We didn’t know randomness could be “subtle”

_{Denyse O'Leary

March 22, 2016

Mathematics, News}

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Erica Klarreich at Quanta has the story of a surprising new result about prime numbers from Kannan Soundararajan and Rober Lemke Oliver. They have found that, given a prime number with a certain last digit, there are different probability for the last digit of the next one (among the various possibilities). This violates usual assumptions that such things are in some sense “random”, indicating just how subtle this “randomness” is. More.

From Klarreich at Quanta:

Two mathematicians have uncovered a simple, previously unnoticed property of prime numbers — those numbers that are divisible only by 1 and themselves. Prime numbers, it seems, have decided preferences about the final digits of the primes that immediately follow them.

Among the first billion prime numbers, for instance, a prime ending in 9 is almost 65 percent more likely to be followed by a prime ending in 1 than another prime ending in 9. In a paper posted online today, Kannan Soundararajan and Robert Lemke Oliver of Stanford University present both numerical and theoretical evidence that prime numbers repel other would-be primes that end in the same digit, and have varied predilections for being followed by primes ending in the other possible final digits.

…

The discovery is the exact opposite of what most mathematicians would have predicted, said Ken Ono, a number theorist at Emory University in Atlanta. …

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EZ, It being budget season here I am busy, but I note past 2 [= 10] the primes in binary all end in 1 [start at 11 then 101]. That is, our digit issues are in part tied to the accidents of our use of decimal numbers. I suppose sexagesimals are truly dead, a pity as the number of things that evenly go into 60 is extraordinary. The randomness of primes is in part linked to that. But the runs of 6 are of course interesting as opposed to the runs of 2. 5 + 6 --> 11 + 6 --> 17 + 6 --> 23 +6 --> 29 [then 35 breaks the string] for instance. KF PS: First 1,000 primes: https://primes.utm.edu/lists/small/1000.txtkairosfocus_{March 23, 2016
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BA77, thanks, but WP strikes again and it will not embed -- nope going text fails. I give the link to your page in my long running thread on infinite past etc. KFkairosfocus_{March 23, 2016
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kf: https://docs.google.com/document/d/1q8uAEdSy9V2tL9JTHuGzBvbQT0tXRuNdaVI1JPjIcZ4/edit?usp=sharingbornagain77_{March 23, 2016
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BA77, can you toss the embed code for that Cantor vid my way? KFkairosfocus_{March 23, 2016
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It is hard to imagine a more convincing proof that we are made 'in the image of God', than finding that both the universe and life itself are 'information theoretic' in their foundational basis, and that we, of all the creatures on earth, uniquely possess an ability to understand and create information. I guess a more convincing evidence could be if God Himself became a man, defeated death on a cross, and then rose from the dead to prove that He was God. But who has ever heard of such overwhelming evidence as that? Turin Shroud Quantum Hologram Reveals The Words 'The Lamb' on a Solid Oval Object Under The Beard - video http://www.godtube.com/watch/?v=J21MECNU Verses and Music: Genesis 1:26 And God said, Let us make man in our image, after our likeness: and let them have dominion over the fish of the sea, and over the fowl of the air, and over the cattle, and over all the earth, and over every creeping thing that creepeth upon the earth. John 1:1-4 In the beginning was the Word, and the Word was with God, and the Word was God. The same was in the beginning with God. All things were made by Him, and without Him was not anything made that was made. In Him was life, and that life was the Light of men. Casting Crowns - The Word Is Alive https://www.youtube.com/watch?v=X9itgOBAxScbornagain77_{March 23, 2016
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A few related notes: Geometric Principles Appear Universal in Our Minds - May 2011 Excerpt: Villagers belonging to an Amazonian group called the Mundurucú intuitively grasp abstract geometric principles despite having no formal math education,,, Mundurucú adults and 7- to 13-year-olds demonstrate as firm an understanding of the properties of points, lines and surfaces as adults and school-age children in the United States and France,,, http://www.wired.com/wiredscience/2011/05/universal-geometry/ “Geometry is unique and eternal, a reflection from the mind of God. That mankind shares in it is because man is an image of God.” – Johannes Kepler "Nothing in evolution can account for the soul of man. The difference between man and the other animals is unbridgeable. Mathematics is alone sufficient to prove in man the possession of a faculty unexistent in other creatures. Then you have music and the artistic faculty. No, the soul was a separate creation." Alfred Russel Wallace - An interview by Harold Begbie printed on page four of The Daily Chronicle (London) issues of 3 November and 4 November 1910. Leading Evolutionary Scientists Admit We Have No Evolutionary Explanation of Human Language - December 19, 2014 Excerpt: Understanding the evolution of language requires evidence regarding origins and processes that led to change. In the last 40 years, there has been an explosion of research on this problem as well as a sense that considerable progress has been made. We argue instead that the richness of ideas is accompanied by a poverty of evidence, with essentially no explanation of how and why our linguistic computations and representations evolved.,,, (Marc Hauser, Charles Yang, Robert Berwick, Ian Tattersall, Michael J. Ryan, Jeffrey Watumull, Noam Chomsky and Richard C. Lewontin, "The mystery of language evolution," Frontiers in Psychology, Vol 