Posts Tagged ‘Exponential’


February 8, 2020


Abstract: What’s Math? And why does it matter?[1] Mathematics uses words denoting high dimensional concepts (defined subsequently). Those dimensions are the vertices of sophisticated logical systems. Logic itself is physics (nature), as basic as it goes. Thus mathematics is a maximally logically concentrated language which speaks of, and with, various conclusions humanity has drawn from the universe (that’s what “abstract” means: drawn away from!) Hence mathematics’ beauty, even poetry, let alone intelligence, from its enormous logical power.

Warning: Some of this essay is very basic, some on the forward edge of human understanding and will be controversial. Readers should jump harder sections. 


Mathematical concepts are hyper powerful because they are neurologically multidimensional and those dimensions are logically equivalent.

Mathematical concepts are hyper powerful because they are neurologically multidimensional and those dimensions are logically equivalent.

The power of mathematics comes from its power to abstract entire trains of thought, and more. This way is not unique to mathematics. Normal language works the same way. But mathematics is just much more powerful. As I will try to explain, the words of mathematics are much higher dimensional. 

If we say “red” (in any human language), we mean electromagnetic radiation within a more or less well defined wavelength range (which can be measured in fraction of a meter, or multiple of an atom). It doesn’t matter in which human language “red” is said: it’s always the same idea: a range of frequencies.[2]

A prehistoric man may have measured “red” as the wavelength of light emitted by blood, or bauxite, or iron oxide. Not exactly the same connotation, but the same general idea: a range of electromagnetic wavelengths. 

“Red” is a concept. So is a “parabola”: a concept too. But the second one is tied in, and it is, a much more complicated logic, with many aspects.

A parabola represents some sort of fixed equidistance, between one point, and a line. A hyperbola, some sort of fixed difference of the distances to two points. Two different subtle notions about distance. The two concepts are in turn full of corollaries and theorems: other unexpected at first sight consequences. Ellipses are the set of points whose sum of the distances to two points are fixed. Turns out that this is the trajectory of an object submitted to inertia counterbalanced by a force proportional to the inverse square of the distance to a central point

However a “parabola” is not just one concept, but many concepts, logics, so-called “theorems”. When you kick a soccer ball (or shoot an arrow, fire a missile or throw a stone), on a planet without atmosphere, it arcs up and comes down again, following a parabola (on a planet with atmosphere, the parabola shrivels a bit into a more complicated curve which can also be computed). A parabola is the set of point equidistant (same-distance) from a fixed line (the directrix) and a point (the focus).

A parabola has this profitable property: Any ray parallel to the axis of symmetry gets reflected off the surface straight to the focus.One can see the interest if one wants to concentrate (say) solar power, or conversely, have a focus of heat send back a beam of parallel heat… or parallel light, as in a lamp. if we slice through a cone, parallel to its side, we also get a parabola. The Ancients knew this. Menaechmus in the 4th century BC discovered a way to solve the problem of doubling the cube using parabolas (not just with compass and straight lines).

With such useful properties, parabolas are all over mathematics and physics, engineering and technology. A celestial body on a parabolic trajectory probably came from outside the solar system (and certainly so if it’s hyperbolic, the next conic section over…) Hence, when mathematicians, physicist, engineers brandish the word “parabola”, they actually brandish lots of elaborated logic, enough to fill up an entire book from senior high school mathematics. We are far here from a simple range of frequencies. So “parabola” is an abbreviation of thoughts.



The dimension of a mathematical concept shall be equal to the number of different neurological networks its various definitions, non obviously equivalent, but mathematically equivalent, call upon

One could object to this definition that it is subjective, that, if we were much more clever, the different definitions of a given mathematical concept would be glaringly obvious, etc. However, we have reached a level of intelligence that is enough to conquer the galaxy (if we don’t self-destruct, a big if, it’s only a question of time). So we have here a particular level of intelligence which is absolutely defined (roughly).

To further dig into the  notion of “subjectivity”: the notion of “mathematically equivalent” is different from “logically equivalent”: mathematics is, partly, a social concept. For example, mathematicians did excellent infinitesimal calculus, getting great results using Descartes Algebraic Geometry, for two centuries without a rigorous definition of “calculus” (and now we have too many notions!) This is no accident, but caused by the “neural networks” definition of mathematics. When we say that mathematical concepts are made of logical assemblies of neural networks, we are also alluding to the saying that the truth is in the pudding. This was practiced before, but not explicitly said, causing confusion. Something was clearly missing. What is mathematics? I say neural networks. Before this, the best authorities on the subject had nothing very deep to say on the subject. An example is Bertrand Russel, an authority in the Foundations of Mathematics (he found a glaring problem in the foundations of Set Theory and replaced it by the Theory of Types… launching an industry of foundations of mathematics…

As Bertrand Russell put it… well before neural networks, but I long meditated that quote, bringing me where I am:

As this essay shows, and I have long held, this quote expresses a thought which, unsurprisingly, turns out to be untrue. Why? Because it excludes the neural network definition of mathematics… which I embrace (as I created it!) it’s unsurprising, because as Russell would have been the first to admit, mathematics works, thus is, he would readily admit, true. Somehow. I show how.

Here is Bertrand more fully quoted: “Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. […] Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate.”

Explanation in a more modern language which Russell, living a century ago, couldn’t have the notion of. Neural networks don’t have to prove they are true, because, as soon as they exist, they are. Mathematics is all about neural networks, proving their equivalences, or building more with them (hence the success of category theory). Hence Russell was wrong: mathematics contains absolute truths, the truths of the neural networks which depict them. 

Bertrand Russell was on the trail which led where yours truly got.

Anyway the point here is to demonstrate, first of all, the role of mathematics in human intelligence, and how it relates to the universe.

That sort of dimensional approach can be extended to other concepts, for example love (sexual, parental, romantic, etc.; love is obviously in some sense very high dimensional… but not in the mathematical sense, because there are no rigorous proofs of the logical equivalence of the various notions of love (said logical equivalences making their own networks)… for the good and simple reason that they are often illusory or false, as they call upon different neurohormonal systems)

Each word is a theory. In normal language, as in mathematics. Neurologically, each word is a network. The concept of elephant is well-known to be made of various attributes, as described by blind men: a tail, tusks, legs like tree trunks, belly like a cave, ear like giant leaves, etc. And it eats trees, doesn’t forget, and can be tamed. So the concept of an elephant is a network.

