Monday, August 5, 2019

We came in with a Bang

I was alluding to the Big Bang, the creation event that the atheist scientist community really hated to have to swallow. I happened to be reading the description of the 1978 Nobel Prize awards, which included Arno Penzias and Robert Wilson for their 1965 discovery of the CMB, the cosmic microwave background, that simmering remnant of the explosion of space-time some 13.8 billion years ago (using the 2018 Planck report figure), and found the Nobel site (hopefully written in 1978) described it "as tempting to assume that the universe was created by a cosmic explosion...though other explanations are possible." In 1978 I considered it well-established that the Big Bang was the origin of the Universe, but as I say, some had to be dragged kicking and screaming to that conclusion. They managed to save their position (atheist, PC, etc.) by creatively proposing that our universe is only a single bubble in a huge froth of universes, the "multiverse." Besides being conveniently unfalsifiable, the concept also provides some justification for the relative lack of progress in theoretical physics for the last half century (or more if you demand equivalence to the discovery of quantum mechanics and relativity at the turn of the twentieth century) in that it can now be claimed (with a straight face) that it is really impossible to discover more about nature now because the remainder of what happened is simply random turns of the dials on each bubble that pops into the Multiverse (this goes well with a string theory diet).

In any case, after my previous post I began to feel guilty about my brief characterization of CMB photons at a single frequency equivalent to 2.725 K temperature. The CMB photons are described by a spectrum, a curve describing about how much power (equivalent to about how many photons there are) at each of a range of frequencies, so it is a little misleading to speak about a single frequency. I accordingly dropped my other work (a physics study I have spent two years on at this point) to add a little more to my discussion.

I was surprised (again, and why is it that most surprises are undesirable? must be my life's experience or the deterioration of this world during my lifetime) to find that the equations and presentations of the data related to blackbody radiation (the CMB radiation is almost exactly that of a blackbody spectrum equivalent to a perfect radiator at 2.725 degrees Kelvin) are ambiguous at best, and possibly flatly wrong on occasion, depending on the author (there are many different forms for the equations and units involved, e.g., intensity at unit area vs integrated over a hemisphere). It is not that the mathematics and physics involved is new, Max Planck got the ball rolling back in 1901 or so, when he invented what became known as the Planck constant, h, in order to avoid the ultraviolet catastrophe. Scientists up to that time had made very accurate measurements of the frequencies of radiation emitted from hot objects, in particular those constructed to be the equivalent of a blackbody. By blackbody I mean an object that does not reflect any electromagnetic energy, like a mirror reflects, instead absorbing everything that comes its way and in turn emitting a very precise spectrum of energy related to its own temperature. At the time they created hollow cavities,  termed "hohlraum", with a tiny hole out of which they could measure radiation intensity and character inside. Almost all the radiation of the hohlraum is trapped inside, bouncing around internally, which has the effect that its internal heat is solely related to its own temperature rather than through exchange processes with the environment by which it might otherwise come to equilibrium.

A few had offered equations to characterize the blackbody radiation, but they had the undesired quality of blowing up a short wavelength, i.e., creating an ultraviolet catastrophe. Planck laboriously (I have discovered through the accounts of many scientists that scientific work requires a lot of effort, which makes me feel a bit better about my own self-inflicted pain in that area) came up with the idea that the radiation could not simply be any number on the real number line, but rather would have to come in integer multiples of this odd constant, 6.62607004e-34 Joules seconds (I have one of his 1901 papers and see he got very close to this present value of h, he giving 6.55e-27 erg second; an erg is 1e-7 Joule). He hoped for some time that someone would find a way to explain this in some other way, but quantum mechanics took off within a few years, changing everything Planck and every other scientist had known in the way of a world view (along with special and general relativity, most people are aware Einstein created that new area, but less aware that he only received the Nobel Prize for his work in creating quantum mechanics, he having found immediate use for Planck's constant in describing the photoelectric effect).

I'll see if I can somehow type the Planck law equation of interest to me (it is a real pain in the neck to work without LaTex math or its online version MathJax):
๐ˆ(ฮฝ,T) = (2โ„Ž๐‚³ / c²) (1 / exp(โ„Ž๐‚/kT) - 1)
Whew that is ugly. Well anyway. That equation gives you the radiation intensity as a function of frequency (that is the ฮฝ, a Greek letter nu) and time (per second) in, depending on what constant units you use, J or W per m^2 (meter squared) per steradian (a chunk out of the surface of a sphere, picture the radiation emerging from the eyes of angry Superman stuck in a big ball, say a hohlraum, and cones of death ray hit the surface). The famous Planck's constant is the โ„Ž, which is privileged to have its own Unicode symbol up here apparently. exp means "exponential", i.e., Euler's number raised to the power of the stuff in parentheses next to it. I would have liked to put a "B" subscript on the k to signal that this is Boltzman's constant, put that was perversely impossible (every letter other than b had a subscript when I looked at the menu here). T is temperature in Kelvin absolute degrees. The frequency ฮฝ is in Hz (cycles per second). I wrote some computer code (I work in a Jupyter notebook environment, where I can type perfectly typeset mathematics in one place and active Python computer code in another, with all the scientific software available for that environment, SciPy, NumPy, SymPy, Matplotlib, etc.---first time in years I have felt like computers were actually fulfilling some of their promise, despite Gates' constant attempts to sabotage anyone's efforts to accomplish that...the Borg operating system, as it is known in the world-wide computing community) to graph the result of a spectrum of frequencies from 100 MHz to 1000 GHz input to that equation above:

And there you have it (hoot, there it is?). I encountered so many errors and misuse of the Planck equations in the literature that I intentionally used the same spectrum covered in a figure in a 2011 paper by P.J.E. Peebles (The natural science of  cosmology, 2011) as a cross check on my methods. Peebles, by the way, was one of the scientists who Penzias and Wilson shared their discovery with in 1965; they published a letter in 1965 Astrophysical Journal, vol. 142, p.419-421 and Peebles, along with his collaborators, published "a possible explanation for the observed excess noise temperature [of the antenna]" You have to consider that this was an age when men in science conducted themselves like gentlemen, with integrity and understatement. It is probably difficult to understand for anyone of the current era. These folks knew they had probably found the simmering heat of the Creation, but described it in clear, non-hyperbolic terms as the Bell Telephone scientists that they were (they had used a re-purposed piece of satellite communication equipment, a giant horn, I kid you not, used to communicate with a couple of early telecommunications satellites). Bell Telephone Labs, that is another loss of that age. Amazing discoveries that came out of their work, most of it freely shared with the world. I was just reading Nyquist's 1924 paper (published in a Bell technical journal) on Certain Factors Affecting Telegraph Speed the other day. That was the work that in many ways started the field of communications theory, his name still used today as the term for the minimum sample frequency to prevent aliasing (the Nyquist criterion). I should add that "the horn" was a microwave horn, not something you might find in the Alps. At high frequencies electromagnetic fields can be guided through suitably configured plumbing as it were.

In any case, I wanted to make it clear that there were more frequencies involved in the CMB than my June 17, 2019 post discussed. I had mentioned 57 GHz. You can see that the peak of the spectrum is around 160 GHz. Penzias and Wilson's 1965 article specifically dealt with the 7.35 cm flux, 4080 MHz or 4.080 GHz (their horn was tuned to that frequency, like trumpets with air, there are favored notes in electromagnetic radiation for the particular construction), which is pretty far out on the left tail of the distribution I graphed above.







Monday, June 17, 2019

Loose associations, binary thinking, time-space, neutrinos, and counting

I was thinking about the tree structure of syntax diagrams (that I discussed in my previous post) and began to wonder about the matter of binary decisions. By binary decision I mean having two choices only, i.e., branch to the left or branch to the right. In the most fundamental expression of such a dualism, mystics must inevitably confront the paradox of God as the spiritual light, but also the cause of nature's black deeps (paraphrasing Carl Jung). You encounter this tension of the pairs of opposites throughout the temporal world, that is to say, the world where we find ourselves conscious and proceeding through time moment by moment. There is no change where there is no time, and anything material, at least anything with mass, experiences time.

For example, neutrino particles (the Standard Model of physics, incorporating the electromagnetic, weak and strong nuclear forces or interactions, classifies all the subatomic particles known to date; there is the familiar electron, proton and neutron, but there are many others, including the electrically neutral and difficult to detect neutrino) must have some small mass and experience time since it apparently does change. The neutrino can be observed to interact in three different "flavors," i.e., electron, muon, tau (the flavors are simply the type of charged lepton it appears with in an experimental observation), and in fact oscillates among those three states (or the probability of catching it interacting with a particular lepton changes periodically with distance) as it travels, which involve the propagation of its composite mass states at slightly different rates.

The propagating neutrino mass states (there are three of those, different from the interaction flavor "state", i.e., the mass states are the true neutrino, the flavor states simply a convention) are energy-momentum four-vectors, rank-1 tensors really. Before Einstein and Special Relativity we conceived of physical objects occupying space, e.g., we could bring two objects into contact and that contact would end where the "space" occupied by each met. We had a vague notion of an abstract "space" in which objects existed, but that notion really required a body of reference (following Einstein's 1921 Princeton lectures on Special and General Relativity).

