The Annual Netanyahuite Lecture: What is Learning

A fainting lecture delivered by the Netanyahuite philosopher, father of the school of philosophy of learning, who was finally invited to answer once and for all the question: What is the "learning" whose name he always bears (and twice in every sentence). In an attempt to preserve the spirit of the oral presentation, the transcript of the lecture was edited into boring paragraphs as long as the exile, reflecting the feverish and oxygen-deprived manner of speech of the deceased, which leads to early brain death

Enthusiastic reactions in the hall (source)

What is learning? Like any complete philosophy, the philosophy of learning must answer the question of what learning is using its own tools, otherwise it will fall into contradiction. That is, one must answer the question of what learning is through learning - learning what learning is. And here the logical problem is already reversed - not contradiction, but circularity. Therefore, the question that immediately arises when we come to deal with learning is: how do we learn what learning is? That is, in learning, we quickly shift from the question of what to the question of how. It is enough for us to know how to learn, and this will already be an answer to the question of what learning is, because learning is not an object or action, but a way.

But what is a way at all? The invented object of the how. How do we learn what learning is? This is the same question as how do we learn, and if we answer it we will answer it as well. If so, we have replaced what is learning what is learning what is learning what is... etc. with how to learn how to learn how to learn how... etc., so what have we done? Have we done nothing and just returned to the starting point?

Well, we have completed a full circle, but have we gone backwards? The problem is not in the shape of the circle, every complete philosophy is circular, and its purpose is to close a complete circle - a worldview that stands on its own. The problem is in a regressive circle, which goes backwards, to infinity, as opposed to a circle that moves forward. The direction of movement in the circle is the determining factor. This is the difference between logical circularity and learning circularity, which is an infinite loop, as in programming. That is: a loop that operates again and again and progresses in the world.



Hence, learning is not logic (in a side note for the advanced, we'll add: in logic this inference is not valid, but should be reversed, but here we are learning what learning is). In logic, for example, if you repeat the same sentence, it has no meaning - but in learning it does. In logic, repetition is just that, while in learning, repetition and memorization are essential. And if we repeat this idea again, in a different way, we will learn it better. Therefore, logic is foreign to the brain, it is a machine from its perspective, while learning is our mode of operation. When something encounters its own mode of operation - it is difficult for it to define it, and in fact it cannot imagine another mode of operation, but only simulate it, that is, recreate it as a machine. Therefore, we do not understand logic, but can only operate it and use it. We will never think in mathematics, and even the greatest mathematicians do not think in proofs, but in learning - they learn mathematics (even proofs can only be learned). Logic always progresses or goes backwards, while in learning one can progress by going backwards in a circular repetition (even in an argument), because although both involve construction, in logic the construction goes backwards to first principles, while in learning the construction goes forward - we start from somewhere and continue. It cannot be justified, it is unidirectional by nature. Nothing can justify its own mode of operation, because even this justification is in its own mode of operation. A computer cannot prove the laws of logic (neither can mathematics), and therefore it is always built on first principles. Not so with learning, where there are no axioms to return to, but starting points to continue from, and there is no meaning to going back to the beginning in learning - this is an illusion (or simulation). One can only learn forward. In the brain, a thought always causes the next thought, and you have no way to think twice in the same brain, because the brain itself changes. Learning itself changes itself. And therefore, although we have not progressed logically here, we have learned here. The fact that the brain is a sequence of changing thoughts does not mean that we cannot progress, on the contrary, it means we can do only one thing: progress. Because this change is not logical (which would mean there was no progress at all here, just jumps), but it is the progress of learning. And even when a computer learns, it simulates learning using logic, and this simulation indeed learns, but it cannot logically base itself. Because logic can go backwards, and its essence as construction is that it can be traversed in both directions, and checked. While learning knows only one direction: forward. Am I repeating myself? Great. That's how you'll understand. In learning, even repetition advances, because there is only one direction. We repeat forward, not backward. The circle rotates forward.



