## Monday, May 10, 2021

### Mathematics, quantitative attitude distinguishes the human civilization from animals

People have replaced Nature's random "trial and error" method with something a bit more pre-planned, and calculations are needed for those civilized plans

Mr Huml (from the Ice Age) was explaining to Mr Mach that it was a cold age and people had a lot of cold. They often caught a mammoth and ate it with a dill+cream sauce.

In the text about the limits of the ancient Greek "science" and "technology", a commenter named giant claimed to argue that one didn't need any science in the modern sense – founded by Galileo – because one may mention a counterexample, namely bronze. He added:
Somehow people figured out how to make an alloy, without understanding atoms, crystalline structure, temperatures, no chemistry at all.
Cute. Indeed, people could make bronze without any of these sciences. But that's because "making bronze" isn't scientifically difficult – surely not to an extent that would be comparable to the science that is needed to build and/or optimize a steam engine (with a gearbox).

What is bronze? It's an alloy composed of (mostly) copper and also some other metal, typically tin (12-12.5%). However, the percentage is this precise because the Wikipedia editors added this number long after the Bronze Age. ;-) That's a percentage where the alloy holds together. In the Bronze Age, not only the folks didn't know about the 12-12.5% interval. In fact, the tin was often not tin at all. Bronze may also be a copper alloy with aluminium, manganese, nickel or zinc; or arsenic, phosphorus, or silicon. And these other materials were indeed mixed with copper during the Bronze Age (arsenic was actually common there).

So what "making bronze" meant was to "create and/or maintain fire" and "melt and mix a few metals". That's it. When the result was good enough – bronze, the alloy, is better because it is a harder material – they used it. And the probability was high enough to "succeed". When you look at the actual behavior of the Bronze Age people without any added nonsense, you will see that the creation of bronze was no harder than eating a mammoth with dill (or a dill+cream sauce, as a Czech cartoon claimed). You just mix two things – which you eat or use or melt separately, anyway – and you find out that the mixing is good enough, too.

I don't know whether other animals are doing anything comparable but if they could cook a mammoth or melt a metal separately, they could also mix two such things. Assuming that you are OK with getting no engineering degree and no Michelin Star, there is nothing hi-tech about bronze or a mammoth with a dill+cream sauce.

Mixing two things doesn't depend on any difficult science – or any "hard science" as a terminus technicus. Animals could mix things. And little babies love to mix things even though they don't understand anything yet! When the probability of getting something yummy is high, they may repeat it. However, mixing metals into something that may be called "bronze" is more likely to lead to some improvement than the probability that by mixing some mechanical or electric pieces, you get a steam engine or a smartphone.

Nature has had billions of years to mutate organisms and evolve new species. Most mutations end up being crippling OR approximately neutral. But some of them end up being qualitative improvements. Those are rare, Nature needed lots of time to get from the simplest bacteria (which weren't the simplest ones!) to homo sapiens, indeed. And humans repeat this "trial and error" in their individual lives; and in the life of whole tribes or human communities.

But up to some level, what humans and humans communities were doing wasn't too different from the lives of other animals. Animals also respect some "most general rules of capitalism", namely that they are willing to do something to survive or eat or eat more or eat better or have sex and other things. But the human race is superior, isn't it? I kindly ask readers who are skunks fighting for the equality of skunks and humans to stop reading now. I don't want any stinky skunks here! OK.

The moment when the behavior of individual humans or human communities becomes "qualitatively better" is the same moment when people start to count or describe and plan things quantitatively. They may have various assets, like marten skins, and they want to have a greater number of those. They start to distinguish 1 skin from 2 skins or 5 skins... and that is needed for trading and the basic forms of barter economy. The economy gets more complex, the number of possible "levels of wealth" grows from the set of numbers 1-5 (marten skins) to real numbers (mass of gold, money), whatever.

Also, people start to measure time more accurately. The distinguish the stages of the day-and-night cycle; and the seasons. They count seconds and hours; days and months; years. That is the temporal part of the four-dimensional geometry (where the units of time, like one second, are helpful). Of course, people have also mastered the 3D space, the units of distance (one meter is a popular choice today), and their powers which are units of area or volume. Weight was later discovered – along with scales. Two objects could be said to be equally heavy. An object could be equally heavy to "N standard objects" and these objects could have been called "pounds" or "kilograms", and we could quantify the weight.