You may recall from high school algebra that we could construct Cartesian coordinate systems where we indicate the position of points (and objects consisting of the loci of such points) in a three-dimensional grid with three axes, ๐‘ฅ, ๐‘ฆ, and ๐‘ง.  A line connecting two points in this grid could be described by an interval ๐‘ ² = ∆๐‘ฅ² + ∆๐‘ฆ² + ∆๐‘ง², where the "∆๐‘ฅ" notation means "the difference in position between the two points in relation to the ๐‘ฅ-axis." Time was thought to be independent of space, a separate concept. We would speak about events occuring at different Cartesian coordinates simultaneously, without a lot of thought about what that implied. Einstein realized that there are no instantaneous effects across distance, the speed of propagation of effects being limited to the speed of light. Accordingly, it became clear that only events had physical reality, the time and the location of the event varying (mathematically, not as opinion) by observer frame of reference.

By 1895, Henri Poincarรฉ commented that the recent results (1887) of the Michelson-Morley experiment, which failed to show any variation in the speed of light with respect to the translational velocity of the Earth, implied a principle of relativity which prevented a moving observer from detecting any difference in the laws of physics compared with another observer at rest (there are a number of qualifications here, but we will leave it in a simple form).  You might have expected, contrarily, that if space was a substance, an aether through which light travelled, like an ocean wave through water, that it might travel a bit slower going upstream in a river as it were, or in the case of the Earth moving in its orbit around the Sun, light might propagate a little slower moving against the direction of motion of the Earth.

In 1904 Hendrik Lorentz summed up his recent work including Poincarรฉ's contributions, on transforms from one frame of reference to another, consistent with the Maxwell electromagnetic equations implying the constancy of the speed of light in all inertial systems. He proposed the Lorentz transformation still used today, time dilation and contraction of bodies along the direction of motion. Interestingly, the contraction concept was originally introduced by Oliver Heaviside (you can find a copy of this work from The Electrician, 1888, on pg 511 of the Heaviside Electrical Papers, Vol. II, published 1894). Heaviside had analyzed the movement of a charged spherical conductor and found that the charge distribution could only remain static only if the sphere shortened in the direction of motion, becoming an oblate spheroid by a factor  (1 −๐‘ฃ²⁄๐‘²)¹⁄² .

For our context, we are interested in the fact that these new ideas required our old view of a separate three-dimensional space and one-dimensional time be replaced by a four-dimensional space-time continuum, which Einstein presented in a unified way in his 1905 papers. Our three-dimensional coordinate equation above becomes instead an invariant space-time interval ๐‘‘๐‘ ² = ๐‘²๐‘‘๐‘ก² - ๐‘‘๐‘ฅ² - ๐‘‘๐‘ฆ² - ๐‘‘๐‘ง², connecting points in a space-time coordinate system (we replaced the ∆๐‘ฅ² notation with ๐‘‘๐‘ฅ², which means about the same thing, a difference, albeit infinitesimal, between two points). Though this space-time interval is measured to be the same by all observers, moving clocks tick more slowly and objects shrink in the direction of their motion, i.e., as we alluded to previously, different observers might not agree about time and location of events, but they can all agree on the space-time interval. The Lorentz transformation permits, e.g., an observer at rest to calculate the time experienced on a moving object. For example, the proper time ๐œ on the moving object, say a rocket headed away from the Earth (say along the ๐‘ฅ-axis) would be ๐‘‘๐œ = ๐‘‘๐‘ก ∕๐›พ, where ๐›พ = ๐Ÿท  ∕ (๐Ÿท −  ๐›ฝ² ) and  ๐›ฝ = ๐‘ฃ ∕๐‘ (the velocity we observe for the rocket, divided by the speed of light).

๐‘‘๐‘ ² = ๐‘²๐‘‘๐‘ก² - ๐‘‘๐‘ฅ² - ๐‘‘๐‘ฆ² - ๐‘‘๐‘ง²  = (๐‘๐‘‘๐‘ก )² [๐Ÿท −  ๐›ฝ² ]  =  (๐‘๐‘‘๐‘ก  ∕ ๐›พ)²   = (๐‘๐‘‘๐œ)²

That is, we observe time in the moving clock frame as ๐‘ก  = ๐›พ๐œ, their clock tick intervals appear longer than ours by a factor ๐›พ, i.e., their clock appears to be running slowly in comparison with our clock at rest. Though we may not agree on clock times or the lengths of objects, we all calculate ๐‘‘๐‘ ² to be the same number, just as we may rotate three-dimensional vectors in Euclidian space without changing their length.

The time-space interval example above used the position (or position-time) four-vector, ๐‘ฅแต˜  = {๐‘ฅ⁰, ๐‘ฅ¹ ,๐‘ฅ², ๐‘ฅ³}where the first component is time (multiplied by the speed of light), ๐‘๐‘ก, and the remaining three variables are the usual Cartesian ๐‘ฅ,๐‘ฆ,๐‘ง axis coordinates (numbered ๐Ÿท,2, 3 superscript to indicate a contravariant vector). We wanted to look at another type of four-vector though.  Above we mentioned the energy-momentum four-vectors that describe the three mass states that comprise a propagating neutrino particle. They look like ๐‘แต˜  = {๐‘⁰, ๐‘¹, ๐‘², ๐‘³}. The ๐‘⁰ component is now ๐ธ, the relativistic energy and the remaining three components are the momenta on the ๐‘ฅ,๐‘ฆ,๐‘ง axes. What we wanted to note was that the neutrino mass states, ๐œˆ₁,๐œˆ₂, and ๐œˆ₃, each propagate as an energy-momentum four-vector, together initially, but because their relative masses differ slightly they travel at slightly different velocities and, amazingly, are believed to separate after a sufficient distance.

We may simplify the analysis by considering only two of the four-momenta (we are not concerned with relativistic frames of reference here, so the ๐›พ factor will be omitted), described by the relativistic energy relation, ๐ธ =  (๐‘² + ๐‘š²)¹⁄² (the square root of the squared momentum and mass). ๐ธ is one of the ๐ธ₁ and ๐ธ₂ and the rest mass ๐‘š is one of ๐‘š₁ and ๐‘š₂ in turn. Using the binomial expansion that can be reduced to ๐ธโฑผโ‚– =  ๐‘โฑผโ‚– + ๐‘šโฑผโ‚–² ∕ 2๐‘ for each ๐ธโฑผโ‚– and ๐‘šโฑผโ‚–. ๐‘ is half the sum of the two momenta, ๐‘ = 1/ (๐‘โฑผ  + ๐‘โ‚– ).

The difference between any two mass state energies becomes ๐›ฅ๐‘š² ∕ 2๐‘ and the relative velocity difference then ๐›ฅ๐˜ท = ๐›ฅ๐‘š² ∕ 2๐‘² or  ๐›ฅ๐˜ท = ๐›ฅ๐‘š² ∕ 2๐ธ², ๐‘ and ๐ธ being very close for the highly relativistic neutrino (it propagates at very near the speed of light). If the size of the neutrino propagating mass states are roughly ๐œŽ (the length of a packet) then the distance at which the mass states separate is ๐ฟ = ๐œŽ ∕ ๐›ฅ๐˜ท. We estimate that neutrinos arriving from a supernova at the center of our Milky Way galaxy might arrive with their mass components separated by up to 41 meters, a detectable difference of about 137 ns (billionths of a second) apart. As far as we know though, none of the current supernova detection experiments are set up to detect separated packets.

Resuming our original thread, the neutrino has mass, therefore it cannot travel at the speed of light, by Einstein's equation for relativistic energy, writing in a form with the rest mass and ๐›พ factor,  ๐ธ = ๐›พ๐‘š๐‘²  = ๐‘š๐‘² ∕ (1  − ๐‘ฃ² ∕๐‘²)¹⁄².

You see in that equation that if v, the velocity of the neutrino, were to equal the speed of light, c, in the denominator, the v2 divided by c2 would equal 1 (since a number divided by itself is 1), and since that is subtracted from 1 beneath the radical, the result of the subtraction will try to go to zero. The denominator of a fraction is not allowed to be zero because that operation is not defined. a/b = c implies a = bc and there is no number c that could be multiplied by zero to get a non-zero number a. 

However, it you graph y = 1/x as x approaches zero (becomes very small in absolute value) from negative infinity (the left, e.g., -3, -2, -1, -0.9, ....-0.000001...) and as x approaches zero from the right or positive infinity (e.g., 3, 2, 1, 0.9, 0.00001...) y goes to either negative or positive infinity (the red line in the graph heads straight up to infinity or straight down to infinity as it nears the y axis):


So that tells you that E, the energy of a relativistic particle (or any particle with mass), would go to infinity as the particle approached the speed of light. Since there isn't infinite energy available (at least not since the moment of Creation at the Big Bang), particles with mass are not allowed (by the laws of this universe) to reach the speed of light (the neutrinos are pretty close though, but have a very tiny mass). If a particle travels at the speed of light, it cannot change because time for the particle does not exist (a change is an event in time that occurs when something passes in time from one state to another).

