MATH IS A LANGUAGE WHOSE WORDS ARE NOT JUST THOUGHTS MADE OF SETS OF OBSERVATIONS, BUT COMPLICATED UNOBVIOUS LOGICAL SYSTEMS, Endowed With High Dimensions:
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.
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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.
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Patrice’s DIMENSIONAL POWER OF CONCEPT THEORY:
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.
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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.
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Math beauty, the beauty of neural networks. Neural networks give us power, and we find that beautiful…
HIGH POWER CONCEPTS HAVE HIGH DIMENSION:
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
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[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…)
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[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.
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[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…
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[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…
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[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).
Patrice Ayme's Thoughts 