The unidirectionality of learning and progression in the brain actually stems from the unidirectionality of time. If there was a time machine, could we go backwards in learning? Could we turn into logic machines, able to return mechanically to the previous stage? No, because going back in time would only be to run the film backwards, not to change it. Learning does not stem only from the direction of time flow, but from its unidirectionality, that is, from its being exceptional (compared to spatial dimensions) in being one-dimensional. For there to be real change in learning - and in its progressive circularity - we would need not a time machine but two dimensions of time. And this we cannot grasp at all, unlike additional spatial dimensions. In fact, the whole idea of going back in time and the desire to go back and choose differently is an anti-learning fantasy. "If only we had known" in Chekhov - but if you went back in time and didn't learn, how would you know? (Even the physical problem is probably not going back in time but the return of information in time, in a way that violates learning). And this is also the difference between the childish Christian return to Eden, before the sin, versus the Jewish idea of learning-based correction, after the breaking.

On the opposite side, if we were to remove one dimension from time, and turn it from a line to a point, learning would also be nullified, and the world would pass beyond our perception, and become pure information - a hard drive (or hologram). For if there was no time in the world there would be no learning, only a given state. And even if in that world there was a magnificent logical structure that captures the whole world, for example all of mathematics was written as given (this was the Platonic fantasy) there would be no learning here (this is what Aristotle understood. And therefore also added concepts of time). Why does the lecture take time, why is it long, why do you need to sit here and listen to me, or sit and read it? Because we have no way to grasp this text itself other than in linear reading, in time, that is, other than through learning. We cannot absorb an entire book at once, grasp or think about it simultaneously, and insert it into us all in parallel, but only slowly unravel the thread and follow it, until the book ends when it all turns into one long thread, which was our performance - our reading - of it, that is our learning (therefore everyone reads differently, because one can learn differently).

And if we could read a book like that, it would be like a computer, that is, to copy it into us, as information - and not learning. The text would become pure information, and lose meaning for us. We would not learn anything from all this abundant information, because it would not change us, that is, change our learning itself, unless afterwards within us we would do another process of learning, that is, of linear reading along it. As if a child learned the Kaddish or the Zohar by heart as a code in Aramaic, and then as an adult who learned Aramaic he deciphers it - and actually reads it for the first time within himself. The touch of the divine in religion stems precisely from this relationship to the text as information (pure and therefore sacred and not human), as God has no dimension of time and perceives the world as whole and given. Hence the infinite attempt to learn the text, and to return it from the transcendent timeless dimension to human, Jewish time.



And if we return (again!) to logic, unlike logic, learning does not engage in the cult of the source, but in the cult of originality (or Jewish innovation), that is, in the search for the next stage - the progression on the path - so that it will not be a logical continuation that is routine and expected, and walking on the same boring path. It seeks the turn in the path, and hence the interest of learning: interest. Interest is what lies ahead of it, and not at its base behind it, the future attracts it more than it is built on the past. Hence the human mathematical instinct (or mathematical curiosity) - to find learning precisely in logic, that is, to overcome logic, using its most unexpected, original and difficult places, and thereby transform logic into learning (of course, from a human perspective). Hence mathematics is a learning decomposition of logic (or in less mathematical language: a learning construction of logic). That is, mathematics is a learning digestion of logic using the brain as a learning machine, as opposed to the computer as a proving machine. Therefore, proofs using computers do not satisfy us, because we have not really learned. We not only want to know the proof as information, but to be able to learn it - and therefore learn from it. Memorizing proofs by heart is not learning, and therefore in learning mathematics there is a need for many exercises. Therefore, our fear of the computer stems from its being a logical machine, but in order for it to have intelligence it must also become a learning system. An entire philosophy book can be logically trivial (and in fact all mathematics is such) but its power is what it teaches - a new learning method.

And if we are looking for originality - and we would never have written this text (or read it, in both senses) unless it was trying to innovate, that is, to teach us something new - we must (unlike in logic) ask what it innovates - to justify it. And not ask on what basis it innovates, where philosophy is very shaky, always, because it cannot be logic, and is not supposed to be, but learning. The stupid hammer for logic is analytic philosophy, or Spinoza's proofs, which do not understand that philosophical construction is learning - not logical (Spinoza's propositions are interesting - the proofs are not). Therefore, contradiction in philosophy is not a disaster - but boredom is. In learning there can be contradiction (for example between Torah and science or between behavioral systems), but not in mathematics. The brain can contain and live contradictions - and even different learning systems. So if so, what is the innovation here? What is the difference between saying that our learning deals with the question of how and not what - and therefore always only demonstrates (and does not define) learning (unlike logic which defines) - and the Wittgensteinian definition of meaning as use?



