Goofy Sicence: Kurzweil's Singularity and Evo-SETI

KURZWEIL'S SINGULARTY AS A PART OF EVO-SETI THEORY

Author:

Maccone

Abstract:

Ray Kurzweil's famous 2006 book “The Singularity Is Near” predicted that the Singularity (i.e. computers taking over humans) would occur around the year 2045. In this paper we prove that Kurzweil's prediction is in agreement with the “Evo-SETI” (Evolution and SETI)” mathematical model that this author has developed over the last five years in a series of mathematical papers published in both Acta Astronautica and the International Journal of Astrobiology.

The key ideas of Evo-SETI are:

1)

Evolution of life on Earth over the last 3.5 billion years is a stochastic process in the number of living Species called Geometric Brownian Motion (GBM). It increases exponentially in time and is in agreement with the Statistical Drake Equation of SETI.

2)

The level of advancement of each living Species is the (Shannon) ENTROPY of the b-lognormal probability density (i.e. a lognormal starting at the positive time b (birth)) corresponding to that Species. (Peak-Locus Theorem of Evo-SETI theory).

3)

Humanity is now very close to the point of minimum radius of curvature of the GBM exponential, called “GBM knee”. We claim that this knee is precisely Kurzweil's SINGULARITY, in that before the Singularity the exponential growth was very slow (these are animal and human Species made of meat and reproducing sexually over millions of years), whereas, after the Singularity, the exponential growth will be extremely rapid (computers reproducing technologically faster and faster in time).

But how is this paper structured in detail?

Well, first of all (Part 1) we describe what the GBM is, and why it reflects the stochastic exponential increase that occurred in Darwinian evolution for over 3.5 billion years. Please notice that the denomination “Geometric Brownian Motion” (taken from Financial Mathematics) is incorrect since the GBM is NOT a Brownian motion as understood by physicists (i.e. a stochastic process whose probability density function (pdf) is a Gaussian). On the contrary, the GBM is a lognormal process, i.e. a process whose pdf is a lognormal pdf.

Next (Part 2) we compute the time when the GBM knee occurs (i.e. the time of minimum radius of curvature) and find what we call the knee equation, i.e. the relationship between t_knee, ts (the time of the origin of life on Earth) and B (the rate of growth of the GBM exponential. This equation holds good for any time assumed to be the Singularity time, either in the past, or now, or in the future.

Then (Part 3) Ray Kurzweil's claim that the Singularity is near becomes part of our Evo-SETI Theory in that t_knee is set to zero (i.e. approximately nowadays, when compared to the 3.5 billions of years of past Darwinian evolution of life on Earth). This leads to a very easy form of the GBM exponential as well as to the discovery of a pair of important new equations:

1)

The inverse proportionality between the average number of Species living NOW on Earth and B, the pace of evolution. In other words, it would be possible to find B were the biologists able to tell us “fairly precisely” how many Species live on Earth nowadays. Unfortunately, this is not the case since, when it comes to insects and so on, the number of Species is so huge that it is not even known if it ranges in the millions or even in the billions.

2)

More promising appears to be another new equation, that we discovered, relating the time of the origin of life on Earth, ts (that is known fairly precisely to range between 3.5 and 3.8 billion years ago) and the average number of living Species NOW.

Finally, the mathematical machinery typical of the Evo-SETI theory is called into action:

1)

The Peak-Locus Theorem stating that the GBM exponential is where ALL PEAKS of the b-lognormals running left-to-right are located, so that the b-lognormals become higher and higher and narrower and narrower (with area =1 as the normalization).

2)

The Shannon ENTROPY as EVOLUTION MEASURE of the b-lognormals, more correctly with the sign reversed and starting at the time of the origin of life on Earth, that is rather called EvoEntropy.

3)

After this point, one more paper should be written to describe… how the b-lognormal's “width” would correctly describe the “average duration in time” of each Species (before the Singularity) and of each COMPUTER Species (after the Singularity)…

4)

…but this is “too much to be done now”, and so we leave it to a new, forthcoming paper.

Evo-SETI SCALE to measure Life on Exoplanets

Evo-SETI SCALE to measure Life on Exoplanets

Author:

Maccone

Abstract:

Darwinian Evolution over the last 3.5 billion years was an increase in the number of living species from 1 (RNA?) to the current 50 million. This increasing trend in time looks like being exponential, but one may not assume an exactly exponential curve since many species went extinct in the past, even in mass extinctions. Thus, the simple exponential curve must be replaced by a stochastic process having an exponential mean value. Borrowing from financial mathematics (“Black–Scholes models”), this “exponential” stochastic process is called Geometric Brownian Motion (GBM), and its probability density function (pdf) is a lognormal (not a Gaussian) (Proof: see ref. Maccone [3], Chapter 30, and ref. Maccone [4]). Lognormal also is the pdf of the statistical number of communicating ExtraTerrestrial (ET) civilizations in the Galaxy at a certain fixed time, like a snapshot: this result was found in 2008 by this author as his solution to the Statistical Drake Equation of SETI (Proof: see ref. Maccone [1]). Thus, the GBM of Darwinian Evolution may also be regarded as the extension in time of the Statistical Drake equation (Proof: see ref. Maccone [4]). But the key step ahead made by this author in his Evo-SETI (Evolution and SETI) mathematical model was to realize that LIFE also is just a b-lognormal in time: every living organism (a cell, a human, a civilization, even an ET civilization) is born at a certain time b (“birth”), grows up to a peak p (with an ascending inflexion point in between, a for adolescence), then declines from p to s (senility, i.e. descending inflexion point) and finally declines linearly and dies at a final instant d (death). In other words, the infinite tail of the b-lognormal was cut away and replaced by just a straight line between s and d, leading to simple mathematical formulae (“History Formulae”) allowing one to find this “finite b-lognormal” when the three instants b, s, and d are assigned.