A mathematical object or concept would often be similar, with various, widely different aspects… but they can be demonstrated to be all equivalent, modulo lots of logic. Math concepts are like the concept of elephant, with various aspects, but logically tied together: where the tail implies the tusks and the trunk, and the ears and the big feet. The number of these neurologically different aspects of one mathematical concept I call the conceptual dimension of that concept

Let me go on with my little example. “Red” is, literally, a one dimensional concept: a color is more or less red, as the frequency varies along the spectrum. Now a dimension of a function is simply described: a function, or a space, of n arguments, or n coordinates, is n dimensional. So how does the brain work? It has inputs and outputs. Inputs are known as senses. The senses are actually made of dedicated processing organs. For example the “visual area” has 17 or so processing sub-organs. Then end result, though, is that “Red” is PERCEIVED AS ONE INPUT. So we will call it ONE DIMENSIONAL. For that reason alone? Not quite electromagnetism literally demonstrates “red” is indeed a range of frequencies, it’s one dimensional in its fundamental input. 

A “Parabola” is high dimensional. Why? It is simple, a parabola has different definitions.  “Different” means that they look nothing like each other. They can be proven to be all equivalent, through a lot of mathematics and other keen observations. However, those equivalences are not obvious. Parabolas were known to have wonderful properties… for twenty centuries… before it was discovered that they described the trajectory of a projectile submitted to gravity. 

By making what he called his “War on Mars”, Kepler was able to prove that Mars followed an ellipse. However, it took another 70 years or so before newton published a more or less finished proof that Kepler’s Three Laws of planetology (including the ellipse) were equivalent to inertia plus the inverse square of the distance law. This is Newton’s greatest claim to fame (and many astronomers and mathematicians in Paris, from which came the gravitation law, would have liked to prove that… so it was not easy to do so). The bottom line is that here we have here two completely mathematically equivalent definitions and one can go from one to other, only through enormously hard work. Another definition of an ellipse, equivalent through more hard work, and that one known for 24 centuries is that it’s a particular section of a cone. 

So “ellipse”, like parabola, is a concept that is at least three dimensional: it is the equivalence of three completely distinct neural networks.   

Much mathematics consists in proving that completely different notions and approaches (different neural networks) are equivalent. For example, in differential geometry, the famous Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions of some operators on the manifold) is equal to the topological index (defined in terms of some topological data/network). That equivalence in turns includes many other theorems, as special cases, and has applications to theoretical physics.


Is mathematics the language of the universe? No. Universe don’t talk, just is. Mathematics is the smartest language of Homo Sapiens, talking about the universe in the most abstracted, thus most powerful, fashion!

Traditionally, it is said that Galileo discovered that, without air, a body would follow a parabola (artillery men had long discovered something like that was true). Galileo said: “Philosophy is written in that great book which ever lies before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.”  

And so it goes, all over mathematics. The exponential is an arsenal of theorems. The square root of (-1) even more so. To understand the square root of negative numbers means to understand the complex numbers, the “largest” field (both of the latter word are themselves mathematical concepts, that is, sets of most significant theorems).  

The word “red” is already a broad abstraction of a vast field of possibilities. But the exponential or the complex numbers, or any mathematical concept can symbolize entire logical systems. Exp and the complex numbers are actually connected by the famous equation: exp (ix) = cos x + i sinx… Where i is the square root of minus one. So, in particular, exp(i) = -1…

Introducing basic, crucial mathematics to the uncouth multitudes is necessary, as Plato himself proclaimed at the entrance of his Academy… Said multitudes absolutely need more intuitive grasp of mathematics to become cogent enough about the world to help sheperd our great leaders toward enough sanity to ensure survival of the species. Nice perspective on parabolas, and what the different coefficients thereof mean. 

Not the easiest method to solve the quadratic equation, of course, as changing variables by taking X= (x+ b/2) as new variable is algebraically irresistible and solves the equation in 4 lines or so. 

Parabolas, and ellipses (both conic sections) were central to Seventeenth Century physics.

However, in the Nineteenth century waves, rose to prominence, first with light as wave, Fourier analysis (decomposing periodic motions into sum of cosines/sines), electromagnetism. it turns out (plenty of theorems) that all these come from the exponential!

Without a thorough grasp of exponentials, phenomena such as the CO2 catastrophe, or pandemics, can only escape the understanding of the commons or god-struck politicians. Exponentials grow at an instantaneous speed equal to their instantaneous value… exactly as a bacterial colony. Most catastrophes involve exponentials. Exponentials also illustrate all sorts of decays and, glued together, the most frequent probability distributions. 


Math beauty, the beauty of neural networks. Neural networks give us power, and we find that beautiful…


All this goes meta. Example: the concept of “Coronavirus” (“Crown Shaped Virus”). Antivirals against some type of Coronaviruses act against others (Remdesivir). So what is logically connected can be collectively treated. This is why broad concepts feed intelligence, thus action power.

By this I mean (rough) equivalences of foundations themselves form high dimensional conceptual objects: Category Theory is, by itself, such an object.

Another, more practical example: Infinitesimal Calculus. Infinitesimal Calculus has many different definitions, more or less equivalent, the earliest dating back to Archimedes, and then another one, which I call the Infinitesimal Geometric Calculus developed the Buridan school in the Fourteenth Century (this is the one Newton used). The more recent definitions of infinitesimals (Robinson and Al.) are from the Twenty-first Century (2006 Karel Hrbacek). This means the field is still fully active research! More dimensions to be added!

This makes Infinitesimal Calculus, according to my definition, a very high dimensional object. Refined, high dimensional thinking was of course hated by the terroristic, mentally simplistic Roman Catholic Church. Accordingly, Infinitesimals were the subject of political and religious controversies in 17th century Europe, including a ban on infinitesimals issued by clerics in Rome in 1632! (Notice that this was long before the birth of Leibniz or Newton, to whom the creation of calculus is often erroneously attributed by Anglo-German tribalists…)

Mathematics is the language whose words are ready made sets of powerful thoughts (for example word-concepts such a “parabola”, or the “exponential” come with an arsenal of thoughts and inner logic). 

By learning to speak and think math, we learn a metalanguage, the most powerful language humanity has written, and keeps writing, whose elements belong to, and depict, the world. Mathematica and, even more, logics are the skeletons of physics, and the latter is how the world is made. To have more advanced thoughts on what the world is made of, they are not just the eyes, but the senses one can’t do without.  