Well, the Wittgensteinian investigation itself is a way of learning, which he demonstrates again and again (but of course does not define). He too can be grasped through learning, and not just the opposite - grasping everything through language, in the dictatorship of the philosophy of language. But the main thing here is that we are not dealing here with linguistic meaning - but with learning meaning, and therefore the how is not how we speak but how we learn, that is, it is not looking for the usual meaning (in which we make use) but for the one that advances us, that has some innovation in it.

Are we essentially basing the entire concept of learning on the specific question word "how" as the basis of the definition, with different question words (for example: why, how, what, whence, etc.) being the basic building blocks? No, because the word "how" has many meanings in language, and most of them are not learning-related (how is the lecture going? Excellent, no one came). But we are not dealing with the specific word, but with one specific and non-trivial meaning of it, the most appropriate and advanced in terms of learning, in which the how seeks how learning is done. The purpose of the idea of how in our learning is not lexical, and not linguistic, that is, there is no definition here, but connecting learning to another idea, because learning never seeks the final step, but one more step to progress. The meaning of sentences in any theoretical text is never to reach some exhaustion of meaning (for example in its first definition or final conclusion), although there are texts that pretend to be such, for example as logical, but every word advances us one step in our learning. If we ever reached the final ultimate meaning, the matter would be sealed and useless, because the use of an idea is always adding meaning, that is, not use that leaves it as it is, but innovation. An idea is a way to continue learning, something that shows how to take the next steps, and enables them (and this of course includes all of Wittgenstein's own sentences, and their importance in their innovation - and in their method precisely).

Therefore, philosophy never leaves anything as it is, but opens it to progress, by showing a way of learning. The stuckness in philosophy is when a new way of learning is not found, and then it remains only as a method, that is, learning becomes mechanical and loses its original vitality, until finally it sometimes really dies. We have interest in the ideas of the Greeks only because we have not exhausted learning from them, and not because of their truth, and on the other hand we have completely lost interest in scholasticism, because we have not found a way to learn from it, and not because it is nonsense (that's how mathematical fields die - or flourish). So when we ask how we learn, we will remember that we are trying to progress in learning, and not in language. Otherwise we will not get out of the previous way of learning. And indeed who is interested in progressing in language, apart from in artificial lexical or philosophical analysis - what really interests the brain is always to progress in learning. Philosophy has always been stuck on what is right and wrong, when what matters is what is interesting and what is boring. If, in a certain learning path, we have an interest in truth, it is only because from truth we can progress. In mathematics, for example, falsehood is a contradiction and leads to the annihilation of learning, hence the problem with contradiction, because from it follows that everything is correct and learning dies. Not because contradiction is invalid in itself, for some logical or transcendent reason - and indeed in other methods it is not invalid (or not with the same sharpness), after all, the brain is not a logical machine - but a learning machine.



If so, if we have abandoned the idea of defining "what is learning", what are we left with? If we cannot get out of learning, because this is our own mode of operation, and therefore we cannot look at learning from the outside and define it from the outside, what can we learn about it from the inside? First of all, every serious philosophy provides a boundary beyond which one cannot cross. But learning does not draw this boundary from within, but constantly expands it. It is a constant struggle against the boundary - from within. If we succeeded in drawing the boundary of learning once and for all, that is, learning to the end what learning is, we would lose its meaning as learning, and it would become a mechanical algorithm. Therefore, from the perspective of a being with higher intelligence than us, our learning can appear, from the outside, as non-learning, just as we can look at the learning of a fly, computer or virus as a mechanical mechanism. Learning is learning only from within. Therefore, we would not learn what learning is if we deciphered the algorithm of the brain, because one can only learn from within, when one does not know everything, and learning exists only from the perspective of inside the system. To learn, one needs not to know - God cannot learn. Only if we operated the algorithm of the brain - as opposed to deciphering and knowing it - could we learn (and perhaps faster than the brain). And what would we learn if we deciphered it? Not what learning is - but how to learn.