5:401 (May 7, 2014).) Luskin comments: “It's difficult to imagine much stronger words from a more prestigious collection of experts.” http://www.evolutionnews.org/2014/12/leading_evoluti092141.html An Interview with David Berlinski - Jonathan Witt Berlinski: There is no argument against religion that is not also an argument against mathematics. Mathematicians are capable of grasping a world of objects that lies beyond space and time …. Interviewer:… Come again(?) … Berlinski: No need to come again: I got to where I was going the first time. The number four, after all, did not come into existence at a particular time, and it is not going to go out of existence at another time. It is neither here nor there. Nonetheless we are in some sense able to grasp the number by a faculty of our minds. Mathematical intuition is utterly mysterious. So for that matter is the fact that mathematical objects such as a Lie Group or a differentiable manifold have the power to interact with elementary particles or accelerating forces. But these are precisely the claims that theologians have always made as well – that human beings are capable by an exercise of their devotional abilities to come to some understanding of the deity; and the deity, although beyond space and time, is capable of interacting with material objects. http://tofspot.blogspot.com/2013/10/found-upon-web-and-reprinted-here.html Recognising Top-Down Causation George Ellis, University of Cape Town Excerpt: page 5: A: Causal Efficacy of Non Physical entities: Both the program and the data are non-physical entities, indeed so is all software. A program is not a physical thing you can point to, but by Definition 2 it certainly exists. You can point to a CD or flashdrive where it is stored, but that is not the thing in itself: it is a medium in which it is stored. The program itself is an abstract entity, shaped by abstract logic. Is the software “nothing but” its realisation through a specific set of stored electronic states in the computer memory banks? No it is not because it is the precise pattern in those states that matters: a higher level relation that is not apparent at the scale of the electrons themselves. It’s a relational thing (and if you get the relations between the symbols wrong, so you have a syntax error, it will all come to a grinding halt). This abstract nature of software is realised in the concept of virtual machines, which occur at every level in the computer hierarchy except the bottom one [17]. But this tower of virtual machines causes physical effects in the real world, for example when a computer controls a robot in an assembly line to create physical artefacts. Excerpt page 7: The assumption that causation is bottom up only is wrong in biology, in computers, and even in many cases in physics, for example state vector preparation, where top-down constraints allow non-unitary behaviour at the lower levels. It may well play a key role in the quantum measurement problem (the dual of state vector preparation) [5]. One can bear in mind here that wherever equivalence classes of entities play a key role, such as in Crutchfield’s computational mechanics [29], this is an indication that top-down causation is at play. http://fqxi.org/data/essay-contest-files/Ellis_FQXI_Essay_Ellis_2012.pdf Moreover, 'Brain Plasticity', the ability to alter the structure of the brain from a person's focused intention, has now been established by Jeffrey Schwartz, as well as among other researchers. The Case for the Soul - InspiringPhilosophy video - (4:03 minute mark, Brain Plasticity including Schwartz's work) - Oct. 2014 - video The Mind is able to modify the brain (brain plasticity). Moreover, Idealism explains all anomalous evidence of personality changes due to brain injury, whereas physicalism cannot explain mind. https://www.youtube.com/watch?v=oBsI_ay8K70 Moreover, completely contrary to materialistic thought, mind has been now also been shown to be able to reach all the way down and have pronounced effects on the gene expression of our bodies: Scientists Finally Show How Your Thoughts Can Cause Specific Molecular Changes To Your Genes, - December 10, 2013 Excerpt: “To the best of our knowledge, this is the first paper that shows rapid alterations in gene expression within subjects associated with mindfulness meditation practice,” says study author Richard J. Davidson, founder of the Center for Investigating Healthy Minds and the William James and Vilas Professor of Psychology and Psychiatry at the University of Wisconsin-Madison. “Most interestingly, the changes were observed in genes that are the current targets of anti-inflammatory and analgesic drugs,” says Perla Kaliman, first author of the article and a researcher at the Institute of Biomedical Research of Barcelona, Spain (IIBB-CSIC-IDIBAPS), where the molecular analyses were conducted.,,, the researchers say, there was no difference in the tested genes between the two groups of people at the start of the study. The observed effects were seen only in the meditators following mindfulness practice. In addition, several other DNA-modifying genes showed no differences between groups, suggesting that the mindfulness practice specifically affected certain regulatory pathways. http://www.tunedbody.com/scientists-finally-show-thoughts-can-cause-specific-molecular-changes-genes/ More interesting still, the three Rs, reading, writing, and arithmetic, i.e. the unique ability to process information inherent to man, is the very first things to be taught to children when they enter elementary school. And yet it is this information processing, i.e. reading, writing, and arithmetic that is found to be foundational to life: Information Enigma (Where did the information come from?) - video https://www.youtube.com/watch?v=aA-FcnLsF1g Complex grammar of the genomic language – November 9, 2015 Excerpt: The ‘grammar’ of the human genetic code is more complex than that of even the most intricately constructed spoken languages in the world. The findings explain why the human genome is so difficult to