Computers for SETI , Kurzweil’s SI NGULARI TY and Evo-SETI Claudio Maccone IAA Director for Scientific Space Exploration, IAA SETI Permanent Committee Chair, Associate, Istituto Nazionale di Astrofisica (INAF, Italy) E-mail : clmaccon@libero.it Home Page : www.maccone.com UK SETI Research Network, Manchester, UK, March 2-3, 2017
700-pages BOOK about “Mathematical SETI”
TALK’s SCHEME Part 1: GEOMETRI C BROWNI AN MOTI ON (GBM) Part 2: Darwinian EXPONENTI AL GROWTH Part 3: 2006: «THE SI NGULARI TY I S NEAR» Part 4: Merging GBM, SI NGULARI TY & Evo-SETI Part 5: Peak-Locus Theorem & EvoENTROPY
Part 1: GEOMETRI C BROWNI AN MOTI ON (GBM)
GEOMETRIC BROWNIAN MOTION (GBM): exponential mean value
WARNING !!! GEOMETRIC BROWNIAN MOTION is a WRONG NAME : This stochastic process in NOT a Brownian Motion since its probability density function is a LOGNORMAL, and NOT A GAUSSIAN ! So, the pdf ranges between ZERO and INFINITY, and NOT between minus infinity and infinity !
GEOMETRIC BROWNIAN MOTION (GBM): exponential mean value : ( ) ( ) − = B t ts ≥ L t e for t ts . GEOMETRIC BROWNIAN MOTION b_lognormal probability density : 2 ( ) σ − 2 t ts ( ) ( ) − − − ln n B t ts 2 − ( ) σ − 2 2 t ts e ( ) σ − = GBM_pdf n B ; , , t ts . π σ − 2 t ts n
Part 2: Darwinian EXPONENTI AL as a GBM in the number of LI VI NG SPECI ES
TWO REFERENCE PAPERS ► A Mathematical Model for Evolution and SETI ► Origins of Life and Evolution of Biospheres (OLEB), Vol. 41 (2011), pages 609-619.
► SETI, Evolution and Human History Merged into a Mathematical Model. ADVISED ► International Journal of ASTROBIOLOGY, Vol. 12, issue 3 (2013), pages 218-245.
Darwinian EXPONENTIAL GROWTH ► Life on Earth evolved since 3.5 billion years ago. ► The AVERAGE number of Species GROWS EXPONENTIALLY: we assume that today 50 million species live on Earth. ► Then:
Darwinian EXPONENTIAL GROWTH exponential mean value curve in time = starting at t ts with a value of 1 (RNA?) : ( ) ( ) ( ) − ≡ = B t ts L t m t e L = − ⋅ 9 ts 3.5 10 years Origin of Life on Earth, ( ) = ⋅ 7 m 0 50 million species=5 10 speciesTODAY, L ( ) ( ) ( ) ⋅ 7 − ⋅ ln 5 10 ln m 0 16 1.605 10 = = = L B . − ⋅ 9 3.5 10 year sec ts
DARWINIAN EVOLUTION is a GBM in the increasing number of Species Evolution as INCREASING NUMBER OF SPECIES 1 10 8 × Number of LIVING SPECIES on Earth 9 10 7 × 8 10 7 × 7 10 7 × 6 10 7 × 5 10 7 × 4 10 7 × 3 10 7 × 2 10 7 × 1 10 7 × 0 − − − − − − − 3.5 3 2.5 2 1.5 1 0.5 0 Time in billions of years
Part 3: 2006 «THE SI NGULARI TY I S NEAR» book by Ray Kurzweil
THE SINGULARITY IS THE KNEE OF THE GBM EXPONENTIAL Number of CIVILIZATIONS since 10 billion years ago 1 10 8 × Number of CIVILIZATIONS in the Universe 9 10 7 × 8 10 7 × 7 10 7 × 6 10 7 × 5 10 7 × 4 10 7 × 3 10 7 × 2 10 7 × 1 10 7 × 0 − − − − − − − − − − 10 9 8 7 6 5 4 3 2 1 0 Time in billions of years
THE SINGULARITY IS THE KNEE OF THE GBM EXPONENTIAL In other words: BEFORE the SINGULARITY , the Darwinian Evolution is very SLOW. AFTER the SINGULARITY , the Computer Evolution is very FAST.
THE SINGULARITY IS THE KNEE OF THE GBM EXPONENTIAL In other words still: REPRODUCTION in Darwinian Evolution is very SLOW. REPRODUCTION, among computers is very FAST.