Therefore, if we were able to grasp and understand finally all the operation of the brain to the end, it would no longer appear to us as learning but as a machine - but there is no real concern here for the loss of our learning. Because the truth is that a system cannot know how it itself learns without reaching an infinite regression. As in the paradox of Achilles and the tortoise - when the tortoise is Achilles' brain - when Achilles learns how the tortoise learns, meanwhile the tortoise will learn how Achilles learns that the tortoise learned, and then Achilles will need to learn how the tortoise learned that he learns, and so on. Each time there will be a rise of one level in the level of method, and the method of the method, and the method of... etc., which is possible, but it is not possible to jump the whole ladder and reach the sky, to some final and highest method - there is no ultimate method. From the perspective of any learning system, there is simply no last method up there (otherwise it is a machine, which is actually defined as: having a defined and finite method). Beyond that, knowing the algorithm itself would not allow us to understand the learning of the brain (just as knowing the evolutionary algorithm does not yet allow us to understand evolution, including of course understanding the learning of the brain), because learning does not reside in the very definition of the algorithm, but in its specific application. That is: in the way of learning, which depends on previous steps, and in fact countless steps - since birth and since the beginning of culture (the beginning of collective brain learning).

A system cannot know how it learns, but it can learn how it learns, because it can progress each time one step in the regression - each additional step of Achilles following the tortoise is learning. Knowledge is the limit of learning, in the infinitesimal sense, that is, knowledge is when learning tends to infinity. If learning eventually converges (perhaps like in scientific knowledge), then one can speak of truth, and if it diverges (like in mathematical knowledge, which in principle has no limits) then in the end there is nothing but mystery, and therefore mathematics is more spiritual than physics and biology. The universe can have one final equation, and the brain can have a final algorithm, but not mathematics. Scientific learning or that of brain sciences can end, but not so mathematical learning, or literary, or Torah learning. This is precisely the difference between natural sciences and humanities, and between nature and spirit - not learning itself, but the existence of its limit, which is final knowledge. Hence biology can have an end, one can understand the human body to the end, but not evolution. And the same relationship exists between science and technology. Therefore, evolution and technology belong to the creative world of infinite learning, which is spirit. Biology includes within it the past of evolution, which can be known, but not its future possibilities, which are open to all directions and not imprinted, and therefore it is spirit and not nature. The material has an end, in principle, and the spiritual does not. Religions defined the boundary diverging to infinity as divine, and secularism claimed that there may be no convergence to infinity but just absurd divergence. And the Messiah is the limit of history, and therefore if he is finite he is the final Holocaust of Judgment Day and the end of history, and if he is infinite he is redemption, which is always the world to come. Knowledge is the final solution.



Philosophy has always erred in that it wanted to know - and not to learn. That is, it wanted to pretend to be science - when it is part of the world of spirit, and more similar to spiritual technology (the Anglo-Saxon tendency) or spiritual evolution (the Continental tendency). Why did it want to pretend to be science? Because the moment there is truth there is a direction towards which it is correct to learn, while in technology or evolution there is no direction towards which it is correct to develop, but here is the wonderful thing about learning - it does not mean that development is arbitrary. The horror of arbitrariness in philosophy stems precisely from its identification of arbitrariness in the myth that preceded it (and particularly the Greek!). Not everything is possible in evolution or technology, and therefore they are neither arbitrary nor predetermined, but a certain maturity is needed to try to move one more step forward, instead of trying to reach the end, in a leap that falls into the abyss - which is the specialization of philosophy. The goal of the philosophy of learning is to take one step forward. That is: to progress. It is aware that there will be philosophies after it, which will progress further than it. But it is not arbitrary, because it progresses from the previous steps of philosophy, and is built upon them. It does indeed rebel against the domineering and castrating father (Wittgenstein), but unlike Wittgenstein himself - it does not commit patricide. It recognizes its complete genealogy, and does not claim (like him) that it has not read Kant. It does not have the tendency (which is a fantasy) towards philosophical proof, but it certainly engages in philosophical learning. How does it do this?