One could call mathematics the Post-Prehistoric Language. [3]

In any case, mathematics is the surest, inescapable way to more powerful thinking. [4] Even the lousiest pseudo-philosophers nowadays know some more important mathematics than Archimedes itself (a truly horrendously offensive thought!)  The more advanced thinking they got imprinted with in primary school, much of it mathematical, helps to explain why even the lousiest official thinkers nowadays are smarter than the Ancients.

When communicating mathematics, one communicates with entire, high dimensional logical systems.[5] Thus the language is hyper powerful: it has huge logical bandwidth.

Patrice Ayme



[1] Plato famously interdicted access to his Academy to all non-mathematicians. The essay above explains why. Top philosophy can’t indulge mental retards too much, out of the lab, to study them. Mastery of contemporary math insures some minimum standard of intellectual capability.

By the way, my neurological network definition of mathematics shows that the Platonic world of math, out there was… all along inside Plato’s head. Or the heads of all mathematicians (including those in kindergarten…) 


[2] Range of frequencies is of course the post-Maxwell description/explanation… Now prehistoric man would have shrugged that he knew red when he saw it in sunsets, blood, bauxite, flowers… That comes down to the same excitement of the brain in the same way each time, a particular pattern: there is no logic to it.


[3] “Postmodernism” means, of course, nothing. because when was “modernism”? When William The Conqueror suggested that the Earth turned around the Sun, before freeing all the slaves of England while his friend the Abbot Berengar was suggesting that Reason was what was meant by God (to the impotent fury of the Vatican)? That was during the Eleventh Century… Whereas, “Prehistory”, defined as what was before the Neolithic (because the Neolithic is entering history, thanks to lots of archeology) is certainly a well-defined notion. Prehistoric men knew concepts such as red, as in bloody sunsets, very well. But they had little notion of parabolas… except of course, in practice, when they threw a projectile onto a prey or predator…


[4] Learning math doesn’t guarantee wisdom, especially not anti-fascist wisdom, to wit, Plato. The deplorable “modern” case being Kant. Kant started as an astronomer, a co-discoverer of the concept of galaxy. He should have stuck to that, instead of helping turn hundreds of millions of germans (over a few generations) into moralizing murder robots.

Many people are full of hatred, and they don’t even suspect it. Worse: the Zeitgeist, the spirit of the times, is to pretend that there is such a thing as good, moralizing people, bereft of hatred. A contradiction in adjecto

Philosophically, of course Kant was mostly an enslaving pre-Nazi robot as his most important characteristic, proving mathematics produces plenty of idiot savants. Nietzsche, an excellent philosopher, was no mathematician, but a philologist (a lover of logic, of the interpretation of the meaning of texts; recently the term hermeneutics is preferred because it sounds more savant)

Descartes, of course was one the greatest minds and a very astute psychologist… and used psychology to further math… by forcing math in more useful logic… something I also advocate in my stance relative to infinity! A lot of top scientists were top philosopher, having to invent new philosophy to invent new physics (Maxwell’s identification of electromagnetism and light, Boltzmann’s murky states and Poincare’s local space and time being obvious examples) And of course the Foundations of Quantum Physics are a philosophical abyss questioning time, space, and reality itself into an uncertain, not to say ethereal, medium…


[5] The dimension of a logical system is the minimal number of axioms in its axiomatics. Don’t look it up: I invented the notion. It boils down to the usual definition of dimension in a manifold (by subtracting, axioms in common).

Super Earths, Or How The Exponential Function Can Matter

April 23, 2018

We live in the times where exponentials have come to rule, as they never ruled before. Ignore at the risk of everything we claim to hold dear. As mathematically challenged Silicon Valley nerds put it, all too simplistically, the coming “singularity” looms. Simple minds do not much understanding create, though, so here a little elaboration…

An example of exponentials in action, is graciously offered by so-called “Super Earths“, giant versions of Earths, hundreds of which have been discovered in our neighborhood.

Before I get into this, a short lesson on the exponential.

The Ancient Greeks thought they knew mathematics, but they were prisoners of linear thinking (especially after the top intellectuals spurned non-Euclidean geometry and arithmetic). The exponential is the most obvious, most crucial to understand, most vital to handle example of nonlinear thinking.

An exponential is any function which grows proportionally to itself.

Our present “leaders” (Putin, Trump, Xi, Macron, etc.), and their underlings have no idea what an exponential is, and that it feeds on itself.

Civilizations get ambushed by exponentials. This is why they so often irresistibly decay: the effect is blatant, be it the Late Roman empire, Tang China, the Maya…  


Socrates:The unexamined life is not worth living“. That was HIS (wise) feeling. His own feeling. Others don’t have to share it. Actually vain, self-admiring, erroneous, hateful people detest nothing more than self-examination. They deeply dislike, hinder those, and what, promotes self-examination.

And tell me, Socrates, you who didn’t like knowledge you didn’t already have, and you thought everybody had, when did you learn about the exponential function? How can you know something that important you never even suspected existed? And, absent that tool of the spirit, you thought you could examine everything? How stupid was that? And you, out there, the ignorant admirers of Socrates and his ilk: you don’t even have the excuse to have been dead for 24 centuries! To extract you from the gutter, seize the exponential!


After discovering a few thousands exoplanets, Super Earths are, so far, more frequent than simple Earths (it may be a bias from our present telescopes, but I don’t think so…). If the Super Earth is slightly bigger than Earth, depending upon the nature of its core, its surface gravity doesn’t have to be much higher than Earth (I computed). However, the present article considers Super Earths were the gravity is much higher than on Earth…

“Super-Earth” planets are gigantic versions of Earth. In some ways, they are more likely to be habitable than Earth-size worlds: their thicker atmospheres protect them better from radiations, either from their parent stars, supernovae, gamma ray bursts, galactic core explosions, etc.. However, it would be difficult for any inhabitants on these exoplanets to access to space. At least with known, or imaginable technologies.

To launch a vehicle as light as the Apollo moon mission capsule, a rocket on a super-Earth such as (potentially inhabitable) Kepler 20b would require more than double the escape velocity.

To leave Earth (“⊕”)’s gravitational influence, a rocket needs to achieve at minimum the escape velocity vesc = s 2GM⊕ R⊕ ∼ 11.2 km s−1 (2) for Earth, and vesc ∼ 27.1 km s−1 for a 10 M⊕, 1.7 R⊕ Super-Earth similar to Kepler-20 b. Computation shows one would need a mass of about 400,000 metric tons, mostly due to the exponential demand of fuel. That’s 5% of the mass of the Great Pyramid of Giza in Egypt (still by far the Earth’s most massive monument, excluding utilitarian walls and dams).  

