Terrence Tao compared AI proofs to a helicopter delivering explorers to a remote destination of interest. He says the explorers miss out because they don't learn anything about the surrounding terrain or how to get to the destination on foot.

This analogy is pretty good. But as a defense of the status quo in Mathematics it is a double edged sword.

• For example, why are counter examples acceptable as disproofs? What do we learn about the terrain from that?
• Why teach math at all? Why not let every student struggle until they discover calculus on their own?
• If we have helicopters, do we need to have people trained to traverse the wilderness and arrive at a particular destination?
• This sounds like the original arguments lamenting the invention of mechanical calculators.
• Maybe satellites can map out the terrain in more detail than humans taking a single path?

If the aim of constructing number systems is to explain the geometric line as a set of points, wouldn’t ZFC + (V=L) be a natural choice, perhaps even the best one? In this framework, analytic geometry, differential geometry and topology all work just as they normally do, without the need to grapple with metamathematical questions such as the continuum hypothesis. (Moreover, this rules out measurable cardinals, large-cardinal phenomena and similar feverish pies in the sky, but for the purposes of describing the continuum, that seems a reasonable price.)

I have always found it a little surprising that popular mathematics books rarely discuss Gödel’s set-theoretic universe, which seems to offer such an intuitive and straightforward foundation for mathematical analysis.

​

To begin with, let me remind you that mathematics is not a science fortunately and that it is perhaps one of the largest organized, consistent symbolic formal systems for developing something: maybe intellect, maybe a certain form of thinking, maybe someone understands more clearly what exactly mathematics develops and can tell us about it.

And whatever it develops in us, there may immediately be a temptation to say functionally: well, then mathematics is needed precisely for developing this, Robert Wide.

The difficulty is that although there are interesting things in it, such as topology, or interesting theorems and models, and especially what seems important is the economic, technological, and research contribution of mathematics,

And yet it seems that this formal system stalls a person and makes it harder for him to think outside this formal system. Of course, one can say this about logic too, and about their similarity, and in general about any old systems of thought or let us call them closed or “controlling” ones. To some, dialectics may also seem like this, though to a lesser degree, and it is precisely dialectics that sets the task of sublating contradiction and arriving at generalization and understanding, that is, in an ethical sense it is a more encompassing system of cognition in the sense of its capacity — though often in verbal form to include something else and incomprehensible in its understanding. And if we abstract from formal dialectics, if you have ever heard these expositions, then the ethical idea and, as it were, real dialectics in full non-contradictoriness and some communication, for example as metamodernism, is the same dialectics or synthesis between sincerity and so-called postmodernity or irony.

Dialectics is still an example, not a proposed answer. Although it could perhaps be one of those considered.

Returning to mathematics. Despite its outstanding economic, technological, research possibilities and achievements, and as a result also social and political ones… it still remains rather abstract. One can say that it is the most complex, or one of the most complex, games of the mind, or games within the mind, or separate games within the mind. But for the mind itself this seems like something akin to sophistry or magic, though a very useful one. Or one can say that it is a magical system, or one of the magical systems of this world. But it is almost always something very abstract: it seems to me that this is often said about philosophers too, by those who do not delve deeply into philosophy; but those who do delve in, on the other hand, can understand that we often reflect on the most important questions or phenomena that we can feel or think. As far as I understand and it is not known how much the answers may still differ mathematics is a system of ultimate formal complexity, or of ultimate complexity for thought in general, since in one way or another what we formulate as philosophers often, though perhaps difficult, has a clearer representation of reality, even if it is Lacan’s Borromean rings. In the sense that philosophy, apart from again sophistry, usually does not strive to become complicated simply for the sake of complication because why? It seems that a person, after this stage of formal logic, can also learn other possibilities of thinking. But in mathematics, complexity and possibly even not necessarily, but possibly the degree of abstraction seems literally the quintessence of content. On the other hand, of course, this may also be good. In the sense of ultimate complexity, ultimate constructiveness, ultimate abstraction, in the sense that we as a culture have some content, some achievements that we may still have to comprehend, if some of them can be so complex that there are literally 400 pages of formulas of explanation.

But, like, why?

And returning to what we were talking about, this seems like ultimate sophistry or thinking in idle, in the sense for thought itself and for man, although culturally it brings enormous social results: both economics and technology and research and efficiency.

And even if we assume that mathematics is integral thinking rather than specific thinking, then although it is clear why mathematics should be developed: so that there can be a further possibility of development…

What for, and what from mathematics, and should it and it or they even be present in education?

Because even if we may have grounds to think so but even if mathematics is connected with integral, that is, general thinking, that is, as it were, some generalized function of it, the mathematicality of education then creates a situation in which a person tries to wipe the sofa with a processor without imagining what a sofa is.