It identifies previous directions and previous methods in philosophy and tries to continue one more step along the way. Each step along the way is arbitrary only seemingly, because if it was truly arbitrary there would be no path but random walking. There is nothing that forces it not to be arbitrary, but in retrospect one can see that a path was indeed created, and directions and trends can be identified, that is: it works. There is evolution and not just mutations. But what causes this to work? Why is there a path, even in philosophy? The path does not stem from the fact that it reaches and converges to truth, as philosophy tried to delude itself (all along the way). The path does not stem from a final, global direction, but from a local direction.

In fact, philosophy is not one path but a stream of paths, where at any given moment there are all kinds of philosophers, big and small, who try to continue it. The small ones continue on exactly the same path, or with small deviations, and the big ones and charlatans try to jump a step forward, and only in retrospect, from those who continued them, is the path revealed. That is, learning appears as learning only from afar, but up close there is chaos. Therefore, the canon crystallizes long after the literature is written, because it crystallizes from the literature that was already written after it. They decided what to continue, and where the path went - and where it did not go. That is, if there is no continuation to the philosophy of learning, and no additional doctrines emerge from it, then it was a curiosity and not part of philosophical learning. Therefore, being the father of a species in evolution depends not only on you, but on the continuation of evolution. But does this mean that the thing is arbitrary and random?

No, on the contrary. The depth is the understanding of where the path and trend truly continue, for the longer term, and not just the shortest. There is always a lot of superficial learning, but the one who identifies the deepest trends within the path, and continues them or responds to them, is the one who creates deep learning. That is, who not only learns, but understands how to learn, and how to learn how to learn, and so on - and at each such stage deepens further inward, into the method of the method of the method and so on. That is, the progression of learning stems from understanding the derivative, and the second derivative, and the third, and so on, and thus the next step can be bigger and advance us more, and sometimes in a real leap. Like solving differential equations by approximations. And this is the depth of the question of how: how to learn how to learn how... endlessly.

Because previous learning is just an example, and the path is a collection of examples for learning. And from each such example one can continue to many possible directions that it exemplifies, but not to every direction to the same extent (which is the postmodernist mistake, which is the loss of learning and path that Wittgenstein caused). It is not entirely arbitrary because the more economical (and therefore more principled) hypothesis created from the examples is more plausible, and the belief in this is the belief that there is indeed a path. That is, that it has a description significantly shorter than just a collection of points that make up the path - that there is learning and not just information. This perception, that there is learning and not just details, and that there is a story and not just events, and that there is a picture and not just pixels, is the human belief, which is not a superstition (or a harmful cognitive bias to religions and conspiracy theories), but a valid mathematical bias - towards learning.



Hence the tendency of philosophy to summarize reality and search for some general principle that will summarize human learning so far, as much as possible, including philosophical learning itself. Philosophy is a summary of the path - the principled path. And whoever succeeded in this became a great philosopher from whom the path continued, or he was one of its fathers, if we think of evolution as a path, and of adaptation to reality that leads to survival as internalizing the depth of reality. The next stage in evolution is not a conclusion of the previous stage but a continuation of it, but not just any continuation, but a deeper continuation of it, and therefore - more advanced than it. Therefore, true innovation does not stem from disconnection from the previous stage, from a hasty leap, but on the contrary from internalizing the previous stage not only superficially but deeply, to the method of the method of the method and so on, that is - precisely from deeper continuity, which is what allows extrapolation.

Hence the need to study precisely the history of philosophy in order to deepen philosophical learning. This is the reason that the amnestic analytical philosophy will probably be erased like scholasticism, in contrast to a little more continuity (relatively) from Continental philosophy, which is more continuous in relation to the history of philosophy. But in general, contemporary academia, which in a sad joke took over philosophy and art and significant parts of the world of spirit, is not destined to produce more than philosophical dwarfs, because of its tight methods of establishment and castration, and it is actually responsible for the degeneration of philosophical learning. We always deal with great philosophers as examples of how to learn, but it is important to also deal with small philosophers as examples of how not to learn, and how variations do not really advance learning, but constitute a smoke screen - the dust of the road that hides it. And on the other hand, it is also important to learn how the establishment of a significant leap is at high levels of abstraction, which are high levels of method, but not at too high levels, where abstraction loses the information in the path that was made, and deepening becomes mystical, and therefore the leap - arbitrary. These are those who try to jump too many steps forward, despite the fact that it is not possible to decipher the depth trends leading there from the information so far, instead of being content with one significant learning step. There is a limit to what you can learn. You cannot see too far ahead not because you are stupid, but because you do not have enough data yet.