Light, on the other hand, is propagated by photon particles, which have no mass and travel only at the speed of light. So the photons from the Big Bang moment of Creation have not changed in the intervening billions of years separating us from the Big Bang? Well, space has expanded under their feet as it were. The photons we detect today as CMB (cosmic microwave background) were emitted at Recombination, about 380,000 years after the Big Bang, when hydrogen and helium atoms are formed, taking up most of the free electrons, making it possible for photons to free stream without many collisions at an equivalent temperature of about 4000 K, with a wavelength of around 3.60 millionths of a meter. When they are detected today, because the Universe has been expanding over the intervening billions of years, they have been stretched out to a wavelength of about 5279 millionths of a meter, a frequency of about 57 GHz (your cell phone frequency is about 1.9 GHz and microwaves lie between 0.1 to 1000 GHz, hence the term cosmic "microwave" background).

Aside from the stretching they are beyond time as it were. Recall from our earlier discussion that we observe time in a moving clock frame as ๐‘ก  = ๐›พ๐œ, clock tick intervals on a moving object lengthening by a factor ๐›พ, where  ๐›พ = 1∕ (1  − ๐‘ฃ² ∕๐‘²)¹⁄².

You can see we get the same problem of the fraction as a whole going to infinity as the velocity equals the speed of light (and photons, having no mass, always travel at the speed of light in vacuum), which you may interpret as time being frozen more or less for the thing travelling at the speed of light (if it takes infinity to reach the next clock tick time has stopped).

GPS satelllites move at 14,000 km/hr and have to correct for relativistic time dilation of 7 ฮผs/day (slower satellite clock). (They also have to correct for the General Relativity opposing effect of the gravitational redshift from the Earth's gravitational field making clocks back on the Earth's surface relatively slower than the satellite clock, i.e., clocks in a "gravitational well" run slower.)

If we are able to send people to nearby star systems someday and reach significant percentages of the speed of light, the travellers (astronauts) will find that people back on Earth have aged more than they have. For example, if they travel 10 light years (round trip and neglect the year or more accelerating at both ends) at 80% of the speed of light they will age 7.5 years and their friends and family back on Earth will have aged 12.5 years (at 0.8 speed of light they would cover 10 light years in 12.5 years and their clocks are at 0.6 of "normal" so they would age only 7.5 years).

I have glossed over the idea of a quantity going to infinity, proposing that once we see a quantity headed in that direction we can safely envision a progression conceptually reaching infinity, i.e., one foot placed after the other in a never-ending march, counting each step (as I suggested above when pointing to the graph of 1/x heading up or down to infinity since we could see it was "headed that way").

We assume that we have always had the idea of adding more things to increase the number of things in hand. I've got one apple and if you give me another, why then of course I have two then. In fact though it takes more than two years for human children to grasp that the difference between "one thing" and "more than one thing" can be more abstractly understood as the successor function which generates all of the numbers with which we count, i.e., the natural numbers (1, 2, 3...some include 0). A child can proudly perform the counting script they have been taught, naming, say, one toy fish, two toy fish, three and so on. However, typically the 2.5 year old child who has just "counted" the toys in that way will hand you one fish if asked for one, but give you an arbitrary handful if asked for any number other than one!

After additional months of experience the child slowly, in a stepwise fashion, learns to understand "two" then "three." Sometime after this comes the great leap forward, where the child grasps implicitly the induction definition of natural numbers, i.e., that each word in the counting routine actually defines how many things you are considering and that each successive count adds one to the number of things (in your set) and that this can be continued indefinitely, with no upper bound (see Evolutionary and developmental foundations of human knowledge, by Marc Hauser and Elizabeth Spelke).

Although a chimpanzee can laboriously learn to associate a number symbol with a particular number of objects, they never (at least not after 20 years of training on one particular subject) progress to the understanding of the successor function, i.e., chimps cannot learn that a new number symbol means that one has been added to the previous set of items. It appears though (from research done by Spelke and others) that humans and some non-human primates both draw on a core neurophysiological basis for (1) representing the approximate cardinal values (about how many items are present) of large groups of objects or events and (2) representing the exact number of object sets or events when there are only a small number of individual units.

It appears that the uniquely human capability to construct the natural numbers (i.e., use the successor function) relies first on the core perception of one versus many, then mapping other number words to larger numerosities, then noticing that the progression in the language (the words representing numbers) of the counting routine corresponds to increasing the cardinal value of the set, the number of units in hand. This (and other research) suggests that natural language ability is involved in the human leap from those core perceptions shared by some non-human species to the natural number concepts unique to humans.

There has been some controversy about extending the concept of infinity, at least in the context of mathematics, namely, the Brouwer-Hilbert controversy about the foundations of mathematics at the beginning of the twentieth century.

L.E.J. Brouwer did not believe that the rules of classical logic laid out by Aristotle have an absolute validity independent of the subject matter to which they are applied. For example, Aristotle defined the Law of the Excluded Middle, which reasonably from our experience in life states that any proposition is either true or it is not true, e.g., Socrates is either a mortal or he is not a mortal, he cannot be something in between the two.

The claim of formal logic is that this law (of the Excluded Middle) applies simply because it is an accepted rule of logic, not because we have seen examples which permit us to infer that it is true in a specific case (e.g., the case of whether Socrates is mortal or not). Brouwer objected to making such an automatic claim via logic when offering a formal proof in mathematics. Brouwer wanted to see a proof that constructed specific examples (actual mathematical entities) rather than simply claiming one or the other of two contradictions must necessarily be true.

This may seem like rather abstract contentions among mathematicians, but if you go with Brouwer (and his intuitionist stance) then you are not allowed to extend presumptions to the infinite (which would cramp our style in the discussions earlier). For example, the induction axiom of mathematics states that if a mathematical proposition P(n) is true for n = 0, and if, for all natural numbers n, P(n) being true implies that P(n + 1) is true also, then P(n) is true for all natural number n. You will recognize our successor function from our earlier discussion. The so-called "animal instinct" here is that it must be true since you can conceive of marching forever, one foot placed after another in a never ending march, thereby defining infinity.

The alternative notion would be Georg Cantor's aleph-null, a completed infinity all at once without laying out the steps leading there. Well, strictly speaking aleph-null or aleph-naught represents any countable infinite set, for example, the natural numbers N. The real numbers, R are also infinite, but not countable. Cantor developed his famous diagonal argument to prove that R, the set of real numbers was not countable, though it was infinite.

Cantor's diagonal argument for showing a set is uncountable goes like this: Enumerate (list) all of the members of the set T of infinite sequences of binary digits (ones or zeros). No matter how you list them there will always be a member of the set that you miss, because you can draw a diagonal slash from left top corner to right bottom infinity corner, pull out the string of digits selected by your slash, then complement each of the digits you obtained, i.e., if there is a "1" replace it by a "0" and if a "0" replace by "1." The sequence you end up with cannot have been in the list because it differs from every string in the list by the nth digit:


In the above example the diagonal slash pulls the red digits 01000101100... and complements each of those to get s = 10111010011... You can see that s cannot have been in the list because you have made it differ by one digit in each listed sequence by complementing the slashed sequence. Therefore it is impossible to count, i.e., enumerate or list all of the numbers (each sequence represents a number in the set) in the set because every time you complete your list, a new unique number pops up in the diagonal slash!

David Hilbert designed a formal definition of mathematics where no intuitive notion about "reality" or actual examples or objects was necessary, but rather just rigorous definitions of symbols and the operations you could apply to them. Hilbert believed you could find a rote procedure for manipulating the symbols of his formal mathematics such that you could decide automatically whether a particular theorem expressed in his symbols was consistent, in effect putting all mathematicians out of work. This would make use of the Law of the Excluded Middle also, by assuming that such an automatic proof machine could decide if any arbitrary string of symbols was a correct theorem or was not a correct theorem (either it was or it was not, proof by contradiction accepted).

In 1900 Hilbert presented a number of questions to the international congress of mathematicians. Questions one and two were (1) was the system of mathematics he offered complete and (2) was it consistent,

A mathematical proof system, a set of axioms, is complete if any statement within its formal language may be proven or its negation proven using only the axioms. Such a system is consistent if it is impossible to construct a valid argument that is inconsistent, i.e., impossible to construct a statement from its axioms which is both true and false. Questions 1 and 2 were answered by Kurt Godel in 1930, proving that undecidable propositions may be constructed in any minimum arithmetic system).

The third question, was the system of mathematics decidable (the so-called Entscheidungsproblem) was answered shortly thereafter by Alan Turing and independently by Alonzo Church. Turing created the concept of an automatic computation machine and proved that there cannot be a general process for determining whether a given formula of mathematics in the symbolic logic of the system is provable within the system. Turing's mental concept used in the proof was rapidly developed into physical digital computers that we use today.

It remains a bizarre paradox that Turing's work, which implied that mathematics really required a mind, something not a machine, was soon used to "support" the premise that the human mind is a kind of computer. John von Neumann, by some accounts the most intelligent human who ever lived (where intelligence means the capacity to do the things measured on IQ tests, e.g., use memory and manipulate symbols, concepts, not the same thing as wisdom), helped design several of the initial digital electronic computers in the 1940's. He left notes for an incomplete book setting out his thoughts about human brains and computers.