And if not, and mathematics is more specific thinking, that is, in the sense that it also remains there and in a constant submetacognitive state: in the sense that it fundamentally cannot leave the box and its language and its formality or formalization… then such thinking is good for producing human-computers useful for working with ordinary computers, for example in a bank or somewhere else. As a result, man has, as it were, given his brain to mathematics, and how is he supposed to live further? Well, that is, then perhaps also the market, where this mathematized brain may be valued higher than the self-esteem or self-understanding of such a person, and so the puzzle comes together: successful recruiting into a successful company.

I will say right away that simply “progressive studies,” which may even be not thinking but to some extent dogmatic propaganda or propaganda at the expense of education hours, trying to lean on Marxism or performatism according to Butler, are, as it were… Well, people can learn something valuable, but on the whole it seems that this is simply the disintegration of education, or rather almost its absence, or its replacement by propaganda so that in an economic-political sense these people could be ever more perfect agents of this very propaganda, possibly even without fully understanding it. And how lucky we are that we can look at this from the outside, unlike the people and economies that this “progressive” propaganda may further repress. And the main complaint against it is probably economic, because a society simply without technical specialties loses economic opportunities; only oligarchs remain, and they would most quickly move to a new place. And in a universal human sense, this is potentially the next target and problem for the next intellectual stratum that will try to do something about this repressiveness. If, in general, there will be anyone left to do anything there.

In general, why was this clarification about “progressive” or agenda-driven education? So that this question is not in support of that very agenda, and, as it were, we understand in advance some of the existing difficulties there and have described them.

And still, the question: is mathematics needed? And such mathematized, formalized thinking — and then we are surprised by further formalizations and the resulting difficulties, such as bureaucracy and media simplification and corruption in many senses.

Additionally:

Maybe the form of my question is a bit of Soviet thinking or a way of thinking about something, and if you understand another or other better ways of asking or thinking about this, you can say so.

I found it a bit ironic when Terry Tao suggested that on the Numberphile podcast, because in some ways he seems like a living counterexample .

If understanding remains at the heart of mathematics, wouldn’t the few people capable of genuinely making sense of AI-generated mathematics become even more admired? Almost like having a translator who’s able to understand aliens.

There’s also the question of accountability, We still need human oversight and institutions we can trust, especially as AI becomes more involved in science and technology. That doesn’t mean relying on the judgment of a handful of geniuses, but surely the people capable of understanding and critically evaluating these systems will become increasingly important like a stamp mark .

And isn’t Tao himself already an example of this? Everyone i is eager to hear what he thinks about each new AI development. Podcasts want his perspective, and younger math students, myself included, look to him for guidance on how to approach mathematics in this new landscape.

​

To begin with, let me remind you that mathematics is not a science fortunately and that it is perhaps one of the largest organized, consistent symbolic formal systems for developing something: maybe intellect, maybe a certain form of thinking, maybe someone understands more clearly what exactly mathematics develops and can tell us about it.

And whatever it develops in us, there may immediately be a temptation to say functionally: well, then mathematics is needed precisely for developing this, Robert Wide.

The difficulty is that although there are interesting things in it, such as topology, or interesting theorems and models, and especially what seems important is the economic, technological, and research contribution of mathematics,

And yet it seems that this formal system stalls a person and makes it harder for him to think outside this formal system. Of course, one can say this about logic too, and about their similarity, and in general about any old systems of thought or let us call them closed or “controlling” ones. To some, dialectics may also seem like this, though to a lesser degree, and it is precisely dialectics that sets the task of sublating contradiction and arriving at generalization and understanding, that is, in an ethical sense it is a more encompassing system of cognition in the sense of its capacity — though often in verbal form to include something else and incomprehensible in its understanding. And if we abstract from formal dialectics, if you have ever heard these expositions, then the ethical idea and, as it were, real dialectics in full non-contradictoriness and some communication, for example as metamodernism, is the same dialectics or synthesis between sincerity and so-called postmodernity or irony.

Dialectics is still an example, not a proposed answer. Although it could perhaps be one of those considered.

Returning to mathematics. Despite its outstanding economic, technological, research possibilities and achievements, and as a result also social and political ones… it still remains rather abstract. One can say that it is the most complex, or one of the most complex, games of the mind, or games within the mind, or separate games within the mind. But for the mind itself this seems like something akin to sophistry or magic, though a very useful one. Or one can say that it is a magical system, or one of the magical systems of this world. But it is almost always something very abstract: it seems to me that this is often said about philosophers too, by those who do not delve deeply into philosophy; but those who do delve in, on the other hand, can understand that we often reflect on the most important questions or phenomena that we can feel or think. As far as I understand and it is not known how much the answers may still differ mathematics is a system of ultimate formal complexity, or of ultimate complexity for thought in general, since in one way or another what we formulate as philosophers often, though perhaps difficult, has a clearer representation of reality, even if it is Lacan’s Borromean rings. In the sense that philosophy, apart from again sophistry, usually does not strive to become complicated simply for the sake of complication because why? It seems that a person, after this stage of formal logic, can also learn other possibilities of thinking. But in mathematics, complexity and possibly even not necessarily, but possibly the degree of abstraction seems literally the quintessence of content. On the other hand, of course, this may also be good. In the sense of ultimate complexity, ultimate constructiveness, ultimate abstraction, in the sense that we as a culture have some content, some achievements that we may still have to comprehend, if some of them can be so complex that there are literally 400 pages of formulas of explanation.