The ancient civilizations have mastered the quantitative approach to time (clocks, calendars); and to space (3D geometry which was used to build structures on Earth and to observe the celestial bodies, too: well, only angles were measured there because the distance to celestial bodies is effectively infinite). Later, the other SI units were added when the people mastered the weight; electric charge and other units of electromagnetism; and the units such as kelvin, mole, candela. Lots of derived units have existed, people have quantified many other variables which are products or ratios of others, I don't plan to review the whole history of science (and the present status of science).

But it's totally crucial for the human civilization that we have mastered some quantitative (and in this sense, mathematical – but it is not the same thing) analysis of things and processes. And this switch to the quantitative thinking probably occurred at the very beginning of "civilizations". Well, back in February, I defined a civilization differently:
A civilization is a conglomerate of lives of humans that live together who occasionally create things and values that continue to live after the creators die.
A human life is ephemeral, like an animal's life, and that's true even if people live in communities. But civilizations have some "added value" which is more than the "sum of the parts" – because people preserve something that survives their lives. That was my definition of a civilization from the text about capitalism. My new definition clearly isn't equivalent because mathematics isn't equivalent to capitalism. However, they may unsurprisingly produce a similar moment for the "beginning of a civilization". Why? Because you really need mathematics to protect values that survive the individual people's lives. To say the least, some damn calendar is pretty needed for making some things "more eternal". I think that according to both definitions of a civilization, "making bronze" could easily be placed before the birth of civilizations – while the steam engines do need a civilization to be born with a sensible probability.

And a small thought experiment - I am not 100% convinced that you need SR/GR knowledge to construct a satellite and rockets to put it in orbit. So first GPS satellites failing due to time dilatation could have been the event leading to discovery of SR/GR.
Right, I often discuss this experiment. Of course the GPS satellites could have been placed to the orbit without knowing the corrections from general relativity. Someone would find out that they didn't work accurately. The location would be incorrect and the deviation would grow by a few miles per day. People could find out an empirical rule to approximately fix this mistake. They probably wouldn't understand why the correction is needed. They could discover the general theory of relativity from the direct observations... of the corrections needed to make their GPS work accurately enough!

Yes, such modifications of the history of science and civilizations are generally possible. The chronology of scientific discoveries and technological inventions that has materialized in our real world isn't the only possible one. The events could have been permuted a little bit. But we are only talking about events that took place sufficiently close to each other (or if two events don't really depend on each other much, you may exchange them even if they took place a long time after one another).

However, "knowing some basic mathematics" and "not knowing any mathematical or quantitative concepts at all" are two very different situations. You simply cannot have a space program (and even more primitive things) without the people who really know how to analyze things quantitatively. The reason is that if you just shoot some rocket somewhere pretty much randomly, in a random direction, speed, other parameters, the probability of a "success" is basically zero. If you want some men to walk on the Moon, you need to compute the speeds and directions rather accurately, and then launch the rocket. In fact, a huge number of parameters must be planned and pre-computed to succeed, including the amount of fuel in various parts of the rocket, the thickness of the walls, lots and lots of other quantities. They are real numbers and require some precision (often much better than 1%) for them to work (surely to find the Moon). The probability that you will smoothly land on the Moon accidentally, by using the method of trial error, is effectively zero.

Engineering and many related civilized activities simply go well beyond the random enough "trial and error" method. Well, science and technology are still following the method of "trial and error" but only some unknown parameters (and qualitative properties) of the new theories or gadgets are "variable". There is a lot of things (properties of the rockets and physical theories) that are not arbitrary at all. They are calculated because people know that they must be calculated to obey some conditions for them to be promising at all! The number of things that are constrained in this way – for engines or theories to work at all – is large and increasing.

Advanced quantum field theory and string theory are quantifying additional things that a person who is unfamiliar with state-of-the-art physics – e.g. an annoying "critic of string theory" – doesn't understand at all. They are quantifying deviations from the lowest-derivative terms in the Lagrangian of the effective field theory. They are quantifying variables that look like the black hole entropy or an estimate of the number of compact dimensions and other things. A person who doesn't understand high-energy theoretical physics of 2021 hasn't even had the idea that such things may be or could be or should be quantified. Many properties of a viable theory are extremely constrained and calculable – but you need to understand the things at a certain level to take all these constraints into account.

If you don't understand these constraints or the correspondingly level of your "theory", you are destined to send random pieces of crap, powered by random burning things, to the air. You will simply never place men on the Moon in this way; and you will never understand any elementary particles' interactions beyond those that you can directly observe. Many answers are extremely constrained, may a priori be points in a finely divided higher-dimensional space, and the probability to get the right answer by "trial and error" is therefore basically zero.