Neumann thought of the computer, whether analog representing numbers by variable physical quantities like voltage produced by electronic circuits representing equations or by the presence of absence of a marker (digital) producing an input stream of pulses and operating on them to produce an output stream of pulses, as devices that performed arithmetic operations on numerical data under the control of logic. Oddly, von Neumann, who was well aware of the distinction between manipulation of symbols (as we mentioned regarding Hilbert and Brouwer above) and the interpretation of those symbols by a human, tacitly assumed that computers manipulated numbers, rather than symbols. Neumann assumed that brains compute (becoming part of the philosophy of mind), but offered no justification for that assertion.

Computers are designed by humans to manipulate symbols which are subsequently interpreted by humans, but it does not appear (to me) that brains perform arithmetical operations on numerical data. David Berlinski offers the analogy that some people are able to accept without consideration the thesis that the human mind is like a computer but would balk at the suggestion that the human mind is like an abacus, though the fact is that there is no difference fundamentally between an abacus and a Turing machine or the digital computers which were developed from Turing's conception. They are all mechanical devices which when manipulated by humans produce symbolic output of use when interpreted by a human. However, as physicist Lee Smolin (a theoretical physicist with contributions in the field of quantum gravity) has observed, neuroscience "is a field that is as bedeviled by outdated metaphysical baggage as physics is. In particular, the antiquated idea that any physical system that responds to and processes information is isomorphic to a digital programmable computer is holding back progress."

That is a good transition back to our original thread, the discussion of the pairs of opposites in the temporal versus the eternal (since perhaps the most fundamental dualism is the Light vs the Darkness speaking in spiritual terms). I suppose some might propose that the middle is not really excluded, i.e., that things are not really one state or another (particularly at this moment in history).





Tuesday, June 13, 2017

Natural Language Processing

I have been studying natural language processing lately (otherwise known as computational linguistics). I began with NLTK (Natural Language Toolkit), an open source natural language processing tool kit. This is a superb guide to practical computational linguistics featuring a free comprehensive textbook (which is frequently used for a single semester course in natural language processing at advanced undergraduate or postgraduate level) and software package running in a Python environment on Windows or Linux.The field covers a wide range, but an example readily available to many people these days is the process by which your smart phone accepts vocal commands from you. This involves segmenting the phonemes (the individual pieces of spoken words, nominally involving a consonant and a vowel), putting breaks in the incoming stream of language sound you make and then attempting to match those with words from a lexicon (large list of possible words).
This is no easy task, but it is followed by the even more challenging pursuit of meaning, attempting to map what you have spoken to actions the phone can take, including the object of such an action. For example, if you commanded your phone "search for Mexican restaurants in Las Cruces" the phone would look for a command in that string of sounds, a command it recognizes. If it successfully recognized "search for" then it would branch in its processing logic to objects of such a command, i.e., what you want to search for.

This would require tagging each word in the utterance (what you just said to the phone) to identify the command and its object(s). The phone would have to recognize that "(Mexican) restaurants" is the search object.

Here is a look at the result of a natural language processor tagging the text string of the utterance we are discussing (we will ignore the details of how the sounds you made became this text stream):

>>> grammar = nltk.data.load('grammars/large_grammars/atis.cfg')
>>> parser = nltk.parse.EarleyChartParser(grammar)
>>> text = nltk.word_tokenize("Look for Mexican restaurants in Las Cruces.")
>>> nltk.pos_tag(text)
[('Look', 'NN'), ('for', 'IN'), ('Mexican', 'JJ'), ('restaurants', 'NNS'), ('in', 'IN'), ('Las', 'NNP'), ('Cruces', 'NNP'), ('.', '.')]

Notice how each word in the utterance (what you said to the phone) has now been tagged with a part of speech label (which we refer to as simply "tag").  'IN' means "preposition."  'NN' means "singular noun," 'NNS' means 'plural noun,' and 'NNP' means 'proper noun' (typically the name of a person or place). These grammatical tags are taken primarily from the Brown Corpus, a landmark publication by Brown University in 1964, featuring over a million words of running text or edited prose published in the United States in 1961.

Before I ran the parser on our target utterance I had to give it a grammar (you can see that I loaded the atis.cfg grammer in the Python IDLE session above (Python is a computer programming language frequently used in science; IDLE is an integrated development environment, i.e., a windowed application that makes it easier to write and test code). The ATIS grammar, developed by Microsoft Research, was extracted from a treebank of the DARPA ATIS3 training sentences. Sentences are typically parsed into treelike structures. Well, I will see if a picture is worth a thousand words here and show you a tree parse diagram of the sentence we are working with (from a parse done later to correct mislabelling of the verb):

It does appear somewhat like an upside down tree, where the tree's root is at the top and its branches become developed as it proceeds down the page, the inverse of an oak tree rooted at the ground and branching above. A treebank (in the context of computational linguistics) is a database of such syntactic or parse trees. Such a treebank can be analyzed to discover typical patterns of syntax, i.e., the way the different parts of speech are normally organized in sentences of a particular language. For example, English sentences typically have the subject first (going left to right) and it usually is a noun or noun phrase. The predicate, the part to the right of the subject typically, is formed around a verb. We form sentences without having to think much about it, having brains that are evolved to learn and process language (I will agree with Noam Chomsky on this and may say more about it later), but it is difficult to program a machine to do this. One of the ways to construct a computer program that will parse sentences is to analyze a treebank and produce rules of grammar that describe the frequent patterns in the treebank (like subject/NN-->predicate/VP).

A context free grammar (CFG) is often used to formally present the rules of grammar in a form a computer can process. For example, S --> NP VP, which tells us that the symbol 'S' on the left, symbolizing 'sentence' can be produced by a noun phrase (NP) followed by a verb phrase (VP). There are many different ways to form sentences (understatement). In fact, that is one of the things that distinguishes human speech from animal communications (well, it used to), i.e., that each utterance is potentially unique, never previously said, created by putting together the blocks of language by the rules of grammar to accomplish the communication of a potentially novel thought. So you really want a computer to help grind through huge treebanks of sentences labelled with their parts of speech (POS) tags and generate grammar rules as much as possible (since the rules will be lengthy, i.e., many lines of the kind of production rule I just showed you above).

DARPA, the Defense Advanced Research Projects Agency (of the United States government), has done linguistics research among other things.  For example, they created the TCP/IP protocol that we use to communicate over the Internet---that protocol was designed to assure an email got to its destination even if a city or two was destroyed by a nuclear attack, TCP/IP being able to try different routes if a particular city disappears). They have also been working on true artificial intelligence (not the chicanery promoted as AI by many software folks and companies, which I won't name since I am blogging on their platform), but they abruptly went "black" on the subject after 2012, now only presenting this effort as using mammalian brain structural hints to create advanced computer chips. Their actual intent is to create true mammalian brain intelligence, which I will prove by reproducing one of their press images from 2012 (which seems to have been removed) describing the SyNAPSE project (to alarm those of you who have watched the Terminator movie series):


DARPA was interested in machine reading and other computational linguistics subjects and produced the ATIS3 training sentences which Microsoft used to produce the ATIS grammar that I gave to the Earley parser I used to analyze the "look for Mexican restaurants in Las Cruces" sentence above. The Earley parser is a chart parser that uses a set of grammar rules (as just discussed) in a dynamic programming environment, trying to predict which of the grammar rules to use next as it moves from left to right across a sentence trying to match up rules with the words and POS tags it encounters. It is important to predict which rule to use next because to simply scan through the entire CFG grammar file for each word of each sentence might take a prohibitively long time. The ATIS grammar I used above has about 5,235 lines (rules).

Well, some of you who have persisted in the grueling task of reading my entire post may be wondering if I noticed that the Earley parser mislabelled the verb 'look' in the sentence. Yes, I did. So I had to obtain a more robust computational package (I am sure I could have gotten better results with NLTK had I spent more time teaching classifiers, but I was in a hurry), my simple sentence being a somewhat unfair target, being a command to a machine and missing a subject (that being understood by most humans to be 'you,' i.e., the person or thing being commanded).

I got hold of a language processing pipeline from the Natural Language Processing (NLP) group at Stanford University and ran their coreNLP pipeline as a local server, using http post to request parsing of the target sentence from my IDLE Python console (assisted in that http post process by use of a nice Python package called Requests, advertised as 'the only non-GMO HTTP library for Python, safe for human consumption' by its ingenious author, Kenneth Reitz, and some interface Python code written by a postgrad AI student at Berkeley, Smitha Milli). The Stanford NLP software is industrial strength but did churn for a minute or two to produce a correct parse:

c:\stanfordcnlp>java -cp "c:\stanfordcnlp\*" -Xmx1500m edu.stanford.nlp.pipeline.StanfordCoreNLPServer -port 9000 -timeout 60000 -annotators tokenize, ssplit,pos,depparse,parse
...
[main] INFO CoreNLP - Starting server...
[main] INFO CoreNLP - StanfordCoreNLPServer listening at /0:0:0:0:0:0:0:0:9000

Then I started a Python IDLE session and requested service from the Stanford server locally:
 
ActivePython 2.7.10.12 (ActiveState Software Inc.) based on
Python 2.7.10 (default, Aug 21 2015, 12:07:58) [MSC v.1500 64 bit (AMD64)] on win32
...
>>> from pycorenlp import StanfordCoreNLP
>>> nlp = StanfordCoreNLP('http://localhost:9000')
>>> text = ('Look for Mexican restaurants in Las Cruces.')
>>> properties = {'annotators': 'tokenize,ssplit,pos,depparse,parse', 'outputFormat': 'json'}
>>> output = nlp.annotate(text, properties)