That means a chemical rocket there should have one hundred times the mass of one here (Apollo’s Saturn V launcher was 3,000 tons). However, that’s not a show stopper: our largest ocean-going ships are more massive than that, and a massive rocket is imaginable. So Hippke is not correct when he says that:

“On more-massive planets, spaceflight would be exponentially more expensive,” said study author Michael Hippke, an independent researcher affiliated with the Sonneberg Observatory in Germany. “Such civilizations would not have satellite TV, a moon mission or a Hubble Space Telescope.

This is of great practical interest. Research has revealed that Super Earths are abundant, and obvious targets for human colonization. They can reach up to 10 times the mass of our own Earth (after that, they retain light gases, and turn into mini Neptunes, unsuitable for direct colonization, although Pandora like scenarios are highly plausible). Many super-Earths apparently lie in the habitable zones of their stars, where temperatures can theoretically support liquid water on the planetary surface and thus, potentially, life as it is known on Earth. Although I have had reservations about this: I view the presence of a nuclear reactor inside the planet as necessary for life, since it provides with a magnetic shield, and the recycling of the atmosphere through plate tectonic, let alone continents… (Being in the water belt and the nuclear belt simultaneously is a miracle Earth’s biosphere profits from.)

This being said, it is true that some ways to access space that we potentially have, won’t happen on Super Earths. Rockets work better in the vacuum of space than in an atmosphere: super-Earthlings might want to launch from a mountaintop. However, the strong gravitational pull of super-Earths would squash down super Alps (it’s a pure application of Quantum mechanics). Super towers won’t be be feasible, either…

Using space elevators traveling on giant cables rising out of the atmosphere depends upon the strength of the cable material. The strongest (per unit of mass) material known today, carbon nanotubes, is just barely strong enough for Earth’s gravity (it is not at this point possible to imagine stronger materials, putting in doubt the feasibility of space elevators on super-Earths). Here is Michael Hippke (Submitted on 12 Apr 2018):

Spaceflight from Super-Earths is difficult:


Many rocky exoplanets are heavier and larger than the Earth, and have higher surface gravity. This makes space-flight on these worlds very challenging, because the required fuel mass for a given payload is an exponential function of planetary surface gravity, ∼3.3exp(g0). We find that chemical rockets still allow for escape velocities on Super-Earths up to 10 times Earth mass. More massive rocky worlds, if they exist, would require other means to leave the planet, such as nuclear propulsion.

Comments: Serious version of the April Fool’s idea (arXiv:1803.11384). Submitted on April 4th 2018
Subjects: Popular Physics (physics.pop-ph); Earth and Planetary Astrophysics (astro-ph.EP)
Cite as: arXiv:1804.04727 [physics.pop-ph]
(or arXiv:1804.04727v1 [physics.pop-ph] for this version)
  1. INTRODUCTION Do we inhabit the best of all possible worlds (Leibnitz 1710)? From a variety of habitable worlds that may exist, Earth might well turn out as one that is marginally habitable. Other, more habitable (“superhabitable”) worlds might exist (Heller & Armstrong 2014). Planets more massive than Earth can have a higher surface gravity, which can hold a thicker atmosphere, and thus better shielding for life on the surface against harmful cosmic rays. Increased surface erosion and flatter topography could result in an “archipelago planet” of shallow oceans ideally suited for biodiversity. There is apparently no limit for habitability as a function of surface gravity as such (Dorn et al. 2017). Size limits arise from the transition between Terran and Neptunian worlds around 2 ± 0.6 R⊕ (Chen & Kipping 2017). The largest rocky planets known so far are ∼ 1.87 R⊕, ∼ 9.7 M⊕ (Kepler-20 b, Buchhave et al. 2016). When such planets are in the habitable zone, they may be inhabited. Can “Super-Earthlings” still use chemical rockets to leave their planet? This question is relevant for SETI and space colonization (Lingam 2016; Forgan 2016, 2017).


Pessimistically, Hippke considered another possibility, a staple of science-fiction which originated in the very serious “Orion” project of the 1950s, an apocalyptic period: nuclear pulse propulsion. It works by detonating thousands of atom bombs below a shield cum shock absorber attached to the vehicle, hurling it through space. This explosive propulsion has much more lifting power than chemical rockets, and might be the only way for a civilization to leave a planet more than 10 times Earth’s mass, Hippke (naively) said.

However, slaying the radioactive dragon he himself brought up, such a nuclear-powered spacecraft would pose not only technical challenges but political ones as well, he said: “A launch failure, which typically happens with a 1 percent risk, could cause dramatic effects on the environment. I could only imagine that a society takes these risks in a flagship project where no other options are available, but the desire is strong — for example, one single mission to leave their planet and visit a moon.”

Unwittingly, Hippke then demonstrates the danger of the single mind (in this case, his!) Indeed the most obvious way to use nuclear propulsion is simply to run a liquid, even water, through the core of a nuclear fission reactor. That was tested, and it works extremely well… and very safely! It’s much less prone to failure than a chemical rocket.  On a planet with ten times the Earth’s surface, there would be plenty of space to do such dirty launches by the thousands.

Besides, it may possible to engineer absolutely giant thermonuclear PROPULSION reactors (thermonuclear fusion is easier, the larger the reactor: the exponential at work again; if we just made a fusion reactor that was large enough, it would certainly work). The radioactivity generated would be neglectable.

So we don’t have to worry about colonizing Super Earths… We just have to worry about weight (that is, surface gravity)….

But, here, now, we have to worry about all those exponentials going crazy. Last I checked, the Arctic ice was running one million square miles below its old minimum: at some point the so-far linear decrease of Arctic ice is going to decrease exponentially, as warming there is highly self-feeding (that’s why it runs already at twice the rate of the rest of the planet…).

And as usual, let’s remember what the arrogant, stupid imperial Romans never learned, and the Maya never reached: inventing completely new, liberating, energizing technologies is how, and the only way how, to break the strangulation from the ecological, political, economical and moral exponentials which smother civilizations. A most recent example is diffuse, dim light solar cells, dye-sensitized solar cells (DSSCs), a tech already in full deployment, which has just made spectacular progress in the lab.

Even language acquisition is exponential… Let alone thought system acquisition. You want to examine life, in ultimate depth? Learn to think exponentially!