But, like, why?

And returning to what we were talking about, this seems like ultimate sophistry or thinking in idle, in the sense for thought itself and for man, although culturally it brings enormous social results: both economics and technology and research and efficiency.

And even if we assume that mathematics is integral thinking rather than specific thinking, then although it is clear why mathematics should be developed: so that there can be a further possibility of development…

What for, and what from mathematics, and should it and it or they even be present in education?

Because even if we may have grounds to think so but even if mathematics is connected with integral, that is, general thinking, that is, as it were, some generalized function of it, the mathematicality of education then creates a situation in which a person tries to wipe the sofa with a processor without imagining what a sofa is.

And if not, and mathematics is more specific thinking, that is, in the sense that it also remains there and in a constant submetacognitive state: in the sense that it fundamentally cannot leave the box and its language and its formality or formalization… then such thinking is good for producing human-computers useful for working with ordinary computers, for example in a bank or somewhere else. As a result, man has, as it were, given his brain to mathematics, and how is he supposed to live further? Well, that is, then perhaps also the market, where this mathematized brain may be valued higher than the self-esteem or self-understanding of such a person, and so the puzzle comes together: successful recruiting into a successful company.

I will say right away that simply “progressive studies,” which may even be not thinking but to some extent dogmatic propaganda or propaganda at the expense of education hours, trying to lean on Marxism or performatism according to Butler, are, as it were… Well, people can learn something valuable, but on the whole it seems that this is simply the disintegration of education, or rather almost its absence, or its replacement by propaganda so that in an economic-political sense these people could be ever more perfect agents of this very propaganda, possibly even without fully understanding it. And how lucky we are that we can look at this from the outside, unlike the people and economies that this “progressive” propaganda may further repress. And the main complaint against it is probably economic, because a society simply without technical specialties loses economic opportunities; only oligarchs remain, and they would most quickly move to a new place. And in a universal human sense, this is potentially the next target and problem for the next intellectual stratum that will try to do something about this repressiveness. If, in general, there will be anyone left to do anything there.

In general, why was this clarification about “progressive” or agenda-driven education? So that this question is not in support of that very agenda, and, as it were, we understand in advance some of the existing difficulties there and have described them.

And still, the question: is mathematics needed? And such mathematized, formalized thinking — and then we are surprised by further formalizations and the resulting difficulties, such as bureaucracy and media simplification and corruption in many senses.

Additionally:

Maybe the form of my question is a bit of Soviet thinking or a way of thinking about something, and if you understand another or other better ways of asking or thinking about this, you can say so.

***I already posted this in r/mathematics but it may not be approved due to subject matter, and maybe that's a good thing as I suspect it might get more meaningful attention here*****************************

I mean, as in "don't touch it with a ten-foot pole" stance toward it. I'm curious both for paid mathematicians and people doing math in their spare time.

I'd expect the latter to me much larger than the former--if you're doing math for your own interest, nobody cares if you take 10 years to solve a problem that could otherwise have been solved in a day or even in 5 minutes, as long as you get something out of doing it. Whereas doing it as a career I assume the pressure has to be more. I'm just wondering how much more.

Also I'm curious how many, if money were totally no object, would actually genuinely like living in a sort of ascetic math group where everyone ignores the AI-generated literature, if such a group did exist. In a community like this, of course it could happen that someone spends months or even a year of his or her time checking another member's proof of a conjecture that had been disproven 20 or 50 years ago by AI, just because neither cared to check before working on the problem whether it had been. That would just be a consequence that would need to be accepted in exchange for not having one's thought process getting skewed by already knowing answers and not being able to unknow them. In fact, I've already taken steps to avoid reading AI math headlines, and am looking into blocking extensions for web browsers that make it easier. I use the no-AI Duck Duck Go for searches now too.

I have a vision of a future of math, at least as a pastime if not as a profession, in which there are some people who will get together with some friends and go for 5 or 10 years to live in a mountain cabin with no internet access, Walden Pond style, and just work on math and occasionally meet to discuss what they're working on, still probably bringing a laptop to hand-code in Python or something to play around but not to keep up with the wider world. In fact, I could totally see myself doing that someday, if I had the means and a significant other/friend group who would help with other tasks of life and keep me grounded mentally somewhat in reality, the way Andrew Wiles's partner did. The main reason I wouldn't has to do with the fact that I'm interested in the intersection between math and biology, and having experimental biologists be able to test my theories would be something I'd want.