...and the server response was:

>>> print(output['sentences'][0]['parse'])
(ROOT
  (S
    (VP (VB Look)
      (PP (IN for)
        (NP
          (NP (JJ Mexican) (NNS restaurants))
          (PP (IN in)
            (NP (NNP Las) (NNP Cruces))))))
    (. .)))
>>>


So, we got the proper tagging and parse of our sentence. I wanted to see a tree visual of this so I laboriously manually entered into NLTK the CoNLL2000 IOB tag lines corresponding to the parse from the Stanford NLP parse:

>>> chunkTest = """
look VP B-VP
for IN B-PP
restaurants NNS B-NP
Mexican JJ I-NP
in IN B-PP
Las NNP B-NP
Cruces NNP I-NP
. . O
"""
>>> nltk.chunk.conllstr2tree(chunkTest, chunk_types=['VP', 'NP', 'PP']).draw()


...and obtained the following visual presentation of the Stanford parse:




CoNLL2000 was the 2000 Conference on Computational Natural Language Learning (CoNLL-2000). Chunking captures larger pieces of a sentence, grouping POS tagged words into chunks like VP (verb phrases), NP (noun phrases) and PP (prepositional phrases). The data file format they used was IOB, which you can see above were each line of a sentence with a word, a POS tag and an IOB chunk tag specifying 'B' for beginning of a chunk, 'I' for 'in a chunk', and 'O' for 'out of a chunk, i.e., not a recognized chunk type.' 

I had better close this post, since I am getting some strange edit behavior and may be exceeding the size limits here. Stay tuned though---I intended to talk more about what is is machines are doing when they process language.

 










Saturday, November 26, 2016

A directed graph representation of a rank 3 sudoku

Sudoku is a single-player game where the player attempts to choose entries from {1,2,3,4,5,6,7,8,9} to complete a partial assignment on a 9 x 9 matrix (rows and columns like a chess board or checkerboard). None of those 9 numbers (digits) may appear twice in any row, column or any of the nine "blocks" (nine 3 x 3 submatrices on that 9 x 9 row x column board). It should be clear that each of the nine numbers must appear once in every row, column and box (since there are nine cells in each and no number may appear twice in any of those subsets).

The Sudoku board can be interpreted mathematically as a graph: The graph will have 81 vertices with each vertex corresponding to one of the 81 squares (or cells) on the Sudoku 9 x 9 grid. Solving the Sudoku puzzle then amounts to a graph coloring problem, where the term "color" has a general theoretical meaning in graph theory, but does originate in early work done with the practical motivation of determining the minimum number of colors required to distinguish the countries sharing a border on a map. The "colors" in the case of Sudoku amount to the 9 digits that have to be correctly assigned to the cells of the Sudoku board given a partial assignment with a minimum of 17 cells filled in (that is a complex mathematical theoretical problem, the MNC problem, i.e., proving that a minimum of 17 cells must be filled in if you are providing a Sudoku game problem with only a single solution rather than several that might work). If you have not played Sudoku (I find puzzles annoying but enjoy researching why they are annoying), here is what a game might look like:

This is a very difficult Sudoku puzzle, the Will Shortz Puzzle No. 301. Just to illustrate what I am talking about, here is the solution to that puzzle (spoiler alert if you were going to try to solve the puzzle first):


Returning to the graph discussion, each vertex is then one of the 81 cells in the 9 x 9 Sudoku. The connections between these vertices are called edges. The rules of the game I gave earlier are used to define the edge connections: (1) any vertices (cells) in the same column are connected with an edge (2) any of the vertices (cells) in the same row are connected with an edge and (3) any of the cells in one of the 9 three by three boxes (you can more or less see them above in the bold lines overlaying the lighter grid) are connected with an edge. There are 810 edges connecting the vertices (cells) by these rules. This is a regular graph, i.e.,the degree of every vertex is the same. That is, the  total number of edges connected to each vertex (cell) is the same and is in fact 20. There is a mathematical equation to give that number for similar graphs of various ranks (the rank of the 9 x 9 Sudoku is 3; the number of rows or columns is therefore n x n or 9): 3n(squared) - 2n - 1. So for n = 3, the degree of each vertex is 3 squared = 9 x 3 = 27 - 2 x 3 = 6 - 1 = 27 - 6 - 1 = 20 (pardon me for not using markup language to make the mathematical notation pretty; it is late as I write and I do not want to force a MathJax download on visitors anyway). Since each edge connects two vertices, the total number of edges is then (81 vertices / 2 ) x (20 edges per vertex) = 810. The coloring problem then becomes how to assign a digit from a selection of 1 - 9 to each vertex such that no two adjacent vertices (two vertices are adjacent if they share a connecting edge, the mathematical equivalent of two cells being in the same row, column or 3 x 3 box) have the same digit.

In any case, I finally reach the purpose of this discussion, which is to share my first computer generated drawing of the vertices and edge connections of an empty Sudoku (that is, none have been assigned digits since the labels would further clutter the already dense graph). The vertices are around the perimeter of the circular graph (they are colored red, in the human sense of color, and they are much too large, but this was my first plot) and they are connected by the spider web of edges as defined above (I hope you enjoy it as much as I apparently do). You can click on the picture below to see the full size picture of the graph:



Saturday, November 12, 2016

All this time and still hearts of darkness

Around 1991 Gordon Sumner (aka Sting) wrote a song, "All This Time," about the impermanence of the things of men in relation to nature. Talking about an old town in Britain, Sumner sings:

The teachers told us, the Romans built this place:
They built a wall and a temple, an edge of the empire garrison town,
They lived and they died, they prayed to their gods, but the stone gods did not make a sound...
And their empire crumbled, 'til all that was left were the stones the workmen found
...And all this time [centuries] the river flowed in the falling light of a northern sun

The dual message of the song was that there is no meaning to life and no eternal being(s) or metaphysical basis for existence. Sumner claims that he was actually writing about the death of his own father, but he has said elsewhere that he is agnostic and believes that religious faith is dangerous. In the song he quotes from the New Testament (adding a sarcastic but amusing paraphrase) then sees "the old man [God] laughing," earlier he mocks the appearance of Christian priests then asserts that he would like to "bury the old man [God and belief in God] at sea," and asks "Father [God], if Jesus exists, how come he never lived [or lives] here?" Roman historian Tacitus writing c. 117 AD in book XV chapter 44 of his Annals recorded that during the reign of Tiberius 14 – 37 AD, the Roman procurator of Judaea, Pontius Pilatus, had executed "Christus." Jesus, the Latinized Greek version of the common Hebrew name, Yeshua or Joshua, was declared by his followers to be the expected messiah, the Greek translation for which is khristos, i.e., Latin Christus. So whatever else you may believe, you may be certain Jesus did live there at any rate. 

It should be noted that Tacitus was no supporter of Christianity and only mentioned this incidentally to reporting that the emperor Nero was widely suspected of having ordered the burning of Rome July 18, 64 AD and therefore found it necessary to fix the blame for the fire on someone else, so chose the Christians. Tacitus added that the Christians were already hated for their "abominations" and "hatred against mankind" and that though Pontius Pilatus had temporarily checked "a most mischievous superstition" (by executing Christus) it had "again broke out not only in Judaea, the first source of the evil, but even in Rome, where all things hideous and shameful from every part of the world find their centre and become popular."

Civilizations are indeed transitory though, as is human existence in this world. Rome had existed as a republic, with elected officials for about 500 years before internal conflicts and civil strife resulted in Julius Caesar being appointed perpetual dictator in 44 BC. He was followed by a succession of mostly evil absolute rulers (emperors) until the fall of the empire in 476 AD.

The Roman empire crumbled from within, the expense of constant war and general profligate spending having broken the financial health of the country and created a huge gap between the rich and the poor. The government had become completely corrupt, the Roman Senate incompetent and powerless and the   thousands of soldiers comprising the personal body guard of the emperor (the Praetorian Guard; before the end of the Roman Republic it had been illegal for military units to operate within the city) began to instead select the emperor themselves, often auctioning off the post to the highest bidder and hastening the replacement process by murdering the emperor.

The Roman people naturally lost trust in their government. The once invincible Roman military forces that had expanded the boundaries of the empire, projecting power into every area of the ancient world and making possible profitable commerce while keeping the barbarian armies at bay began to be made up of foreign troops without any real loyalty to Rome. Large foreign migrations overwhelmed the empire. By 476 AD Rome abdicated to the Germanic warlord Odoacer and that was the end (the fact that every citizen presumably had the right to bear arms did not seem to protect them from invasion by a large professional army).

I happened to read Julius Caesar's commentary on The Gallic Wars recently. Caesar was a brilliant military commander (who, as we mentioned earlier, eventually became dictator of Rome, promoted by a grateful and hopeful people) who brought most of Gaul, present day Europe south of the Rhine, under control 58 – 50 BC.