The coming “singularity” looms. How to manage it? First by understanding what makes it tick, exponentials.

Patrice Aymé



March 1, 2018


Plutocracy Is Intrinsically Exponential:

A phenomenon exponentiates if it its instant rate of growth is proportional to its own value (by definition). So the bigger it gets, the faster it grows and it grows as fast as it is big. (The exponential is the most important function in math after +, x, /. In particular, once equipped with square root of (-1), trigonometric functions, so any cyclic phenomenon, can be deduced from it, and described, by it. Here we extend the exponential to morality, spirituality, intelligence…)

The paradigm of the exponential is bacterial or viral growth. The growth of a population (and it could be tumor cells, or rats) is, before running out of resource, and without a predator, or other external abating agent, proportional to said population.

One of the greatest progress of humanity, in the last five centuries, has been to develop the tools for, and build an understanding of, the exponential function. It is everywhere. Including looming as the greatest cause of civilizational collapse, moral, ecological, intellectual, epistemological, etc. For a civilization, evil is the deadliest infection of them all. It grows proportionally to its presence, so it exponentiates (we will explain why).

A particular case of evil growth, is the takeover of civilization by plutocracy. It is the main cause of the collapse of civilizations.

If left unchecked, forces of evil will rise. And, if unchecked, they will rise exponentially. Thus it’s important they are not left to rise. Thus evil power, in Greek: Pluto kratia is is not just a moral phenomenon, not just a moral implosion. It is also a mathematical phenomenon, and that makes evil not just a human factor, but a law of physics.


Proof That Evil Exponentiates:

Humanity is intrinsically good: otherwise it would not rise children, thus would not exist. Good, in first order, means you do the work, and don’t fight back.

Hence, those who don’t play by the rules of goodness, decency or common sense, get an advantage: they can exploit, and meet no resistance. That advantage is self-feeding: the more it’s used, the more advantageous the advantage it provides with.

This is observed with lionesses in a pride: some do all the work, other restrict their contribution to showing up, and eating whatever is killed by those who did the work; that’s explained because just the appearance of a big group is impactful… The lazy ones work by just showing up!

Thus, as in pride of lionesses, bad behavior already existed in prehistoric human tribes. But it was intrinsically contained, as in a lion pride: if evil exploitation is too abused, the group would collapse, and thus so would the perpetrators with it. So, there was not that much exponentiation in prehistory. However, with the rise of civilization, something new appeared, enormous power. Enormous power is the core reason for civilization: it makes it useful, it makes it dangerous. Generally a civilization’s power is translated into what is called “money”. (Objecting that crypto currencies, Inca knots and modern future derivatives are not money is silly: those apparently different media all translate in money; and even into gold, in the case of the Inca.)

  1. The exponentiation of money is well-known: it is the law of compound interest. The more money one has, the faster one’s money grows (not relatively, but absolutely). It’s basic mathematics. The same holds for anything money can buy, like real estate. That’s why anybody with serious money in the past used to be called a “rentier” (someone who enjoys a rent).
  2. However money transforms into power onto other people, and reciprocally: money and power are equivalent. To prove something with one, is the same as proving something with the other.
  3. Thus, any form of power will, left unchecked, also exponentiate, because it is readily transformed into money and other (“real”) property (which will exponentiate). One lends only to the wealthy. Not just the wealthy in money, power, but also wealthy in the capability of using evil ways. rich, (A particular spectacular example of money translating into power occurs when generals pay their armies; the history of the Mediterranean, Europe and China are full of such behavior, including when it brought the agony of the Roman Republic).

Abuse of power and abusive power cannot be checked by the love stuff, the gentleness and the low hormones vegans and fanaticized pacifists. Hence plutocracy, the power (kratos), not just of wealth, but of Pluto itself, the god of hell, the god of bad behavior, tends to feed on itself (Pluto = Hades, Angra Mainyu, Satan, Shaitan, Le Malin…)

If one analyzes what happens, it is clearly the concentration of power, in a few hands which causes the exponentiation of power.

Thus, the greater the power, the less concentrated one should allow it to become. In other words, technological progress requires ever more direct democracy. The alternative is exponentiation of evil.

Hence ever more democratization is a necessary consequence of the pursuit of civilization. Without ever more democratization, evil and plutocracy grow, until they overwhelm everything, as demonstrated in various Dark Ages (the “Invasion by the Peoples of the Sea” (33 centuries ago), the Greek and European Dark Ages being the three most famous cases).

Patrice Ayme

Note: More generally, most catastrophes tend to exponentiate, for the same reason as avalanches exponentiate. Thus one exponential loss of control, such as the rise of plutocracy, can launch others. Thus civilization collapse in the Roman Principate in turn launched a number of other catastrophes which, themselves exponentiated (for example “plagues”, which tend to happen when society is itself collapsing; three famous examples are the “plague” which destroyed Athens once the Peloponnesian war started, the plagues which devastated Rome around the Third Century, and Constantinople in the Sixth Century; the counter-example is the “Black Plague” of 1348 CE: it killed half of the population, appearing eleven years after the start of the “100” year war, however, it didn’t disorganize the European states governments, which reacted strongly, taking anti-epidemiological measures; thus exponentials couldn’t develop, and, differently from Athens, Rome and Constantinople, European society rode the plague as if nothing had happened…)


December 16, 2013

[Just a reminder, and refrain.]

Liberty, Equality, Fraternity,” allow civilization to function optimally (as they put all brains in parallel). The Greeks were rightly obsessed by them, especially Liberty, and various notions of Equality (several of which we have lost).

However, time and time again, in history, civilization collapsed into Slavery, Inequality, Hatred, Mayhem. How? Generally, through the plutocratic phenomenon. That’s when greed takes over any other inclination, to become the organizing principle of society.

90,000 pages of the tax code of the USA have been written to insure that the greedier, the less taxes one pays (99.99% of the People use only .01% of that tax code: the inverted percentages are no accident, but design).

Absent a progressive taxation that is sufficiently steep enough, the exponential growth of wealth guarantees that the plutocratic phenomenon will occur. The mathematics of interest enforces this. In simple language, the richer they are, the easier it is for the rich to get richer. (This was known throughout the Neolithic by societies which left a trace; it’s not known in the present USA, because the plutocrats have captured the collective cognition and reflection).

