While writing the preceding post came the desire to write another one about the Infinite, but this time, as Domphy would say, about the second Infinite of Blaise Pascal, the cosmological Infinite. So here is for you a scientific contemplation of it. Just bear in mind that it doesn’t claim by any means to be definitive ! feel free to interact.

A few years ago, I was a physics student at the Ecole Normale Supérieure de Lyon, a French institution belonging to the specific name of “Grandes Ecoles”. With a fellow student and friend, we had taken the habit to work together and to figure out our homework assignment (most of the time on sunday eve…) in a little classroom on the top floor of the School. To give you the atmosphere there, we were all extremely nerdy, I would say, just starting to enjoy life. I was, at the very least ! We had spent 2 to 3 years of our life under incredible pressure, in a strict hierarchical system dating from Napoleon (yes, no evolution on that side, but isn’t it the systematic fate of all strict hierarchies?… !) to ingurgitate a drastic amount of Maths and Physics. For most of us we had not taken time to live at all. This beautiful evening, my friend and I were working on some thermodynamic assignment. Its was quite a fascinating topic, the law of entropy, the second principle, the third principle of thermodynamics. At some point, out of the blues, my friend tells me : “ Catherine, do you know what is the most fundamental law of physics, the most simple law of physics that has been forgotten for centuries and that explains all the evolution in the Universe ?” “ Go for it, ” I said. I was expecting some kind of a very elaborate brainy joke, as we were very used to play with, but then my friend said in a very serious tone ;

**“The most fundamental and important law of physics is that Exponentials always saturate. There is no such thing as a true exponential growth going forever in this universe. This life doesn’t accept exponentials for ever, they saturate and life moves on …”**

I was stunned. He had this look of triumph on his face, which maybe was a pure mirror reflexion of the amazement on my face. I was finding that he had been ultimately brilliant with this remark, ultimately simple and brilliant … with these simple words I felt he had enunciated indeed a very fundamental law, so simple that I had never found it in any physics textbooks.

Exponentials do saturate, sooner or later, always… everyone knows this fact, but why is it so fundamental ?

To get the context for this, lets remind ourselves what is the Exponential law. For exponential growth ( also called geometric growth), we find the following definition : “**Exponential growth** (including exponential decay when the growth rate is negative) occurs when the growth rate of the value of a mathematical function is proportional to the function’s current value.” In one word, it could be said that the exponential growth is a multiplication of change. It simply means to change at least as much as what one currently is. When we think of it, it is a lot of change, and enormous amount of change… and yet it is precisely this law which governs all the phenomena of evolution. Suppose that at time t, you are in a certain state of development. Any scale for evaluating this development will work. Then exponential growth means that at time “ 2t” you decide to grow to a fraction of what you already are, to double your development, and at time “3 t” you will triple it, and at time “4t” quadruple it,and….etc, etc,

……………until what………… ?

It is precisely the point made by my friend, until this scary growth “ saturates” and leads to a complete change of state, a macroscopic change, a cosmic change, a change at the biggest possible scale. As if the change rate was so big that the only way that nature finds ( and this, always…) to deal with exponential growth is to spread the change onto the multiplicity of all involved organisms. The most remarkable thing about the exponential law, is that it is present in all the processes of evolution, from the most simple one to the most elaborate one, and also it is the law governing the spreading of diseases, the spreading of cancer for example. For example below, the multiplication of the cells in the embryo, is one such example.

If we plot the result of the growth (Figure on the left) as a function of the time (x in the Figure1 on the left) we can see the typical shape of the Exponential. It is very felt at the beginning, as if nothing had happened at all, the curve could easily be mistaken for a constant, and then after a typical laps of time has occurred, the curve starts to get growing in a very steep and crazy way very quickly with an almost vertical growth. At this point after the typical time, it is a very “passionate” curve, doesn’t care about anything else int he world, and simply grows grows grows drastically. It is fascinating that nature has chose this very typical law of growth for almost all its natural phenomena. And the most fascinating of all, is that nature doesn’t sustain the exponential very long in it passionate state, balance is restored by moving to “another state”, most of the time discontinuously but not always. In other words,as my friends was stating, the exponential law will saturate at some point and the system move to a completely different (other) state.

The exponential moves very quickly towards the Infinite, it is the pure mathematical expression of the **Infinite as multiplicity**, as the Infinite as Evolution, and at some point it **has to** saturate, to invent a **New State**, simply because the world is finite. All is happening as if the Evolutionary drive is direction coming from the Infinite and wants to realize the Infinite in this world. In its Ultimate Urgency, Evolution “ find” that the sonly appropriate law for such a passionate surge is an Exponential, is the law of multiplication of itself, of absolute replication of its own present development, over and over and over… until the finitude of the world is catching up speed and the exponential collapses, or saturates.

And then, “sometimes”, but only “sometimes”, something quite miraculous happens… sometimes, the only way out of this situation is the emergence of the New, the emergence of a completely new state of matter, of consciousness, of Life.

If this is the true picture, the most important questions of Science should be,

**“What determines the saturation of the exponentials? Is there a systemic way to predict which states will be generated after the saturation ? is there, more humbly, a systematic way to understand whether the saturated state will be “New” or “not New” ? even more humbly maybe, is there any insights to know what is the dominate factor which determines the emergence of the New?”**

all fantastic questions that I will try to address in future posts, if you have any idea before launching on this program, you are most welcome !

Your capacity to create new posts seems to grow exponentially… That is great!

some very preliminary thoughts..:

Exponential growth seems to be the easiest way for physical (material) systems to manage/optimize their energy, when they are under desequilibrium/ put under some pressure for change .

The physical system under pressure has to move towards a new one , in which energy will be optimised to include the change .Within the system, the change is processed through a chain reaction, which can also be often measured by an exponential growth of local physical reaction. That chain reaction allows the physical system to transform itself in the most simple way into a new optimized state, which when reached , leads to a “saturation” of the exponential growth .

At a deeper level, exponential growth is also revealing the very simple and systematic way the universe is managing itself, through the principle of causality.