After Caesar's commentaries I finally got around to reading Joseph Conrad's 1899 novella, Heart of Darkness (I had been intending to read this since I saw Francis Ford Coppola's adaptation of the story in his 1979 film, Apocalypse Now). With synchronicity, Conrad's narrator in the story, Marlow, began talking of when the Romans had first arrived in Britain some nineteen hundred years earlier. In his commentaries Julius Caesar had written an engaging account of his invasion of Britain 55 and 54 BC. Marlow imagines the experience of the Roman legionaries at "the very end of the world, a sea the colour of lead, a sky the colour of smoke...going up this river [the Thames]...sand-banks, marshes, forests, savages,---precious little to eat for a civilized man...Here and there a military camp lost in a wilderness...death skulking in the air, in the water, in the bush...They were men to face the darkness...They were no colonists...they were conquerors, and for that you want only brute force...your strength is just an accident arising from the weakness of others. They grabbed what they could get for the sake of what was to be got. It was just robbery with violence, aggravated murder on a great scale, and men going at it blind—as is very proper for those who tackle a darkness."

Caesar had decided to proceed into Britain because he discovered that in most of the wars of Rome with Gaul the peoples of Britain had helped the Gauls with supplies and other support. In his commentaries Caesar described the Britains: "Most of the inland inhabitants do not sow corn, but live on milk and flesh, and are clad with skins. All the Britains, indeed, dye themselves with wood, which occasions a bluish color, and thereby have a more terrible appearance in fight. They wear their hair long, and have every part of their body shaved except their head and upper lip." He goes on to describe the difficulty in fording the river Thames and the Brits defense of the banks by "sharp stakes fixed in front, and stakes of the same kind fixed under the water covered by the river." Caesar pursues the local forces, who are concealed "in intricate and woody places." He remarks that "the Britons, when they have fortified the intricate woods, in which they are wont to assemble for the purpose of avoiding the incursion of an enemy, with an entrenchment and a rampart, call them a town." 

Returning to Heart of Darkness, Marlow, talking to his shipmates aboard a vessel anchored on the Thames, begins to tell them about a trip he made up the Congo River into the "heart of Africa." 

Going up that river was like traveling back to the earliest beginnings of the world, when vegetation rioted on the earth and the big trees were kings. An empty stream, a great silence, an impenetrable forest. The air was warm, thick, heavy, sluggish. There was no joy in the brilliance of sunshine. The long stretches of the waterway ran on, deserted, into the gloom of overshadowed distances. On silvery sandbanks hippos and alligators sunned themselves side by side... There were moments when one’s past came back to one, as it will sometimes when you have not a moment to spare for yourself; but it came in the shape of an unrestful and noisy dream, remembered with wonder amongst the overwhelming realities of this strange world of plants, and water, and silence. And this stillness of life did not in the least resemble a peace. It was the stillness of an implacable force brooding over an inscrutable intention. It looked at you with a vengeful aspect...

The reaches opened before us and closed behind, as if the forest had stepped leisurely across the water to bar the way for our return. We penetrated deeper and deeper into the heart of darkness. It was very quiet there. At night sometimes the roll of drums behind the curtain of trees would run up the river and remain sustained faintly, as if hovering in the air high over our heads, till the first break of day. Whether it meant war, peace, or prayer we could not tell. ...

We were wanderers on a prehistoric earth, on an earth that wore the aspect of an unknown planet. We could have fancied ourselves the first of men taking possession of an accursed inheritance, to be subdued at the cost of profound anguish and of excessive toil. But suddenly, as we struggled round a bend, there would be a glimpse of rush walls, of peaked grassroofs, a burst of yells, a whirl of black limbs, a mass of hands clapping of feet stamping, of bodies swaying, of eyes rolling, under the droop of heavy and motionless foliage. The steamer toiled along slowly on the edge of a black and incomprehensible frenzy. The prehistoric man was cursing us, praying to us, welcoming us—who could tell? We were cut off from the comprehension of our surroundings; we glided past like phantoms, wondering and secretly appalled, as sane men would be before an enthusiastic outbreak in a madhouse. We could not understand because we were too far and could not remember because we were travelling in the night of first ages, of those ages that are gone, leaving hardly a sign—and no memories....

The earth seemed unearthly. We are accustomed to look upon the shackled form of a conquered monster, but there—there you could look at a thing monstrous and free. It was unearthly, and the men were—No, they were not inhuman. Well, you know, that was the worst of it—this suspicion of their not being inhuman. It would come slowly to one. They howled and leaped, and spun, and made horrid faces; but what thrilled you was just the thought of their humanity—like yours—the thought of your remote kinship with this wild and passionate uproar. Ugly. Yes, it was ugly enough; but if you were man enough you would admit to yourself that there was in you just the faintest trace of a response to the terrible frankness of that noise, a dim suspicion of there being a meaning in it which you—you so remote from the night of first ages—could comprehend. And why not? The mind of man is capable of anything—because everything is in it, all the past as well as all the future. What was there after all? Joy, fear, sorrow, devotion, valour, rage—who can tell?—but truth—truth stripped of its cloak of time.

Marlow continues up the Congo and eventually finds Kurtz, the legendary ivory trade agent the company had tasked him with locating. Kurtz turns out to be somewhat of a sociopath, having apparently taken on a god-like role with the local natives, decapitating and mounting heads on poles among other behavior impressive to the jungle folk, for the primary purpose of extorting all of the ivory from them that could be collected and sent back up the river (it was and still is a very valuable commodity). But he was nevertheless revered by Marlow and others:

That was not the point. The point was in his [Kurtz] being a gifted creature, and that of all his gifts the one that stood out preeminently, that carried with it a sense of real presence, was his ability to talk, his words—the gift of expression, the bewildering, the illuminating, the most exalted and the most contemptible, the pulsating stream of light, or the deceitful flow from the heart of an impenetrable darkness.

Later, Marlow watches Kurtz die:

Anything approaching the change that came over his features I have never seen before, and hope never to see again. Oh, I wasn’t touched. I was fascinated. It was as though a veil had been rent. I saw on that ivory face the expression of sombre pride, of ruthless power, of craven terror —of an intense and hopeless despair. Did he live his life again in every detail of desire, temptation, and surrender during that supreme moment of complete knowledge? He cried in a whisper at some image, at some vision—he cried out twice, a cry that was no more than a breath: “‘The horror! The horror!

Marlow returns to civilization [sic] and delivers some of the dead man's personal letters to Kurtz' fiancee. In an emotional scene, the grieving woman tells Marlow, "What a loss to me—to us! To the world."  She begs him to tell her the last eloquent words of Kurtz as he died. Marlow replied, deciding he had better create a better last memory for her, "The last word he pronounced was---your name." I couldn't help but laugh as I read this, recalling Peter Eckermann's words on the death of Goethe (Johann Wolfgang Goethe, d. 1832; he is revered by the Germans, in large degree because of his noble appearance---I attempted to read Goethe's Wilhelm Meister's Apprenticeship and found it so insipid as to be impossible to continue; I grant his Faust had a potent theme, i.e., selling one's soul to the devil for power over the physical world): 

The morning after Goethe’s death, a deep desire seized me to look once again upon his earthly garment. His faithful servant, Frederick, opened for me the chamber in which he was laid out. Stretched upon his back, he reposed as if asleep; profound peace and security reigned in the features of his sublimely noble countenance. The mighty brow seemed yet to harbour thoughts. I wished for a lock of his hair; but reverence prevented me from cutting it off. The body lay naked, only wrapped in a white sheet; large pieces of ice had been placed near it, to keep it fresh as long as possible. Frederick drew aside the sheet, and I was astonished at the divine magnificence of the limbs. The breast was powerful, broad, and arched; the arms and thighs were elegant, and of the most perfect shape; nowhere, on the whole body, was there a trace of either fat or of leanness and decay. A perfect man lay in great beauty before me; and the rapture the sight caused me made me forget for a moment that the immortal spirit had left such an abode. I laid my hand on his heart – there was a deep silence – and I turned away to give free vent to my suppressed tears.

Well, I am not sure I do feel the remote kinship Conrad spoke of, but I am increasingly aware of the silent jeer of the river (Sumner's wrote “all this time the river flowed, endlessly...like a silent tear,” but it seems to me more a jeer to humanity in 2016).

Monday, April 18, 2016

Synchronicity and transcription in development

I awoke for the final time at 0839 Monday April 18, 2016 with the remnants of a vivid dream still in my consciousness:

I was seated in a waiting room with many connected chairs. It apparently was a large hospital waiting room. A man was seated next to me. He was approximately of Cheri's [Cheri is my late wife] older brother D___'s body proportions and I had the impression he was wearing black plastic glasses. I had the thought later that this could have been her second husband, the father of her two daughters, although I had never met that individual. I think I asked where Cheri was and the D___ person told me that she was in "proline" (I seemed to hear it pronounced proe-lee-in with the accent on the first syllable, but this may have been a waking interpretation of the characters that appeared clearly in the dream in my mind). He added that "only loved ones were permitted into that area to see her." This annoyed me and I dismissed it as an absurdly inappropriate qualifying comment, since I was undoubtedly one of her loved ones, if not her primary loved one. I got up and went into adjoining large rooms, pale yellow institutional cinder block construction with linoleum floors, white or light, I think. There were many people seated or moving about everywhere. I think I asked a woman dressed in 1950's era nursing uniform (starched white hat and skirt) where was the "proline area, i.e., where was Cheri. I saw Cheri in my mind's eye around this time in the dream. She was young as when I first met her in 1972, lithe, with blond hair and energetic lovely eyes. The nurse told me that Cheri had already been moved to the area for the procedure. It came to my mind that the procedure was some kind of mental surgical operation. The concept was somewhat confused, but it didn't seem abnormal to me, nor was there an atmosphere of extreme concern surrounding these events.