The plutocratic phenomenon, plutocracy, comes in two phases: first, an oligarchy confiscates most of the wealth and most of the economy. This is happening now, and is facilitated by the capture of political power. (1)

Secondly, this unjust rise of wealth for a few is accompanied by an implosion of the character of the oligarchs, down into the black hole of the Dark Side. Why? As their rule gets ever more unjust, the richer and more powerful they get, the plutocrats need ever more devious means to stay in power by force and lies. That, in turn, makes plutocrats ever more satanic. (2)

In other words, past a point, the only way plutocracy can grow, is by harnessing the Dark Side, big time. Satanism grows on itself, just as, in the early stage, wealth grew on itself.

This growth of global and overreaching Satanism is one of the (overlooked) consequences of plutocracy.

What does this mean in practice, in the situation we are?

The first thing to do is to break the enslavement of money creation to Wall Street (instead of Main Street). In normal circumstances, most of the money is created by private banks under a mandate from the government. Franklin Roosevelt knew this, and that’s why he passed the Banking Act of 1933 (3).

Secondly wealth ought to be detected: put in place a worldwide system to detect where and what all of wealth consists of. Break through all these shell companies and holdings.

Third, tax the hyper rich enough to block the confiscation of all wealth and power by just a few. Republican president Eisenhower instituted a 93% tax on the upper tax bracket. The same, worldwide, inflicted to all and any wealth above the billionaire level, would instantly re-establish civilization, and its prospects.

Patrice Ayme


Notes: A good way to get out of plutocratic service is default on debt (by the states). Contrarily to what many of my contradictors on the Internet, especially the Krugman blog, claimed, the USA did default on its debt several times (about 6 times). As usual when there is a default, it’s later erased from collective memory by semantic calibration.

For example FDR’ 1933 default, or the Argentinian or Greek defaults are called by other names. Generally one calls default “restructuring the debt”.  

A reason for the popularity of Wall Street style financing in the USA is the stability of the country, an island, just like Britain. Continental financial exchanges have a less good track record, due to invasions. The debt default problem is tightly related to this.

(1)    The seizure of an economy by oligarchs, under order from the USA, happens. This could be observed in Russia in the 1990s under Yeltsin, when Harvard academics, in a crafty plot, advised the naïve president, that a modern economy could not exist without a plutocracy.

(2)    Satanism: this is why Obama tells us that, to lower healthcare costs, one has to bring those who want to make a profit, and that’s why, when the democrats could do it, they refused to even consider enabling Medicare to negotiate the prices of drug in G.W. Bush Medicare Part D plan…

(2)’    Satanism: Just watch the NSA and the like. It’s not so much about “terrorists” that they are after, than just establishing a police state hooked on large corporations, that is, the plutocracy (which owns and control them). Don’t forget the CIA fabricated Bin Laden.

(3)    The so called “Volcker Rule” is a pale reminder of FDR’s Banking Act of 1933: it just requires banks to inform customers when they steal their money.

And now for what Dominique Deux said about the necessity to find out, where, what, how and who about wealth:

Dear Patrice, as usual your comment, far from dwelling in Utopia, is full of small seedlings which only need some nurturing and watering to bloom into perfectly feasible programs/policies. 

One such is “Break through all these shell companies and holdings.” 

Merely stopping to consider this opens huge venues. 

First of all, we’re supposed (ordered) to believe that this is a technically impossible task, and stop there. That is pure unadulterated BS. Government spying is both rife and efficient worldwide, but tracing back an oil tanker’s ultimate owner, or a holding’s ultimate paymasters, is claimed to be impossible due to that tired, low-tech trick of shell post boxes with copper plates in tax havens. Aw, come on. 

Fact is, the smart intel and law enforcement community is carefully kept busy chasing red herrings, and barred from joining that specific fray. No means, no results, it’s as simple as that. 

Year in, year out, hundreds of extremely proficient finance professionals are churned out by business schools and universities worldwide. French graduates, due to their high mathematical level, are among the best and at a premium, but they’re not the only ones by far. 

Let Governments start large recruitment programs, with a view to be as replete with financial intel capacity as they are now with anti-terrorist and economic spying capacity. This means decently paid analysts with decent career prospects. Tracing back clandestine ownership should no longer be a judge-ordered effort, using up spare capacities on a few idiots who made the mistake of getting conspicuous, but a blanket policy, with enough hardware and personnel to spare.  

The costs would be offset, by several orders of magnitude, by the obvious fiscal gains. Other advantages would unfold, some unforeseen. 

Of course the bright school and U alumni are all vying to become seven figure traders, sell swords in gilt armor for the masters of the world. Yet a two pronged approach of (a) offering decent earnings, status and government careers, along with the satisfaction of working for the common good, and (b) ruthless and exemplary treatment of the “golden boys” who somehow fall foul of justice, would make quite a lot of these bright young men (and women) think twice before selling their souls. 

(It’s a bit like prostitution: when attractive young women are given a real choice between gratifying work and selling their bodies, only a hardcore minority will keep to the latter. But the choice has to be real – no Wal-Mart till pseudo-jobs.) 

When I say “Governments” I do not mean all of them. A few relevant ones would be enough. Along with international agencies. 

You’re fond of reminding us of the greatness of WWII warriors. So, here it is: “Where there’s a will, there’s a way”.


March 14, 2013


Many racist theories fester around the “disappearance” of Neanderthals. The latest one, from Oxford University, claims that Neanderthals’ big, beautiful eyes, and their big muscles caused their demise: Neanderthals were too busy looking at things, while flexing their muscles. The “idea” is that larger eyes would have crowded the Neanderthal brain out, making them relatively stupid. How stupid can Oxford get? In particular eyes made them incapable of having social groups as large as those of Homo Sapiens Sapiens.

Big Eyes Do Not Kill

Big Eyes Do Not Kill

Sapiens girl on the left, Neanderthal girl on the right (reconstitution published in Science Magazine a few years ago).

I have long argued that the strength of democracy came from having many brains working in parallel. There is little doubt that larger social groups bring a higher cultural intelligence, hence higher individual intelligence. So I agree about that bit of logic. Yet, ironically, to reach the conclusion that Neanderthals’ social group were less numerous, the simple fact that Neanderthals were bigger, is enough. There is no need for hazardous demeaning allegations about Neanderthals’ brains.

That big eyes made Neanderthals stupid contradicts some facts that were thought to be established:

1) Sapiens Neanderthalis’ brains were significantly larger to start with. See Wikipedia.