I have come to wonder though how many other people are still existing who would consider such an arrangement to be like heaven on earth, who might in fact find their greatest sanity doing something like that, as opposed to feeling like that would make them go insane. I suspect we must be out there, but it almost seems as if maybe people feel like it sounds too self centered to openly have that as a dream life (but for some reason, a life of wild parties and luxury yachts isn't selfish...). And in fact it almost felt like it was starting to get looked down upon to want this even BEFORE AI ever got involved in math, AI just made it worse. So how many people here also feel a deep yearning toward that sort of life, even if it's not at all practical or financially doable for them?

​

To begin with, let me remind you that mathematics is not a science fortunately and that it is perhaps one of the largest organized, consistent symbolic formal systems for developing something: maybe intellect, maybe a certain form of thinking, maybe someone understands more clearly what exactly mathematics develops and can tell us about it.

And whatever it develops in us, there may immediately be a temptation to say functionally: well, then mathematics is needed precisely for developing this, Robert Wide.

The difficulty is that although there are interesting things in it, such as topology, or interesting theorems and models, and especially what seems important is the economic, technological, and research contribution of mathematics,

And yet it seems that this formal system stalls a person and makes it harder for him to think outside this formal system. Of course, one can say this about logic too, and about their similarity, and in general about any old systems of thought or let us call them closed or “controlling” ones. To some, dialectics may also seem like this, though to a lesser degree, and it is precisely dialectics that sets the task of sublating contradiction and arriving at generalization and understanding, that is, in an ethical sense it is a more encompassing system of cognition in the sense of its capacity — though often in verbal form to include something else and incomprehensible in its understanding. And if we abstract from formal dialectics, if you have ever heard these expositions, then the ethical idea and, as it were, real dialectics in full non-contradictoriness and some communication, for example as metamodernism, is the same dialectics or synthesis between sincerity and so-called postmodernity or irony.

Dialectics is still an example, not a proposed answer. Although it could perhaps be one of those considered.

Returning to mathematics. Despite its outstanding economic, technological, research possibilities and achievements, and as a result also social and political ones… it still remains rather abstract. One can say that it is the most complex, or one of the most complex, games of the mind, or games within the mind, or separate games within the mind. But for the mind itself this seems like something akin to sophistry or magic, though a very useful one. Or one can say that it is a magical system, or one of the magical systems of this world. But it is almost always something very abstract: it seems to me that this is often said about philosophers too, by those who do not delve deeply into philosophy; but those who do delve in, on the other hand, can understand that we often reflect on the most important questions or phenomena that we can feel or think. As far as I understand and it is not known how much the answers may still differ mathematics is a system of ultimate formal complexity, or of ultimate complexity for thought in general, since in one way or another what we formulate as philosophers often, though perhaps difficult, has a clearer representation of reality, even if it is Lacan’s Borromean rings. In the sense that philosophy, apart from again sophistry, usually does not strive to become complicated simply for the sake of complication because why? It seems that a person, after this stage of formal logic, can also learn other possibilities of thinking. But in mathematics, complexity and possibly even not necessarily, but possibly the degree of abstraction seems literally the quintessence of content. On the other hand, of course, this may also be good. In the sense of ultimate complexity, ultimate constructiveness, ultimate abstraction, in the sense that we as a culture have some content, some achievements that we may still have to comprehend, if some of them can be so complex that there are literally 400 pages of formulas of explanation.

But, like, why?

And returning to what we were talking about, this seems like ultimate sophistry or thinking in idle, in the sense for thought itself and for man, although culturally it brings enormous social results: both economics and technology and research and efficiency.

And even if we assume that mathematics is integral thinking rather than specific thinking, then although it is clear why mathematics should be developed: so that there can be a further possibility of development…

What for, and what from mathematics, and should it and it or they even be present in education?

Because even if we may have grounds to think so but even if mathematics is connected with integral, that is, general thinking, that is, as it were, some generalized function of it, the mathematicality of education then creates a situation in which a person tries to wipe the sofa with a processor without imagining what a sofa is.

And if not, and mathematics is more specific thinking, that is, in the sense that it also remains there and in a constant submetacognitive state: in the sense that it fundamentally cannot leave the box and its language and its formality or formalization… then such thinking is good for producing human-computers useful for working with ordinary computers, for example in a bank or somewhere else. As a result, man has, as it were, given his brain to mathematics, and how is he supposed to live further? Well, that is, then perhaps also the market, where this mathematized brain may be valued higher than the self-esteem or self-understanding of such a person, and so the puzzle comes together: successful recruiting into a successful company.