Therefore learning takes generations. The continuity of the path does not depend only on what is inherent in it itself - within it, but on what happens in the continuation of learning, which is not random and arbitrary, but also not known in advance. Just like how a hypothesis of what a dog is after pictures of four dogs is not arbitrary, but also not certain. In the more principled and higher and abstract methods in philosophy, we always remain with a handful of philosophical paradigmatic leaps from the history of philosophy as examples, and things become very speculative, because there we are already forced to stop. That is, even if philosophy continues for a million years, still the most principled possible turns in the path will be few, and will constitute a limit and upper bound on what strength of derivative (the hundredth derivative, the thousandth, etc.) can be talked about. There is a limit to the depth of learning, which stems from the length of learning.



Therefore learning, which is not derived in advance unambiguously from examples of the past but continues them and is not arbitrary, is the solution to the psychological problem, which was never really a philosophical problem, of free will, because from within the system (and not from outside) our progress is learning. That is, from our perspective as a learning system, our mode of action does not really work at all through "choice", and therefore is neither free nor predetermined, but through learning. And this is enough for us psychologically, because this is us. Choice is simply the application of our learning judgment - the act of learning. After all, we do not want to choose at all randomly, but in a learning way, and for this to be the meaning of our choice. What bothers us is precisely the "just", that is, the lack of learning. There is no meaning outside of learning.

Is everything predetermined outside the system, outside our point of view, or is everything random, or something else? This is a meaningless question, that is, one cannot learn anything from it - which is the interpretation of meaninglessness. Even an answer to it will not teach us anything. Because we cannot not learn. We cannot, for example, progress randomly, even in the simplest walk, but only find methods that will simulate what seems to us as randomness. To someone who looks at the universe from outside of time, even a completely random universe is predetermined. But because you are within learning - thinking outside of learning is impossible. Even completely opaque thinking, which learns nothing from the world, is impossible. You cannot even be a perfect idiot, even if you want to, just as you cannot be perfectly wise. Because there is no objective universal "reason" or "rationality" that exists somewhere, but only learning. Our current reason was simply learned - whether by evolution or by culture.

Therefore the desire and pretension to know in advance, and hindsight wisdom (for example moral), are anti-learning. The Enlightenment was the pretension to know, and postmodernism the pretension not to know, when both are impossible for the human brain - we are not capable of anything but learning. The preoccupation with certainty in the history of philosophy is a fantasy of the brain to exit learning once and for all - an attempt of the system to get out of the system. Therefore the certain is meaningless. If God is certain He is meaningless. If existence is certain it is meaningless. We learn nothing from it, and it has no value. Meaning is always learning potential.



Mathematics, for example, is not certain, but learned, and in fact it is the product of very intense learning over generations - and full of mistakes (and how common are the mistakes we make in it as students!). Hence its value and usefulness and rigidity and immunity to contradictions - from learning it and not from its certainty. In every paradox and contradiction and logical and conceptual problem found in the history of mathematics, an enormous learning effort was invested, and only what stood up to the highest learning standards for rigidity - was included within mathematics (which represses these histories). Mathematics is not a perfect body of knowledge of marble that we carved out of stone (which of course was there before as an idea...), but a clay sculpture, where every time a piece of human learning was rigid and durable and dry enough - it was added to it. The power of mathematics is that what stood up to these learning standards already produces from within it things that stand up to a similar standard (there is no perfect - perfect is an illusion), because the most central power of mathematics is that its method itself had to pass such rigid standards. Therefore the definition of mathematics is not what we managed to learn with certainty, but what we managed to create a learning method for without contradiction. Mathematics is the most successful method in the world, and this is exactly the reason why it is so useful in the world. Precisely because it is a learning tool.

Is the very existence of such a method a wonder, that is, something that cannot be explained and cannot be learned why it is so? If we learned something, then learning is the explanation for its existence. We have no access at all to other, transcendent, non-learning (and in particular: certain) explanations. We have no reasons that are not learning (philosophy and science always failed in pursuit of reasons, when what the brain always sought was to learn. Kant was wrong in the category). If evolution learned a human, or a computer, then this learning is the explanation for their existence. And we cannot have any other explanation. Philosophy and reason need to go through a process of internalizing their own learning, and thus we will not encounter the arrogance of knowledge again but the humility of learning (no leader knows what needs to be done, no man knows what the woman needs, etc.).