2) Many very clever Homo Sapiens Sapiens have small brains. Famously Anatole France, an intellectual, had only a 1,000 cubic centimeters brain. Homo Floresiensis, the “hobbit” species living on the island of Flores, Indonesia, until it was wiped out recently, was extremely intellectually capable, although it had really small (and completely different) brains.

3) In the Middle East, Neanderthals and Sapiens went back and forth through the same large caves over 50,000 years. So whatever happened, it was not in evidence for 50,000 years.

So, of course, I have my own theory. That’s what philosophy is all about: trying to guess what really matters most, and how that most significant data logically articulate. Then scientists, politicians and writers can swoop, figure out the details, and attribute themselves the glory.

What could have happened by around 28,000 years ago that caused the demise of Neanderthals? At the time, the last fierce glaciation was gaining ground. (It reached its maximum 25,000 years ago.) Some have argued, absurdly, that the Neanderthals could not take it. That’s beyond silly, as Neanderthals had evolved, from half a million years ago, precisely to handle extreme cold.

Neanderthals were stocky, powerful, and they had thrived through hundreds thousands years of glaciation, mostly on a meat diet, hunting big game. But they also knew how to cook plants, and eat them.

27,800 years ago, Cave Bears were exterminated. That huge animal who lived in caves, primed real estate Sapiens Sapiens and Neanderthals craved for. Could the disappearance of Cave Bears be logically linked to the disappearance of Neanderthals? Yes. That’s a consequence of my theory. More advanced technology played a direct role. So did size: Cave Bears disappeared, because they were larger than European Brown Bears (called Grizzlies in America), according to the mechanism below, differential exponentiation.

How did men kill Cave Bears? With technology. We do not know exactly what weapons men had at their disposal. However, technology had improved, and kept improving. Recently it was found that Sapiens Sapiens (Homo SS; I hope one gets the joke) in Africa had invented bows and arrows 80,000 Before Present (BP).  (About 60,000 years earlier than previously thought!) Before bows and arrows, the propeller had been invented, and was used in Europe. The propeller took advantage of angular momentum to send a sort of mini lance further and stronger than by hand.

Why did the Neanderthals and Denisovans (another human species from Central Eurasia) lose their edge? Advancing technology is the obvious answer. When technology of clothing and weapons was sufficiently advanced, the physiological advantage that the Neanderthals genetically had, disappeared. Homo Sapiens Sapiens could thrive just as well through winter.

At that point, Homo Sapiens Sapiens from Africa could be as successful as the Neanderthals through the freezing wastelands of Europe. OK.

But the Homo SS outbred the Neanderthals, so they became genetically more successful. How do I explain that?

Simple. However, the explanation involves the exponential function, the same function found all over, and that the mathematician Rudin called “the most important function in mathematics”. The exponential also explains the plutocratic phenomenon, and that is why it’s so dangerous. The exponential always rules extinction events, that’s why one day a species is all over, like the American Pigeon, or the Tasmanian Tiger, and the next day, it’s gone.

So visualize this. Neanderthals were bigger than Homo SS, just like the Polar Bear is bigger than the Black Bear. Bigness is an adaptation to cold. Southern Europe’s Brown Bears are smaller than those found in Kamchatka, or Alaska (also known as Grizzlies: the Grizzly is an emigrated European Brown Bear!) Bigger makes warmer inside. That’s why the most massive animal that ever was, the Blue Rorqual, at up to 180 tons, is nearly twice the mass of the largest dinosaur (it’s not just that it’s floating, but also that water is cooler than Jurassic air, I hold).

To simplify, let’s use a bit of exaggeration (that’s reasoning by exaggeration, one of my preferred tactic of thought; the one humor exploits, and why joking helps thinking). Let’s assume Neanderthals were twice more massive than Homo SS (certainly, in the average, Cave Bears were twice the mass of Brown Bears).

Now let’s consider an habitat where Homo SS and Neanderthal bands roamed. They will tend not to mix, for obvious racist reasons. The racial hatred between Neanderthals and Homo SS has got to have been colossal. People who look too different are not even sexually attracted to each other (and where Neanderthals and Homo SS were in contact in the Middle East, for 50,000 years, there is no evolution of an interbred species, an indirect proof that there was no love lost there!)

The density of human mass is going to be roughly the same all over, because that density depends only upon the resources available (mostly meat on the hoof, and fur in burrows in glaciating conditions).

Thus, there would have been apartheid. But the Homo SS would have been twice more numerous, where they reigned (from my assumption of twice the mass). So now graft on this a catastrophe; a drought, a flood, a very tough winter, a volcanic super disaster, whatever. The climate was highly variable, starting about 40,000 years ago, just when Homo SS appeared. Some have stupidly argued that Neanderthals were too stupid to adapt to this changing circumstances. Like this paralyzing stupidity struck them just when Homo SS were around. My explanation is more subtle.

After a catastrophe in said habitat, say one of these numerous habitat in Europe isolated by glacial mountain ranges, or seas and lakes, most of the human population would be wiped out, Homo SS, just as Neanderthals. There would tend to be always a small remaining population, because the greatest limit on man is man himself: as a population gets wiped out, resources rebound, and life of the survivors tend to get much easier (that’s what happened in Europe after the Black Death of 1348 CE; if nothing else, survivors could ask for higher salaries from their plutocratic masters, and they did).

So say 90% of the population of the habitat was wiped out. As suddenly resources are no limited, the human population will rebound exponentially. The equation is: N(t) = N(0) exp(Rt). “R” is the “Malthusian” parameter, the rate of growth. Now it’s going to require twice the resources to feed a Neanderthal to sexual maturation (under our outrageously simplifying assumption that Neanderthals are twice the mass). Thus one may assume that R(Homo SS)/R(Neanderthal) is 2. The end result is that the quotient:

Number Homo SS/ Number Neanderthal = A exp(2t). (Where A is the ratio of the populations H SS/Neanderthal after the catastrophe.)

Thus the population of H SS would exponentially grow relative to that of the Neanderthals, resulting in a quick extinction. And in no way this is happening because Homo SS were superior. Just because they were more gracile.

Another factor is that Sapiens had a larger reservoir of population to the south of Europe and the south of North Africa, in sub-tropical Africa, so Sapiens could come up from the south in great numbers, especially when the climate was cool enough for the deserts, including the Sahara to be covered with savannah-park… an environment for which more gracile Sapiens was more suited, and in which the preceding argument about mass of the body would apply.