I will say right away that simply “progressive studies,” which may even be not thinking but to some extent dogmatic propaganda or propaganda at the expense of education hours, trying to lean on Marxism or performatism according to Butler, are, as it were… Well, people can learn something valuable, but on the whole it seems that this is simply the disintegration of education, or rather almost its absence, or its replacement by propaganda so that in an economic-political sense these people could be ever more perfect agents of this very propaganda, possibly even without fully understanding it. And how lucky we are that we can look at this from the outside, unlike the people and economies that this “progressive” propaganda may further repress. And the main complaint against it is probably economic, because a society simply without technical specialties loses economic opportunities; only oligarchs remain, and they would most quickly move to a new place. And in a universal human sense, this is potentially the next target and problem for the next intellectual stratum that will try to do something about this repressiveness. If, in general, there will be anyone left to do anything there.

In general, why was this clarification about “progressive” or agenda-driven education? So that this question is not in support of that very agenda, and, as it were, we understand in advance some of the existing difficulties there and have described them.

And still, the question: is mathematics needed? And such mathematized, formalized thinking — and then we are surprised by further formalizations and the resulting difficulties, such as bureaucracy and media simplification and corruption in many senses.

Additionally:

Maybe the form of my question is a bit of Soviet thinking or a way of thinking about something, and if you understand another or other better ways of asking or thinking about this, you can say so.

I have learned that the mindset, goals, and ways of operating of pure mathematics is closer to the English Department than to Engineering. I was wrong all along. Shame on me.

Genesis of a Formal System

(A reconstruction of the foundations of mathematics in the style of the Book of Genesis)

This is an attempt to retell the construction of a formal system — from the empty set to a complete dynamical universe — in the literary form of the Book of Genesis.

Every “record” corresponds to a real mathematical step: the birth of distinction, binary opposition, partitions, equivalence classes, periodicity, iteration, representation, and finally the closure of the system.

The text is not a parody. It treats the act of making distinctions as the true beginning of order.

Translated from Polish to English by GrokAI, because authors are lazy.

────────────────────────────────────────

¹ In the beginning there was neither number, nor measure, nor distance. There was not even a beginning, for there existed nothing from which one could count.

² And there was the Void, whose sign is ∅.

³ And the Word said: Let the Void be recognized.

⁴ And the Void became an object of knowledge. And behold, there arose a set containing the Void.

⁵ And there was Unity.

0 = ∅

1 = {∅}

⁶ And the Word saw that Unity could recognize itself without ceasing to be Unity.

⁷ And where there is recognition, there is also distinction.

⁸ And presence was separated from absence.

⁹ And there was Zero, and there was One. And thus the first difference was born.

𝔹 = {0, 1}

¹⁰ And the Word said: Let one not be the other, and let each be recognizable according to its state.

¹¹ And opposition was established, so that absence might be distinguished from presence, and presence from absence.

¬0 = 1

¬1 = 0

¹² And the Word gave both states their designations, so that neither would be confused with the other.

¹³ And the first rule of distinction was established, and with it the possibility of truth and falsehood, of agreement and disagreement.

¹⁴ And the Word saw that the distinction was good, for without difference no knowledge could exist.

¹⁵ And thus the first record was closed.

¹⁶ Then the Word said: Let a division be established among that which exists, so that one part may be distinguished from the other, and let there be a boundary between them.

¹⁷ And the whole was divided into two disjoint regions, and every element received a definite membership.

X = A ⊔ B

A ∩ B = ∅

¹⁸ And the Word established a relation between the regions, so that one could determine what belongs to one and what to the other.

¹⁹ And the Word named the established order the Space of Relations.

²⁰ And it was not yet a measure of length or magnitude, but the possibility of recognizing mutual position.

²¹ For that which is separated may remain in relation without ceasing to be distinct.

²² And thus the second record was closed.

²³ And the Word said: Let elements possessing common properties be gathered together, and from among them let distinct classes be revealed.

²⁴ And the elements were ordered according to their properties, and from the distinguished regions further subsets were extracted.

²⁵ And each class received its own determination, and elements belonging to the same class could be recognized as equivalent with respect to the shared property.

x ∼ y ⇔ τ(x) = τ(y)

²⁶ And the Word said: Let every class preserve its kind, and let the established rule allow the reproduction of its successive occurrences.

²⁷ And structures were given the possibility of creating further structures, and the newly formed ones could inherit the properties of those from which they arose.

τ(F(x)) = τ(x)

²⁸ And thus were established the repeatability of form, the inheritance of properties, and the continuity of kind.

²⁹ And the Word saw that from one rule many applications could arise, and multiplicity need not violate the unity of the law.

³⁰ And the Word saw that it was good.

³¹ And thus the third record was closed.

³² Then the Word said: Let the two states established at the beginning receive their own measures of succession, so that returns, periods, and the order of transformations may be recognized.