THE SINGULARITY IS THE KNEE OF THE GBM EXPONENTIAL In other words still: REPRODUCTION in Darwinian Evolution is very SLOW. REPRODUCTION, among computers is very FAST.
THE SINGULARITY TIME IS NOW = = t t 0 GBM_knee SINGULARITY or just a few decades from NOW, i.e. the same when compared to 3.5 billion years of Darwinian Evolution
Part 4: erging GBM, Mer SI NGULARI TY, & Evo-SETI
KNEE EQUATION : relates the time along the GBM exponential when the knee occurs t_ GBM_knee , to the time of the Origin of Life ts , and the exponential increase rate B ( ) ln 2 B = = − t t ts GBM_knee SINGULARITY B
KNEE EQUATION : is derived in the paper by finding the radius of the osculating circle to the GBM exponential and setting to zero its derivative.
MERGING THE TWO EQUATIONS = = 0 t t GBM_knee SINGULARITY ( ) ln 2 B = − t ts GBM_knee B YIELDS: ( ) ln 2 B = ts B
FINDING B FROM ts ( ) ln 2 B = ts B AND THE KNEE-CENTERED GBM EXPONENTIAL : Bt e ( ) ( ) ≡ = L t m t L 2 B
THE AVERAGE NUMBER OF LIVING SPECIES TODAY IS 1 ( ) ( ) ≡ ≡ = 0 0 0 L m m L 2 B BUT BIOLOGISTS ARE VERY UNCERTAIN ABOUT THIS NUMBER: MILLIONS OR BILLIONS ?
IMPORTANT NEW EQUATION relating the time ts of the Origin of Life on Earth to the current number m 0 of Species living on Earth. − ts ( ) = ⋅ 0 log 0 m m 2 1) if ts = -3.5 Gy then m 0= 132.4 M 2) if ts = -3.8 Gy then m 0= 143.1 M
Part 5: Peak-Locus Theorem, b-Lognormals ENTROPY & COMPUTERS AFTER THE SI NGULARI TY
PEAK-LOCUS THEOREM (PLT): ► The Peak-Locus Theorem (PLT) shows that the “Running b-lognormal” always has its PEAK ON THE DARWINIAN EXPONENTIAL, and is more and more peaked as long the time increases.
PEAK-LOCUS THEOREM (PLT): ► The Peak-Locus Theorem (PLT) shows that the “Running b-lognormal” always has its PEAK ON THE DARWINIAN EXPONENTIAL, and is more and more peaked as long the time increases.
ENTROPY( p ) of the Running bLog ► The Shannon ENTROPY of any probability density is ∞ ( ) 1 ( ) ( ) ∫ = − ⋅ H f x log f x dx . X X ln 2 −∞ ► The Shannon ENTROPY of the Running b- lognormal is a function of its peak time p and of its µ and σ : ( ) 1 1 ( ) ( ) = − π σ + µ + H p ( ) ln 2 p p . ( ) ln 2 2
EvoENTROPY( p ) of the Running bLog ► Because of the Peak-Locus Theorem, ENTROPY reads ( ) 2 1 B 1 = + − + H p ( ) ln 2 B B p . π ln(2) 2 2 ► Then EvoENTROPY of the Running b-lognormal is a function of its peak time p and of B and ts : ( ) − B p ts ( ) ( ) ( ) = − − = EvoEntropy p H p H ts . ( ) ln 2
EvoENTROPY of the Running bLogn. is the MOLECULAR CLOCK EvoEntropy of the LATEST SPECIES in bits/individual 30 EvoEntropy of the LATEST SPECIES in bits/individual 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 − − − − − − − 3.5 3 2.5 2 1.5 1 0.5 0 Time in billions of years before present (t=0)
CONCLUSIONS We developed here a new mathematical model embracing all 1) of Darwinian Evolution (RNA to Humans) and SINGULARITY (i.e. computers taking over humans) happening NOWADAYS. 2) Our mathematical model is based on the properties of lognormal probability distributions. It also is fully compatible with the Statistical Drake Equation, i.e. the foundational equation of SETI, the Search for Extra-Terrestrial Intelligence. 3) Merging all these apparently different topics into the larger but single topic called “Big History” is the achievement of this paper. As such, our statistical theory would be crucial to estimate how much more advanced than Humans the Aliens would be when SETI scientists will succeed in finding the first ET Civilization.
Thank you very m uch !
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