As I finished writing a brief note in my medical journal regarding the dream, the words of John 3:6 – 8 came into my mind, "...you must be born again." I smiled to myself, considering the Phoenix Fire Mystery, i.e., the reincarnation hypothesis. I went about my morning and got online later to search for the word "proliene," (since I thought I had heard "proe—lee—in" when I felt the word in the dream earlier) but saw nothing interesting in the Google SERP (search engine results page). Google responded with "did you mean proline" and gave me a link to some site doing lien-related legal work and the like. I dropped the subject and continued with my day. I wondered later if I should email Cheri's daughter, C___, and assure everything was ok with her and her family, but decided that this would be intruding without reasonable benefit for her (I like hearing about how she is doing, but realize I am not very relevant now).

Later I returned to the computer and resumed work I was doing on a proposed article explaining how it is that a spider "knows" how to spin a web innately from the DNA of its genome. I digressed into the detail of DNA transcription and was reviewing the DNA codons, the base pair codes for amino acids that constitute the genetic code. Suddenly the codon "CCC" and its product "proline" stood out on the page. My heart skipped a beat and my eyes widened. "CCC" mRNA codes for an amino acid called "proline." I immediately realized that the actual DNA template coding for CCC in mRNA would be "GGG," the DNA base sequence being complementary (and antiparallel, but that is irrelevant when the codon is three identical codes). Cheri-Cheri-Cheri associated with Gary-Gary-Gary (Cheri had always addressed me by Gary, my first name) it seems, cued by the dream word "proline."

After that discovery, I looked at the amino acid proline. It was curious that proline is the only amino acid that stains yellow rather than blue in chromatography (via ninhydrin), recalling the prominent yellow color of the walls in the dream. Of deeper significance, I discovered (following the dream suggestion) that proline is considered to be a phosphorylation marker, normally found immediately prior to the amino acids serine and threonine (in a polypeptide chain) to mark them as phosphorylation targets in cellular signal transduction, for example, to activate a responder which then binds to DNA to modify the expression of of a gene. Why was that of particular significance to me?

My research (for the article) the last few weeks had been concerned with development, in particular the development of the neural substrates required for the demonstration of innate behavior sequences. The DNA constituting the genome of the developing organism cannot "know" where a particular embryonic cell will be at a particular place and time, but the interaction of the DNA of each cell with the information that is generated dynamically as the embryo develops (the nonlinear evolving external and internal environment of the cell, which is in large part other cells and the mechanical and chemical influences they generate) results in a fairly consistent resulting organism (and behavioral repertoire), i.e., the phenotypic expression of genotype.

I use the term "bioworldline" to signify the trajectory a cell or other biological entity follows in the phase space consisting of all possible states of the entity. For example, with axes for time, position (three dimensionally spatially in the developing embryo) and relevant biochemical parameters (like phosphorylation target status modulating transcription status of local DNA), following the bioworldline of a cell in the multidimensional phase space so defined would be equivalent to a detailed fate mapping of the cell through the process of development. It was therefore of interest to me that my dream pointed me to another entry point in the analysis of development.

These kinds of synchronicities (odd coincidences) are for me rather like a portion of Carl Sagan's plot in his novel, Contact, where the character Ellie looks at a computer computation of the digits of pi on the suggestion of an extraterrestrial who tells her that even more advanced beings than himself have left messages embedded in transcendental numbers like pi. Out at 10e20 decimal places (10 followed by 20 zeros, a very large number of decimal places), the value of pi becomes a string of 1's and 0's which cause a circle to be drawn on a computer screen (when arranged as a suitable array of pixels, the process of which I won't describe further). That, of course, implied that intelligence had been embedded in the very fabric of reality---as does synchronicity.


Sunday, January 24, 2016

A New Thing Under the Moon

I woke at 0525 hrs today, Sunday January 24, 2016. It was dark and cold, sunrise still over an hour hence.  I was groggy, but smiled, examining as art the fragments of the dream I had been living before I awakened. It was very rare for me, cursed to relive in one form or another nightly the many years of battle (regrettably, not in the military), to have a dream that, being as it were a lovely sight, might be the work of an Immortal awake in us while we sleep, to paraphrase the Upanishads (Fifth Vallรฎ, verse 8):  An old friend of mine, a guitarist I had played music with over the years, was seated at a table in the kitchen of his family home when his father came in and sat down and smoked a cigar, blissfully. I adjusted a wall mounted grandfather clock near the table which was off-kilter, but the old man said he preferred it that way. I knew he was dead, but it seemed irrelevant. The scene changed and my late wife, Cheri, appeared, young and bright-eyed. I was very happy to see her. My old friend attempted to call to her, being her first husband, but she only had eyes for me. My reverie was interrupted as my eye caught a glimmer of pure white light from a tiny gap in the closed blinds on my western window. I crawled from my bed cautiously, old bones stiff and questionable and opened the blinds. My face was bathed with pure white light from a huge full moon hanging in almost a direct line of sight to my eyes as I stood looking to the western horizon.

I returned to my bed, lying in the moonlight and switched on the tiny Sansa Clip Zip mp3 player Cheri had given me for my birthday in 2013, choosing a song, The Wind, she had written for me almost 40 years ago. I had asked her to sing it for me in 2012 and she had delighted in recording multiple takes in the little indie recording studio I had thrown together with a computer running a digital audio workstation, leaning close to the microphone and holding the headphones close on her ears to monitor her performance in real time. I had added piano, bass and other instruments subsequently, using only an acoustic guitar track as an initial guide for the vocal. Her seductively feminine cabaret voice, the hint of grit conjuring up smoke haze in a crowded Paris bistro in the 1930's, sang to me, "and you come and you go like the wind, and you touch me like a soft breeze, seems like I never want it to end…."

I knew I must follow her song with Reflections, the lyrics written by my old friend (who had appeared in the dream also) shortly after Cheri and I had found each other again in 2009.  He and I had produced and recorded the song in 2010: "Pieces of my life come back to me, Like an interwoven tapestry--Sometimes happy, sometimes so sad; All things considered, the good outweighs the bad..." As the moon continued its descent towards the lower edge of my window I selected one final song, Claude Debussy's hauntingly beautiful 1888 piano piece, Claire de Lune, the title a reference to Paul Verlaine's poem of the same name ("Your soul is a chosen landscape where charming masqueraders...go, playing the lute and dancing and almost sad beneath their fanciful disguises...they do not seem to believe in their happiness and their song mingles with the moonlight, "au claire de lune").

Later in the day I powered up a computer I hadn't used in a while and was pleased to discover a brief novel I had read last year but had misplaced. I was unable to recall the title or the author and no matter what search terms I assembled I could find nothing of the work online or locally. Fortunately, I had apparently left a copy on another computer. The story, The Outcast, was written by William Winwood Reade in 1875, shortly before he died. It was a very unusual work and it had remained a part of my mental landscape (to the extent that mental life can be separated from existence generally). I remembered a strange tale of a man who had lost his mind following the death of a woman he had loved intensely, the man having dreamed that godlike beings had created Earth and ultimately mankind as simply one of a series of works of art presented for the amusement of their kind. The presentation had met with considerable criticism by the demigods, well, I had better quote from the book so as to assure this is not considered evidence of my own mental deterioration (the following is Reade's account of the criticism leveled against one of the demigods for the artistic presentation which is the history of Earth and mankind):

"….though the work is by no means deficient in power, and contains some original ideas, there is….a roughness of style and execution which bear the stamp of inexperience. However….it is chiefly on moral grounds we think this production ought to be condemned.

The work is simple in conception, and modest in design. We have not here as in some ambitious compositions, a number of inhabited worlds contributing each its part to the story. One system only is placed upon the stage, and the action is confined to one planet of that system.

At first the world was presented to our view as a fiery cloud. It became compressed to a Sun, which advanced through Space, rotating on its axis, and cast off certain pieces from itself like tires from a wheel. These cooled into planetary bodies, and one of them, called by its inhabitants The Earth, was the scene of the drama which we shall now endeavor to describe.

[Reade's narrator describes the evolution of geology and life on Earth then continues with the increasing evolution of man.] At first wealth, culture, and power belonged exclusively to the dominant caste, while the masses labored in subjection. But by means of useful inventions knowledge was widely diffused, and the passion for liberty entered the bosom of the people. One nation after another shook itself free from the tyranny of kings and the tyranny of priests. When class restrictions were removed, all could hope by honest labor to better their condition, and all striving for their own ends assisted the onward movement of the world. At a later period the social equality of men extinguished personal ambition, and the Welfare of the Race was the aim of those who labored for distinction. Fame could only be obtained by adding something to the knowledge or the happiness of men. Finally war ceased ; the malignant forces of Nature were subdued, vice and disease were eradicated, the earth became a pleasure garden, and men learnt to bear without repining a painless death in extreme old age.

