolyn Watt (2007), Introduction to Parapsychology; London: McFarland; Dean Radin (1997), The Conscious Universe; The Scientific Truth of Psychic Phenomena: New York: HarperEdge; and Stanley Krippner (Editor) (2010), Debating Psychic Experience: Human Potential or Human Illusion?; New York: Praeger.

4) Sse Elliot Benjamin (2006), Integral Mathematics: An AQAL Approach; Integral World website; www.integralworld.net

5) The determination of how reasonable it is for these events to be determined by chance factors is my own subjective evaluation, and other people may think quite differently about this, as is obviously the case with the Lanes.

6) Note that a few of the synchronicities in Table 2 may be considered to be “pseudo-synchronicities,” in particular the 469 and 497, but I have included them to serve as a comparison with the other more bona-fide synchronicities.

7) See David and Andrea Diem Lane (2010), Mysterium Tremendum; Integral World website: www.integralworld.net

8) See Charles Tart (2009), The End of Materialism: How Evidence of the Paranormal is Bringing Science and Spirit Together; Oakland, CA: Noetic Books.




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