decipher –,,, http://www.sciencedaily.com/releases/2015/11/151109140252.htm As well, as if that was not 'spooky enough', information, not material, is found to be foundational to physical reality: "it from bit” Every “it”— every particle, every field of force, even the space-time continuum itself derives its function, its meaning, its very existence entirely—even if in some contexts indirectly—from the apparatus-elicited answers to yes-or-no questions, binary choices, bits. “It from bit” symbolizes the idea that every item of the physical world has a bottom—a very deep bottom, in most instances, an immaterial source and explanation, that which we call reality arises in the last analysis from the posing of yes-no questions and the registering of equipment—evoked responses, in short all matter and all things physical are information-theoretic in origin and this is a participatory universe." – Princeton University physicist John Wheeler (1911–2008) (Wheeler, John A. (1990), “Information, physics, quantum: The search for links”, in W. Zurek, Complexity, Entropy, and the Physics of Information (Redwood City, California: Addison-Wesley))bornagain77_{March 23, 2016
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You misunderstand the quote. It is not that Godel expected humans to understand all of the many and varied nuances of mathematics. It is Godel was pointing out that if the human mind were merely material then mathematical intuition would be impossible for humans. Yet mathematical intuition is innate to humans. Thus the human mind must be immaterial. You can see a more in depth look at Godel's thinking towards the end of this video: Cantor, Gödel, & Turing: Incompleteness of Mathematics - video (excerpted from BBC's 'Dangerous Knowledge' documentary) https://www.facebook.com/philip.cunningham.73/videos/vb.100000088262100/1119397401406525/?type=2&theaterbornagain77_{March 23, 2016
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ba77 Re: Godel quote I vote for mathematics being too big for the human mind. Many people feel that Poincare was the last mathematician who could work in all the sub-fields. Mapou The distribution of the prime numbers has nothing to do with evolutionary theory.ellazimm_{March 22, 2016
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Darwinists are in luck. They have been looking for a "subtle randomness" to explain how random mutations can give rise to the amazing complexity of living organisms without getting bogged down by the combinatorial explosion.Mapou_{March 22, 2016
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of related note: Is Integer Arithmetic Fundamental to Mental Processing?: The mind's secret arithmetic Excerpt: Because normal children struggle to learn multiplication and division, it is surprising that some savants perform integer arithmetic calculations mentally at "lightning" speeds (Treffert 1989, Myers 1903, Hill 1978, Smith 1983, Sacks 1985, Hermelin and O'Connor 1990, Welling 1994, Sullivan 1992). They do so unconsciously, without any apparent training, typically without being able to report on their methods, and often at an age when the normal child is struggling with elementary arithmetic concepts (O'Connor 1989). Examples include multiplying, factoring, dividing and identifying primes of six (and more) digits in a matter of seconds as well as specifying the number of objects (more than one hundred) at a glance. For example, one savant (Hill 1978) could give the cube root of a six figure number in 5 seconds and he could double 8,388,628 twenty four times to obtain 140,737,488,355,328 in several seconds. Joseph (Sullivan 1992), the inspiration for the film "Rain Man" about an autistic savant, could spontaneously answer "what number times what number gives 1234567890" by stating "9 times 137,174,210". Sacks (1985) observed autistic twins who could exchange prime numbers in excess of eight figures, possibly even 20 figures, and who could "see" the number of many objects at a glance. When a box of 111 matches fell to the floor the twins cried out 111 and 37, 37, 37. http://www.centreforthemind.com/publications/integerarithmetic.cfm "Either mathematics is too big for the human mind, or the human mind is more than a machine." Kurt Gödel As quoted in Topoi : The Categorial Analysis of Logic (1979) by Robert Goldblatt, p. 13bornagain77_{March 22, 2016
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I'm not sure I would have described the distribution of the final digits of prime numbers as being 'random'. After the first and third prime (2 and 5) there are only four possible final digits: 1, 3, 7 and 9. And there are a lot of 'prime pairs' (prime numbers who only differ by 2) which, I would think, would make it harder for a prime ending in '1' to be followed by a prime ending in '9'. Still, an interesting result. Number theory is fascinating. Lots of cool results.ellazimm_{March 22, 2016
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The randomness of the LORD is subtle.Mung_{March 22, 2016
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Terence Tao discusses the paper here. In the comments, John Baez asks the obvious question: Why wasn't this discovered earlier, to which Tao replies:
There were at least two papers in experimental number theory that noticed some manifestation of the Lemke Oliver-Soundararajan bias on a qualitative level: a 2011 paper of Ash, Beltis, Gross, and Sinott, and a 2002 paper of Ko. But I think the difficult task was to separate this bias from previously known biases in the distribution of the primes, such as the Chebyshev bias discussed in the blog post, or biases arising from the prime tuples conjecture (which, for instance, predicts that a prime gap of 6 is twice as likely to occur as a prime gap of 2). To realise that this particular bias is significantly stronger than what can be easily explained from these previously identified biases seems to require a certain amount of theoretical mathematical training.
daveS_{March 22, 2016
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