Not only scientists and engineers but also "civilized humans" (and "human civilizations") in a very general sense differ from animals and animal communities (and primitive humans) by quantifying many things that were previously not quantified, by a mathematical approach to things. This approach allows them to plan their acts – and design engines and theories – that would be almost impossible to be planned or designed by the most primitive "trial and error". The probability of doing things right would be too tiny.

We are really touching the topic of "evolution" vs "intelligent design". Amazing engineering features are observed in the animals (and even plants) in Nature. Could they have arisen from the "slightly filtered" method of "trial and error"? Yes, they could because the natural selection is powerful and billions of years are long. But one must realize that by producing intelligent humans (and perhaps even some clever enough animals and mechanisms in them), Nature allowed the method of "trial and error" to be replaced with something that is capable of some planning, of some "intelligent design". A funny fact is that even if the human brain were a result of a completely random "trial and error", it is capable of doing things that morally contradict the "just trial and error" picture of evolution simply because humans are intelligent and may calculate and plan things – so that the probability of their success is vastly higher than it would be with the "totally random trial and error".

And yes, I think that it is right to say that humans aren't the first creatures that exhibit this trait. The reason is that the term "totally random" doesn't have any universal meaning. You always need to describe what is the probability distribution that you actually talk about – and different models, species of organisms, engines, theories... may pick different distributions. The word "random" has several (or infinitely many) meanings and one needs to distinguish them. It may mean a qualitative description of "everything that isn't quite determined or deterministic". But within the non-deterministic things, it may mean the "maximally random" outcomes following from some uniform probability distributions. But those are basically impossible for infinite-measure sets (they couldn't be normalized to unity) so you must always add some information about the measure.

I realized that this is a minefield that may get many people confused when I was looking at the sophisticated proofs by the most noted "Intelligent Design" theorists in the world. Some of their arguments against Darwin have argued that "there was not enough time for XY mutations to produce this or that trait of organisms". I noticed two main bugs of this argumentation – in both ways, we might say that they were clever aspects of Nature whom they have underestimated.

One of them was they consider things like the "complex eye" to be irreducible. It is only useful if many different parts are born together, and this is extremely unlikely to occur simultaneously, so someone had to design the complex eye. This argument is wrong for the complex eye because even the separate or imperfect parts of the eye increase the probability that a life form survives. But this is just one loophole that doesn't kill all ID arguments against Darwin.

In other scenarios, I needed something else, too. They found a period in which the mutation rate was too slow to allow some quick adaptation to new conditions. However, one assumption in that argument was unjustified and almost certainly wrong: namely the assumption that the rate of mutation was a universal constant (for a class of similar species). Well, Nature may be clever and when it's better for the survival of the "gradually transforming species", it may speed up (or slow down) the rate of mutations. How does it happen? Well, the rate of mutations is a variable that depends on the DNA (and perhaps some other genetic-like information), too. Some of us mutate a bit more quickly than others. When the faster mutations lead to something useful for the species, the "faster mutaters" become "more likely survivors" and the population of survivors is increasingly dominated by the "faster mutaters". The species as a whole adapts – and the mutation rate and other quantities are adapting to maximize the survival chance. It's important that Nature isn't repeating exactly the asme all the time. It is actually changing all things, evolving species, and modifying things like the rate of mutations, too.

This point is somewhat analogous to the birth of the mathematically literate human civilizations. A collection of advances in the brain were needed to make this step possible. At some moment, the evolution of such smart monkeys was just a matter of "trial and error". An alternative was to evolve chimps into something that has some traits from pigs (we actually have them as well, but that is another story). But once the mathematically literate humans were born, Nature saw that it was a good idea and these early human mathematicians started to do things in a more pre-planned way – more cleverly. Because of this advance, the assumption of the "random trial and error as expected from stupid animals" was never a good assumption for biology and ecology again. Some people were simply smarter than the pigs (many are not, sadly).

Too bad. Just like the braindead New Left incorporates the anti-capitalist delusions of Marx and similar drunkards and Old Left crackpots, they are also embracing something that was too nutty even for Marx, a hatred for mathematics and the quantitative approach to "problems" in general. 2+2=4 is so racist and sexist etc. Well, the previous sentence is an extreme form of this lunacy but the contemporary mankind is drowning in unintelligent opinion makers who are bragging that they are disoriented by mathematics (and that's apparently how things should be!) and instead of being punished, they are actually benefiting because billions of mediocre (but, recently, unjustifiably self-confident) people are similarly innumerate. The New Left is reversing the basic advances of the human civilization – according to all the major definitions of the human civilization. They are an existential threat for the human civilization.

And that's the memo. (I don't plan to proofread it and fix errors because the economy of this work doesn't add up, sorry.)