The need for de-mystification of mathematics is more urgent than the need for de-mystification of faith or the state, and this mystification stems from the fact that mathematics is too difficult to learn for people (and even for mathematicians), precisely because of the high standards it sets. What we can barely learn and continue forward to the next step - touches on mystery for us. But this mystery, in our pride, we do not attribute to our lack of comprehension, but to the field itself. The mouse that learns the maze attributes mystery to it - and finally invents a minotaur for it. The mystification of mathematics, which began in philosophy from the Pythagoreans and their spiritual descendant Plato, created a long-standing anti-learning bias in philosophy. While Greek mathematicians were still struggling unsuccessfully with the primary conceptual problem of incommensurability, Plato had already built a mathematical world of ideas, which remained a philosophical ideal to this very day, which influences not a little on analytical philosophy - not to mention the romantic perceptions of mathematicians themselves. But the power of mathematics is not in its idea, but in its method. Its learning is the longest in human history, and therefore it is so deep. Mathematics should not teach us about knowledge - but about learning. But this does not mean that we are supposed to imitate its method in pastiche (as in analytical philosophy), because then everything that succeeds in this it will annex to itself (logic), and everything that is bad in this will remain philosophy. The irony of fate is that the most successful learning example has become an anti-learning weapon.



Mathematics, as a learning tool, is what created the scientific revolution and exact science and the scientific method, and the delay in its use in biology created the lag of biology in relation to the rest of science. Darwin was the first to describe an algorithm, in a crude way, in the fields of biology, and thus contributed to turning it into a science, and hence his great importance - as an algorithm developer. That is, the mathematical development, and particularly that of Descartes who showed how to perceive physics in coordinate space (that is, in a mathematical tool), was the historical factor for the rise of the modern era. The artificiality of the mathematical method, which unlike, for example, language learning or behavioral laws is not natural to the human brain, is what created the artificial age, whose peak is the computer. That is, in fact, mathematics represents a learning algorithm different from the human one, and therefore we do not fully understand it, but this does not mean that it is not learned, and exists somewhere outside our learning. And on the other hand, the fact that it is learned does not mean that it is arbitrary, and that we could have invented it as we wished, although of course historically it could have developed in other directions. Mathematics is not predetermined and not random, because both of these modes of description look outside the learning system, while from our perspective it is learned and develops - just like history. And like in history, one can identify trends in mathematics, and depth trends, whose continuation led to the creation of new mathematics.

In mathematics, every proof and definition is an example of learning, and every theory, as a collection of them, is a path. From each such example one can continue to many possible and different directions, according to the mathematical method (not the logical, almost everything that is logically correct is not mathematically interesting - because it does not teach anything). That is, from each example, by its very nature as an example, one can learn different things and learning can progress in different directions - does this make mathematics arbitrary? No, because everything is according to its learning method, which itself was also learned, for in mathematics there are different methods, and in them too there are innovations, which are of course important and principled mathematical innovations. Learning creates possibilities, which are not all possibilities (history too is neither arbitrary nor predetermined), and this is the only causality that exists. Not the necessary, bidirectional one, where one can go in both directions to the same extent (and therefore if you go back one logical step, you can go forward again and reach the same place), but only unidirectional causality (direction), which is learning-possible, but not all-possible (and therefore arbitrary). Far from it - generally learning allows a tiny part of all possibilities, with severe restraints on exponential misbehavior, which creates linguistic grammar.