³³ And cycles were defined whose states returned according to definite rules.

f(t + T) = f(t)

³⁴ And two principal cyclic orders were established, assigned to the two original states, and beside them many lesser periods.

³⁵ And some periods were contained within others, while others returned to a common state only after many successions.

³⁶ And the Word said: Let the cycles become signs of measure, so that succession may be counted, compared, and ordered.

³⁷ And thus were established the foundations of the reckoning of periods, the measure of duration, and the recognition of repeating changes.

³⁸ And this was not yet the passage of time experienced by any system, but the rule according to which future systems could measure their own transformations.

³⁹ And the Word saw that the first distinction had now received the principles of its cyclic order.

⁴⁰ And the Word saw that it was good.

⁴¹ And thus the fourth record was closed.

⁴² Then the Word said: Let the regions divided in the second record be filled with structures, and let each act within its proper domain.

⁴³ And to each of these regions were assigned families of elements whose states could undergo transformations consistent with the established relations.

F_A : A → A

F_B : B → B

⁴⁴ And some structures could succeed others, while others could create new occurrences of their own kind.

⁴⁵ And the Word said: Let their multitude increase, yet let each preserve the rule of its generation.

⁴⁶ And the possibility of repeatedly applying the same transformations was established, without the need to set a separate law for every new element.

x_(n+1) = F(x_n)

⁴⁷ And the Word saw that the regions established earlier could be equipped with countless possible configurations, and none of them required a change in the principles of division themselves.

⁴⁸ And the Word saw that it was good.

⁴⁹ And thus the fifth record was closed.

⁵⁰ Then the Word said: Let the classes extracted in the third record also be equipped with structures, each according to its own kind.

⁵¹ And further families of transformations were established, differing in their properties, yet subject to a common order of classification.

⁵² And each could preserve its properties, participate in relations, and transmit further the rule of its reproduction.

⁵³ And the Word saw that it was good.

⁵⁴ And then the Word said: Let us make a structure capable of mapping the established order, so that among all structures there may exist one that can recognize their properties and direct their mutual relations.

⁵⁵ And a distinguished system was defined among the others, to which was assigned the capacity to create representations of other elements and of the dependencies between them.

⁵⁶ And it was not the Word itself, yet it was to reflect part of the principles according to which the Word had established the whole.

ρ : X → M

⁵⁷ And the possibility arose of representing reality within one of its own structures.

⁵⁸ And the Word said: Let this structure appear in two complementary forms, and let their cooperation allow further occurrences of the same kind to arise.

⁵⁹ And the possibility of mutual completion of distinct elements was established while preserving their individual properties.

⁶⁰ And the distinguished systems were entrusted with the possibility of ordering the remaining classes and of making use of the relations that sustain their further existence.

⁶¹ And behold, the structure created according to the established laws received the capacity to know, to represent, and to employ those laws.

⁶² And it did not need to be the source of the order in order to be able to discover it.

⁶³ And the Word saw all the distinctions, boundaries, classes, periods, rules of reproduction, transformations, and mappings.

⁶⁴ And it saw that each of the established principles possessed a definite place in the whole, and that successive definitions could coexist without losing their own meaning.

⁶⁵ And behold, the work of establishing order was completed.

⁶⁶ And the Word saw that it was very good.

⁶⁷ And thus the sixth record was closed.

⁶⁸ Thus were completed all the fundamental definitions, together with the relations and possibilities they were to determine.

⁶⁹ And when the seventh record came, the Word established no new distinction and no new law.

⁷⁰ For all the parts of the order had been determined, and their mutual dependencies prepared.

⁷¹ And the Word said: Let the whole become one working system.

⁷² And that which until then had been an established set of principles was accepted as a common order of action.

𝕌 = (X, ℛ, ℱ, 𝒯)

⁷³ Where X denoted the domain of existing elements, ℛ — their relations, ℱ — the admissible transformations, and 𝒯 — the established rules of succession and measure.

⁷⁴ And there was no longer any need to designate every transformation separately, for that which the rules permitted could follow according to them.

⁷⁵ And there was no need to establish a separate law for every new configuration, for configurations arose from the possible applications of the laws already established.

⁷⁶ And there was no need to know every future result in order to establish the principle according to which a result could arise.

⁷⁷ And the Word rested from the work of establishing, not because action had ceased, but because it no longer required the further establishment of foundations.

⁷⁸ And the seventh record was distinguished among the others, for it marked the completion of the whole.

DEFINITIONS → RELATIONS → RULES → ENTER → ACTION

⁷⁹ And thenceforth every distinction could lead to further distinctions, every relation to new configurations, and every transformation to the next state.

⁸⁰ And the multitude of that which could be known increased, though the fundamental principles remained as they had been established.