We suppose that the moral purpose of this drama is to teach the doctrine of Improvement, and to illustrate that tendency to Progress which pervades the universe. The evolution of mind from matter, by means of natural law, shows the innate power of that tendency or force, and the efforts by which Man achieves his own comparative perfection, are no doubt intended as a protest against that habit of quiescence and content which is perhaps the natural failing of Immortals. We think that the satire on theology is wholesome and just. Nothing could be more ludicrous than to see these ephemeral beings, these creatures of a moment, building little houses in honor of the First Cause, and glibly explaining mysteries which we do not profess to understand. This may serve as a warning to certain presumptuous philosophers who fabricate theories respecting the Supreme; for how can we know that we are not in the same relative position to beings of a higher race as those pigmies we create to ourselves? At least it is certain that our intellects, great as they are, or great as we think them to be, are unable to explain primary phenomena, or to solve the problems of Cause, Existence, and Futurity. So far then we go with our author ; and in numberless ways he has justly derided the follies of our race.

[However] ... it is most degrading that these men who are made in our image, who in their exterior form and mental faculties partly resemble ourselves, should be suffered to retain both in body and mind so much of the lower animals. ...Secondly, the development of matter to mind, of quadruped to man, of savage to civilized nations, is laudable enough as an idea; but how has it been carried out? As regards the first stage of the progress, we have only to praise and admire; but how has progress been produced in the animated world ? We are almost ashamed to explain a law, which, in its recklessness of life and prodigality of pain almost amounts to a crime. In cold forethought, the Creator so disposed the forces of nature that more animated beings were born than could possibly obtain subsistence on the earth. This caused a struggle for existence, a desperate and universal war; the best and improved animals were alone able to survive, and so in time, Evolution was produced. We shall not  deny that there is a kind of perverted ingenuity in the composition of this law; but the waste of life is not less clumsy than it is cruel. By means of this same struggle for existence, man was raised from the bestial state, and his early discoveries were made.  Afterwards, ambition of fame, and later still more noble motives came into force, but that was towards the conclusion of the drama. At first, every step in the human progress was won by conflict, and every invention resulted from calamity. The most odious vices and crimes were at one time useful to humanity, while war, tyranny, and superstition assisted the development of man. we do condemn this confusion of evil and good, and maintain that nothing can be more immoral than to make crime the assistant of progress, and vice the seed of which virtue is the fruit.

Again, Death is a useful and perhaps indispensable appliance in works of this kind, but so potent a means of exciting sympathy should be employed with moderation. Now what do we find here ? The law of evolution is the law of death. Massacre is incessant; flowers, animals, and men die at every moment; the earth is a vast slaughter-house, and the ocean reddened with blood. Nor, incredible as it may seem, is that the worst. With a talent for torture which rouses our wonder only next to our disgust, the Creator has smitten the animated world, even to the insects, with numerous painful and lingering diseases, while the intellect is also afflicted with maladies peculiar to itself...What can be said for such a world? What kind of defense or excuse can there be for its Creator? It is true that he made men himself, but that does not justify his cruelty. The Supreme has endowed us with the power of producing and destroying animated forms, but so terrible a gift should not be abused. We should never forget that though these little creatures live only for a moment, they are yet sentient beings, and their torments while they last are real and intense."

At any rate, I was struck by the odd coincidence (I am always pointing out instances of synchronicity to others, that being one of the means by which humans can see that there appears to be a mystical or dreamlike connection among the events that comprise human existence---waking or dreaming) that the title that Reade's madman gave the excerpt I quoted above (something I did not recall previously) was, A New Thing under the Moon. The madman tells his doctor about the thesis he has written, saying, "I merely assert that my theory of Cause and Creation is the best that has ever been propounded. It explains all the facts of history and nature, is in harmony with science, and is supported by analogy. Above all, it is quite original; nothing like it has ever been imagined before; and though Solomon wisely observes that there is no new thing under the Sun, there may be a new thing under the Moon."

Sunday, November 1, 2015

Of Rio Grande Sunsets, Valkyries, Eleusinian Mysteries, and Druids

On October 21, 2015 1742 hrs MDT I took the following picture out my window looking to the west out across the Rio Grande Rift (the crust of the North American continent began breaking apart about 40 million years ago in a roughly south to north line running from El Paso, Texas up through Las Cruces and Albuquerque, New Mexico into southern Colorado; the Rio Grande river began to take advantage of this rift valley beginning about 780,000 years ago, alternately cutting deeper into the land and filling it back up with sediment deposits). [Text continues following the photo.]



























The color of the sunset suggested fire to me. The unusual three clouds in the center left of the photograph suggested something supernatural. I thought about the sun sinking into the fire and had in mind somewhat the archetype of the ebb and flow of life in myths involving night and day, winter and spring. It began to come to me that the sun was sinking to temporary imprisonment and would be reborn in the morning, the goddess of the dawn. The three clouds at left center seemed to be flying beings and it was impossible for me not to think of them as Valkyries (the fact that Wagner's Ride of the Valkyries was playing on my stereo at the time seemed a hint as well).

To maintain some thematic or linguistic unity (with the Old Norse valkyrie), I decided to refer to the setting sun as Ostara, the Germanic referent to the constellation of concepts of dawn goddess, morning star, hearth and sun Proto-Indo-European linguistic concepts (e.g., Aurora, Roman goddess of the dawn, Istara in Anatolian dialects, Ostara/Eostre Germanic mythology). The Valkyries were the choosers of the slain, selecting who might die in battle and who might live, taking their chosen to the afterlife hall, Valhalla. In around 1870 Richard Wagner composed his Ride of the Valkyries as part of the Ring Cycle, where Wagner sought to unite the arts of music, poetry and drama to powerful effect. Interestingly, Wagner had in mind a magic fire concept at the close of Die Walkure (the opera containing the famous Ride of the Valkyries) where "Wotan surrounds Brunhilde with shrieking flames in order that their terrors may deter cowards from waking her" (p. 425 A Popular History of the Art of Music, by W. S. B. Mathews).

So, with a brief tale taking shape in my mind and the context seeming to be Old Norse/German, I decided that I should use a form of alliterative poetry common in the ancient literature of those peoples, basically each line consisting of two half-lines or verses with a pause in between, two accented syllables more or less per half line, the first half line alliterates (repeats same beginning word sound)  on the two stressed syllables and with the first stressed syllable in the second half line. The following four lines came to me:

Fall Ostara must to fire; fast the valkyries;
Select they the slain; shine they evermore
Thus done are things; thereafter shown and sung
Onward not the One; Only before, nothing more

The first two lines above I have already discussed (as to their semantic referents). Line three above is drawn from an account of the practice of the Eleusinian Mysteries in Ancient Greece, intended to "elevate man above the human sphere into the divine and to assure his redemption by making him a god and so conferring immortality upon him" [Nilsson, Martin P. Greek Popular Religion "The Religion of Eleusis" New York: Columbia University Press, 1947. pages 42–64, cited in a wikipedia article on the Eleusinian Mysteries]. The Greater Mystery participants walked to Eleusis along the Sacred Way and are believed to have taken part in rites inside a great hall (the Telesterion), rites which may have involved "things done" (a dramatic reenactment of the Demeter/Persephone myth, recalling my earlier mention of death/rebirth seasonal cycles), "things shown" (the display and viewing of sacred objects) and "things said" (secret commentaries that accompanied the things shown). I synthesized these ideas in line three above.

Line four above comes from research I was doing in June of 2015 regarding an old Druid song the Viscount Theodore Villemarque published in 1845 in his Barzaz-Breizh collection of old Breton songs he had spent his life compiling. Villemarque laboriously recorded by hand in notebooks the oral tradition songs in very obscure Old Breton dialects he heard from the local peasants in Brittany. This was an old Celtic region, sharing tradition and people with Cornwall, across the channel. The Druids were Celtic mystics, the Celtic word derouyd signifying interpreter of the gods, or one who speaks from the gods. They were suppressed by the Romans and later the Christian church, but some of their songs have been transmitted orally for 20 centuries or more, often as instructive or entertaining chants for their young. Villemarque got the melody and lyrics for one of these Druid songs, known variously as "Les Series" or "Ar Rannou," from a young peasant of the parish of Nizon whose mother had taught it to him ("to form his memory"). My gut feeling as a musician about the score he wrote after hearing the young peasant sing it to him makes me believe the melody he published in 1845 is probably close to the old Celtic melody that would have originally been known pre-Christian era in Brittany/Wales/Cornwall. It is given in the key of Bb, appearing to be used as the G minor scale (the relative minor of Bb) in the song melody given. These lyrics/thoughts have survived the 20 odd centuries embedded in a number chant (the lyrics comprise a dialog between a Druid instructor and his charge), a not uncommon way for the common people to keep alive complex history and thought without writing, i.e., the lyrics given do seem to be Druid era thoughts and memories. To return to the origin of line four in my alliterative verse tale above, the first verse of Les Series/Ar Rannou translates to English from the Old Breton as:
Druid: My pretty, my white child of the Druid, pretty one, what do you want? Of what shall I sing?
Child: Sing to me the cycle of the number one, until I have learned it for today.
Druid: There is no cycle for the number one, only the unique need, Ankou the bringer of death, the father of pain, nothing before, nothing more.

That became my line four above, Onward not the One; Only before, nothing more.