Of course, it's not just about the number of possibilities, but about the way they are chosen, which is the method, which not only looks at the current junction, but continues the direction of travel from all the way before it. Therefore even if there is more than one turn that continues this direction - it cannot turn to every possible direction. And therefore turning back is actually impossible. Moreover - it's not just that if you go back, and try to learn again, you can reach a different place, but in learning you simply cannot really go back after you've learned. If you learned the Pythagorean theorem, it changed your method itself, even if you forget the Pythagorean theorem (that is: going back is in reciprocal relations with the method). Even quantum physics has already reached this, but philosophers, who have never done real mathematics - stick to their guns. They are stuck in a logical-grammatical-linguistic (which is, historically, very new) view of mathematics - and not learning. And therefore their theory of mathematical innovation - and learning in general - is poor, and similar to evolutionary mutation. And then there is also room for Foucauldian theories that everything is politics/power relations/propaganda/advertising/influences/fashions in development in the world - just because of the arbitrary view. And so art becomes a collection of mutations, because it has lost its method and its learning, and therefore its meaning in the world. But mathematics, as the strongest method in the world, continues to turn the world in its learning, and does not behave according to the anti-learning theory, neither the postmodernist nor the Platonic one. Learning is neither arbitrary nor predetermined (why the denial of the vast space between these two possibilities? Perhaps because precisely both of them are not learning? How difficult it is for philosophy to come to terms with the incompleteness in learning, and to replace arrogance - with a step).

Could there be different mathematics in another universe? Even in our universe it could have developed in other directions. If it was natural for our brain to perceive in non-Euclidean geometry, we might never have discovered Euclidean geometry. But could Euclidean geometry itself have been different, in another universe, will ask both the Platonic idealist and the postmodernist? But again, the moment we found another geometry, and even in our universe, we called it non-Euclidean. But is it possible that in another mathematical learning 1+1=3? The truth is yes, in a group with a single element, but what are you really asking: is it possible for there to be a contradiction in learning where what enters into it is only what is without contradiction? You yourself are not even capable of asking this question, which you want so much, because it is a question outside of learning. If you find a mathematical possibility without contradiction that is not in current mathematics, then at that moment it will be included in our mathematics (and congratulations, you are a great mathematician, and perhaps also a forgotten philosopher, and see Frege today), and if you try to find a contradiction in current mathematics, then again, if you succeed, you will remove the part with the contradiction from the realms of mathematics (and see Frege then).

All the wonders in the world, and especially the wonder of mathematics, try to get out of learning. Nature is a wonder - if there is no evolution. The universe is a wonder - if there is no development. A masterpiece is a wonder - if you have no idea how it was created. Poetry is a wonder - because you are a romantic who denies the method of its writing. Even if you yourself wrote, you are capable of hiding this from yourself - but there is a method. And in fact this is exactly what you claim - that the method is not conscious (oh, the muse). The purpose of the sense of wonder is not for you to get stuck in it, but to arouse the brain to learning - with the help of interest. Love too is a wonder only because the lover is not aware of its method that caused him to fall in love, and whose purpose is to arouse in him enormous interest in the partner. And he, indeed, considers her the most interesting thing in the world, and learns about her obsessively, until finally she of course bores him. And in a happy relationship learning never ends. Therefore if you are boring and your life is boring, try to find yourself a lover who does not learn too quickly. But because love creates such enormous interest, it is very difficult to learn against it itself. Hence the phenomenon of unrequited love, in which the lover - usually a reasonable person - simply does not learn, and on the other hand the enormous patience of lovers for long and obstacle-filled learning, like in mathematics. Indeed, mathematicians too are in love with it, and therefore they are such romantics. Love is boundless interest - a learning obsession (yes, your children are the most interesting in the world!). And therefore philosophy too is the love of wisdom, because it tries to learn something that is sometimes impossible to learn, or certainly to learn to the end. But one must remember that the enemy of love is not disappointment, but boredom. Therefore philosophy is allowed to fail in answering a question, but it must perform learning in this failure. We learn also - and perhaps mainly - from failures.



Therefore, after we have removed the anti-learning spells, all that remains is to ask how we learn, that is to learn how we learn, because of course there cannot be a non-learning answer to this question. But at this stage of the lecture, after all the unnecessary introductions in retrospect (that is, only after we have learned them, as always), and after I was left alone, the only thing left is to understand that in fact all our thinking, our entire spiritual and cultural world, try to answer various and strange answers to this question: how do we learn. And all their progress is hidden in new answers, each of which constitutes one more additional step - in learning how to learn. If so, what is learning? Have we answered the question? No. Have we learned? Yes. And in that we learned, we have answered all possible questions the one and only possible answer - an example of learning, from which one can learn further.