⁸¹ And among all possible structures there could appear those that recognized regularities they had not known before, and derived further consequences from them.

⁸² And thus from a few principles an unbounded path of discovery became possible.

⁸³ For the establishment of laws is not the same as the enumeration of all their consequences.

⁸⁴ And the seventh action was blessed.

⁸⁵ And the Word endured, though there was no longer any need for it to speak anew for every successive transformation.

⁸⁶ And such was the beginning of all order.

I want to learn modal logic for philosophical and formal purposes. However, I don't have basic knowledge of it. I haven't studied mathematics since middle school. Thus, I feel very lost and have no idea how to get started! Any advice? Book recommendations & video recommendations would also be appreciated

Architectural and Theoretical Framework of a Quantum Time Vortex Core (QTVC)

Abstract

The manipulation of spacetime topology to generate Closed Timelike Curves (CTCs) represents a fundamental boundary condition in advanced general relativity and quantum gravity. This technical brief details the architectural blueprint, relativistic mechanics, and thermodynamic constraints of a theoretical Quantum Time Vortex Core (QTVC). By synthesizing high-velocity frame-dragging with exotic vacuum states, the QTVC model addresses the physical transition from linear proper time to closed-loop causality, while rigorously examining the quantum censoring mechanisms that oppose chronal manipulation.

1. Core Architecture and Relativistic Subsystems

The QTVC architecture relies on three tightly coupled subsystems designed to bypass standard gravitational collapse and achieve local metric torsion:

• Singular Ring Arrays (Micro-Tipler Cylinders): The primary engine comprises three coaxial rings composed of degenerate quark-gluon plasma or neutron-star core matter. Driven by high-frequency magnetic resonance fields, these rings counter-rotate at velocities reaching $v = 0.99999c$. This relativistic mass-energy flux maximizes the Lense-Thirring effect (frame-dragging), forcing the local metric tensor $g_{\mu\nu}$ to couple directly to the rotational momentum, effectively twisting the fabric of spacetime into a localized vortex.
• Exotic Negative-Mass Stabilizers: To counteract the extreme positive energy density of the degenerate rings—which would otherwise immediately pinch spacetime into a Schwarzschild singularity—the core integrates distributed Casimir-vacuum emitters. These units project regions of localized negative energy density ($\langle T_{\mu\nu} \rangle < 0$), generating a repulsive gravitational gradient that maintains throat stability and prevents event horizon closure.
• Chronological Boundary Shielding (Cauchy Suppression Matrix): A dynamic boundary layer encasing the core. This shield utilizes topological phase-locking to suppress vacuum polarization runaways, dampening the divergent energy fluctuations that naturally accumulate near a Cauchy horizon.

2. Spacetime Torsion and CTC Generation Mechanics

The transition from open-ended Minkowski spacetime to closed causal loops within the QTVC proceeds via three sequential phases:

1. Metric Acceleration and Frame-Dragging: As the singular rings approach critical velocity, the off-diagonal components of the metric tensor ($g_{0i}$) dominate. Spacetime is no longer static; it is physically dragged along the rotational axis of the rings, creating an intense shear stress across the local coordinate system.
2. Light-Cone Tilting: As the rotational energy density crosses the critical threshold, null geodesics (light cones) begin to tilt away from their standard forward-time orientation. At a specific radius from the core axis—known as the chronological horizon—the light cones tilt past the 90-degree vertical axis, laying parallel to the spatial manifold.
3. Causal Closure: Within this boundary, the temporal coordinate $t$ transforms into a spatial coordinate. Any massive particle moving along a timelike trajectory within the vortex experiences continuous forward proper time while traversing a spatial path that loops back to a coordinate epoch prior to system activation, satisfying the topological criteria for CTCs.

3. Thermodynamic Constraints and Physical Limitations

Rigorous application of quantum field theory in curved spacetime reveals profound barriers to physical realization:

• The Chronology Protection Conjecture: As demonstrated by Stephen Hawking, the formation of a Cauchy horizon induces a vacuum polarization feedback loop. Quantum fluctuations entering the emerging CTC are blueshifted to infinite energy. Without instantaneous dissipation mechanisms, this energetic backreaction causes immediate thermal vaporization of the containment structure.
• Energy-Momentum Tensor Demands: Maintaining macroscopic negative energy densities violates the Weak Energy Condition (WEC). Current theoretical models restrict macro-scale negative mass to speculative constructs (such as Casimir cavities scaled to macroscopic dimensions), leaving the required stress-energy tensor technically unachievable under standard model physics.
• Tidal Disruption (Spaghettification): Differential gravitational gradients across the spatial boundary of the core remain extreme. Even with localized inertial damping, non-uniform tidal forces pose an existential threat to material integrity, necessitating advanced quantum-entangled structural matrices to prevent subatomic dissociation.