Next the crucial Peak-Locus Theorem comes. It means that the GBM exponential may be regarded as the geometric locus of all the peaks of a one-parameter (i.e. the peak time p) family of b-lognormals. Since b-lognormals are pdf-s, the area under each of them always equals 1 (normalization condition) and so, going from left to right on the time axis, the b-lognormals become more and more “peaky”, and so they last less and less in time. This is precisely what happened in human history: civilizations that lasted millennia (like Ancient Greece and Rome) lasted just centuries (like the Italian Renaissance and Portuguese, Spanish, French, British and USA Empires) but they were more and more advanced in the “level of civilization”. This “level of civilization” is what physicists call ENTROPY.

Also, in refs. Maccone [3] and [4], this author proved that, for all GBMs, the (Shannon) Entropy of the b-lognormals in his Peak-Locus Theorem grows LINEARLY in time. The Molecular Clock, well known to geneticists since 50 years, shows that the DNA base-substitutions occur LINEARLY in time since they are neutral with respect to Darwinian selection. In simple words: DNA evolved by obeying the laws of quantum physics only (microscopic laws) and not by obeying assumed “Darwinian selection laws” (macroscopic laws). This is Kimura׳s neutral theory of molecular evolution. The conclusion is that the Molecular Clock and the b-lognormal Entropy are the same thing.

At last, we reach the new, original result justifying the publication of this paper. On exoplanets, molecular evolution is proceeding at about the same rate as it did proceed on Earth: rather independently of the physical conditions of the exoplanet, if the DNA had the possibility to evolve in water initially. Thus, Evo-Entropy, i.e. the (Shannon) Entropy of the generic b-lognormal of the Peak-Locus Theorem, provides the Evo-SETI SCALE to measure the evolution of life on exoplanets.

Evo-SETI?!

Evo-SETI Entropy identifies with Molecular Clock

Author:

Maccone

Abstract:

Darwinian evolution over the last 3.5 billion years was an increase in the number of living species from 1 (RNA?) to the current 50 million. This increasing trend in time looks like being exponential, but one may not assume an exactly exponential curve since many species went extinct in the past, even in mass extinctions. Thus, the simple exponential curve must be replaced by a stochastic process having an exponential mean value. Borrowing from financial mathematics (“Black-Sholes models”), this “exponential” stochastic process is called Geometric Brownian Motion (GBM), and its probability density function (pdf) is a lognormal (not a Gaussian). Lognormal also is the pdf of the statistical number of communicating ExtraTerrestrial (ET) civilizations in the Galaxy at a certain fixed time, like a snapshot: this result was found in 2008 by this author as his solution to the Statistical Drake Equation of SETI. Thus, the GBM of Darwinian evolution may also be regarded as the extension in time of the Statistical Drake equation. But the key step ahead made by this author in his Evo-SETI (Evolution and SETI) mathematical model was to realize that LIFE also is just a b-lognormal in time: every living organism (a cell, a human, a civilization, even an ET civilization) is born at a certain time b (“birth”), grows up to a peak p (with an ascending inflexion point in between, a for adolescence), then declines from p to s (senility, i.e. descending inflexion point) and finally declines linearly and dies at a final instant d (death). In other words, the infinite tail of the b-lognormal was cut away and replaced by just a straight line between s and d, leading to simple mathematical formulae (“History Formulae”) allowing one to find this “finite b-lognormal” when the three instants b, s, and d are assigned.

Next the crucial Peak-Locus Theorem comes. It means that the GBM exponential may be regarded as the geometric locus of all the peaks of a one-parameter (i.e. the peak time p) family of b-lognormals. Since b-lognormals are pdf-s, the area under each of them always equals 1 (normalization condition) and so, going from left to right on the time axis, the b-lognormals become more and more “peaky”, and so they last less and less in time. This is precisely what happened in Human History: civilizations that lasted millennia (like Ancient Greece and Rome) lasted just centuries (like the Italian Renaissance and Portuguese, Spanish, French, British and USA Empires) but they were more and more advanced in the “level of civilization”. This “level of civilization” is what physicists call ENTROPY. [T]this author proved that, for all GBMs, the (Shannon) Entropy of the b-lognormals in his Peak-Locus Theorem grows LINEARLY in time.

At last, we reach the new, original result justifying the publication of this paper. The Molecular Clock, well known to geneticists since 50 years, shows that the DNA base-substitutions occur LINEARLY in time since they are neutral with respect to Darwinian selection. In simple words: DNA evolved by obeying the laws of quantum physics only (microscopic laws) and not by obeying assumed “Darwinian selection laws” (macroscopic laws). This is Kimura׳s neutral theory of molecular evolution.

The conclusion of this paper is that the Molecular Clock and the b-lognormal Entropy are the same thing.

And, on exoplanets, molecular evolution is proceeding at about the same rate as it did proceed on Earth: rather independently of the physical conditions of the exoplanet, if the DNA had the possibility to evolve in water initially.