It is known that the climate fluctuated violently where Neanderthals lived and had evolved for. The consequence was potentially lower population expansion when exposed to invasion by Sapiens whose genetic reservoir (the South) was mostly a refuge from said wild temperature fluctuations (so, although the climate would massacre the Neanderthals in the north, the Homo SS in the south would be ready to expand much more, from a larger population, and thus expand into the north; this is a question of comparing two exponentials again, the one starting from a larger population grows faster) .

When the temperature fluctuated up, Sapiens populations could invade relatively
recently more sparse Neanderthal habitat, which had brought Neanderthals to a near-extinction event. This, accentuated by interbreeding, would have led to quick Neanderthal gene replacement (replacement, because of interbreeding).

Why didn’t the replacement of most Neanderthal genes by Homo SS genes happen before? Because the advancement of technology. That was partly led by Neanderthals, but whatever Neanderthals invented was transmitted to the larger Homo SS population, and made Neanderthal genetic advantage redundant. Moreover, the relatively smaller Neanderthal population, all other mental things being equal, would have been exposed to the Tasmanian Effect (See my enormous essay, the Tasmanian Effect, which considers various traps small populations can fall into, among other mental, and thus demographic disasters).

On top of that, technological advances insured that life was becoming possible for Sapiens in Eurasia. As Sapiens encroached, Neanderthals kept on living in more difficult, fluctuating places, thus propagating the extinction.

Hence the mystery of the evolution of contemporary man is smoothly explained. Just a bit of math. QED.

Europeans & Asians: Not Just African

Europeans & Asians: Not Just African


Patrice Ayme


Note 1: what of the mentally deliquescent and racist article in the Proceedings of the Royal Society? First, they sank so low as to using orbit size as a proxy, that Neanderthals had larger visual systems than contemporary AMH [Anatomically Modern Humans]. That’s about as intelligent as saying that, because special forces use night vision goggles, they have got to have bigger visual systems.

The main woman author also found the same physiological feature, bigger eyes, in the past, about people presently living at high latitude. She contentedly asserted that, because light levels are lower in the north, people living in the north (40,000 years at least for Homo SS) have bigger eyes. Amusingly, she did not draw, in that case the conclusion that Norwegians and the English are therefore more stupid. Somehow, though, in her lack of smarts, she applies that controversial reasoning to Neanderthals. Does she have giant eyes?

Seriously the Oxford study rests on a central fact that contradicts one of established facts about Neanderthals. Indeed it claims Neanderthals’ brains were not any larger than Homo SS.


Note 2; what catastrophes am I talking about? Well the climate fluctuated wildly, to start with. Second, A Campanian ignimbrite volcanic super-eruption around 40,000 years ago, followed by a second one a few thousand years later, certainly crashed Neanderthal populations (based on logic, and evidence from Mezmaiskaya cave in the Caucasus. Mitochondrial DNA analysis of a specimen there is C14 dated 29,000 years BP, one of the latest living pure Neanderthals). After such a catastrophe, the exponential rebounds of populations would have advantaged Homo SS, as explained above.


Note 3: OK, I exaggerated with the mass ratio. (Mathematicians often do this, considering an exaggerated case to understand the mean, through the tails.) But the real mass ratio would be aggravated because, Neanderthal was built in such a way, relative to gracile Homo SS, that they consumed more calories per day (some paleontologists have come up with 300). So there is no doubt that the effect above will play a role, even if the mass ratios were not as bad. Notice the mechanism above would tend to extinguish the Neanderthal traits that were most characteristic of the subspecies.


Note 4: A preferred trick of Neanderthals’ haters is to exhibit Archaic Neanderthals‘skulls, and compare them to those of modern men. The skull of an Archaic Neanderthal of 400,000 years ago should not be compared to a modern human, less than 40,000 year old! All the more since Neanderthals’ brain size augmented faster than the brain size of Homo Sapiens Sapiens.


Note 5: NEAR-EXTINCTION THEORY: WHY DINOSAURS DISAPPEARED: SAME MATHEMATICS! Part of the mechanism above generalizes for other species in competition. It provides with a disappearance mechanism after ecological turbulence, according to species’ ecological footprint. The reasoning can be generalized to other species’ extinctions. Let’s recapitulate the preceding, while generalizing it:

1) it is hard to transform a near extinction event into total eradication (see the Black Plague of 1348 CE). Indeed, the more the extinction, the easier it gets for the survivors, as resources rebound (this is similar to the famous lynx-rabbit oscillation).

2) However, larger animals (Neanderthals, DINOSAURS), or animals with a higher metabolic load (Neanderthals) are going to be to be left behind exponentially, during the rebound phase.

In the case of Neanderthals the periodic catastrophes could have been of climatic origins (waves of cooling, warming and unstable climate as the earth underwent various tipping points, one way or another, into the occasionally severely glaciated period between 60 K and 11 K BP. A severe volcanic catastrophe or two would have added near extinctions episodes.

In the case of dinosaurs, the massive Deccan eruptions, over millions of years, culminated with the most acute episode, more or less contemporaneously with a massive asteroid impact (!). According to the exponential extinction theory above, the back and forth of near extinctions would have put a severe extinction pressure on the dinosaurs and the like, as smaller, more efficiently active mammals and birds would have put huge pressure on dinosaurs and flying reptiles (same in the sea).

T Rexes had to grow by three kilograms a day, for years (same for Triceratops, etc.). A huge energetic demand on the land… While smaller mammals could go through generations, adapting to changing circumstances…

By eating dinosaurs’ eggs to start with, mammal and bird population would have exploded very fast back up at any relief in the hyper volcanism catastrophe. And the more they rebounded, the greater the pressure on dinosaurs (This would have happened in addition to other extinction pressures, such as cooling.)


Note 6: A first reading of the ideas above may lead one to wonder why it is that small species do not overwhelm big ones, when they are in competition. But, in normal circumstances, one has an equilibrium ecology, the equivalent of equilibrium thermodynamics. the effect above does not apply. The effect above, exponential extinction, occurs only during non equlibrium ecological dynamics, as found during near-extinction events (hence the importance of near-extinctions). It’s the equivalent of non equilibrium thermodynamics (when Prigogine suggested the latter, he was viewed as nuts; until he got the Nobel Prize). An example of this situation would be a proximal super-nova eruption showering Earth with radiation.

Artificial Turf At French Bilingual School Berkeley

Artificial Turf At French Bilingual School Berkeley

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