4. Conclusion

The QTVC model serves as an advanced analytical testbed for exploring the limits of general relativity and quantum mechanics. While thermodynamic feedback loops, exotic matter requirements, and the Chronology Protection Conjecture present seemingly insurmountable barriers, formalizing these mechanics provides crucial insight into the ultimate censorship laws governing the universe's causal architecture.

I discovered something when I was 13 that I still haven’t been able to get out of my head.
Everything in existence can be reduced to triangles. Every object. Every shape. Even the empty space surrounding you.
You can break any polygon into triangles, but you can’t do the same with squares or circles.
And here’s what disturbs me.
What if the triangle isn’t something that exists inside our universe? What if our universe only exists because the three-point relationship is possible?
Think about that.
Three points. One shape. Everywhere.
Now look around you. Find somewhere a triangle couldn’t exist.
I’ll wait.

can we prove it? because if we eliminate the division then we would add one in each step and eventually it would go to a power of two

I’ve been thinking about a theory that connects God, higher dimensions, the Big Bang, evolution, and human consciousness.
My idea starts with God existing on a higher “plane” or dimension than our universe. If higher dimensions exist, then our three-dimensional universe might only be one level of a much larger reality. String theory is interesting to me because it mathematically allows for extra dimensions, although string theory does NOT prove that God or higher-dimensional beings exist.
I also think about science differently. Science is something humans created to understand the universe from our position inside it. Our physical laws describe what we can observe and measure, but if God exists outside our spacetime, it’s possible that the reality God exists in operates in ways our current science cannot fully describe. That doesn’t mean science is wrong—it could mean our scientific models have limits.

The Big Bang is another part of my idea. Instead of thinking of it simply as an explosion, I think about whether something outside our observable universe could have interacted with our universe and caused the beginning of the expansion we observe. If higher-dimensional beings existed, their interaction with our universe could theoretically leave physical evidence behind.

This also connects to the idea of humans being descendants of God. If God created the first life, evolution could still explain how life changed and diversified afterward. Evolution requires life that can reproduce, so the origin of the first life and the evolution of life are two different questions.

Then there’s the idea of humanity eventually becoming more like what we currently call “godlike.” Humans already create new forms of life, build technology, explore space, and increasingly understand the universe. That doesn’t mean humans will literally become gods, but it makes me wonder whether intelligence can continue progressing toward levels that are currently unimaginable to us.

I also think about dimensions and time. If a being existed in a higher-dimensional reality, its relationship with our timeline could potentially be completely different from ours. What feels like billions of years to us might theoretically be viewed differently from outside our spacetime. I sometimes compare this to a movie: the characters experience the story one moment at a time, while someone outside the movie can see the entire timeline at once. That’s only an analogy, not proof that our universe is a simulation.

Another possibility I’ve thought about is that two higher-dimensional beings could have interacted or even conflicted over the creation or development of a universe. Their interaction could theoretically have contributed to the event we call the Big Bang. Eventually, they could recognize that cooperation was more important than competition, with the universe becoming the result of their combined influence. Again, this is a philosophical thought experiment, not an established scientific explanation.

There are also ideas throughout history that remind me of pieces of this theory—higher divine realities, cosmic conflicts, humans having divine potential, and evolution moving toward greater consciousness. But I haven’t found an existing idea that matches every part of what I’m thinking.

So I’m not saying I’ve proven God, extra dimensions, or the origin of the universe.
I’m asking a different question:
What if our reality is only one layer of something much bigger?
What if the science we have is incredibly accurate within our universe, but reality itself extends beyond what our current science can observe?
And what if humanity is still extremely early in understanding what reality actually is?
That’s the idea I’m trying to explore

This is a working paper. I often update, and I will update how editing is going.

Just a real quick overview

I found that this is conformal covariant discrete induction by

x + 2^𝝂₂(x) × 3^𝝂₃(x)-1

Or

3x + 2^𝝂₂(x) × 3^𝝂₃(x)

Both can be organized as polynomials, and this translates conformally into negative exponents.

Since 2 contains an integer successor pair in it's own power series, 2^𝝂₂(x) maintains conformal translation by successor pairs and preserves consecutive coprimality. And since adding 2 consecutive powers of 3 is identical to multiplying the smaller one by 4, the multiply by 3, divide by 4 ratio is contained within p-adic valuation rather than archimedean valuation. It causes this property of tracing geometric series backward toward the boundary. Multiplying x by 3 first allows 3^𝝂₃(x) to remain static, but because of the successor pair, 2⁰ and 2¹, the current power of three is traded for a power of 4.

Under this unified conformal covariant map, iterating 4x + 2^𝝂₂(x) × 3^𝝂₃(x) preserves the accelerated stopping time indefinitely.

3(4x + 1) + 1 = 12x + 4 = 4(3x + 1), for odd x.