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I am perfectly aware. I have brought up Gödel's implications with other people from the math department, and I am yet to get a response different than "this is philosophy not math" or "this does not mean anything in the the field I care about" That said, I still think the effort of being precise in your language when speaking specifically about mathematics would benefit you for the aforementioned reasons. For the sake of intellectual integrity. One of the reasons I moved away from physics and into mathematics. My belief is that the current QM interpretation is in fact what is the bottleneck in fundamental physics. Again, I totally understand. My uni's Theoretical Physics department is all String Theory BS.

It seems like you are assuming a straw man of my position. Without trying to be combative, I challenge you to quote me where you seem to think I trivialize the implications of Gödel or that here are no larger deeper implications. All throughout this discussion I stressed the fact the metaphysical and epistemological ramifications of Gödel's theory align with your understanding of the limitations of formal systems of reasoning. Nowhere have I claimed math is the "ultimate tool" or that one can describe reality by evading paradoxes. I fundamentally use "Consciousness" as my ontology and not some physicalist or mathematical belief system. My position is that all formal systems, philosophies and such are not "territory" and can as such not "cover it", because Reality cannot be grasped through a "subset" of itself. No ultimate TOE can exist other than the statement that "All is Mind". Math is orthogonal to spirituality, similar to how "enlightenment" work does not substitute development on the spiral. However, that doesn't mean that all systems are created equal. If we were to have relied on Terence Howard's "math" we wouldn't have left the caves. The current iteration of math has allowed for all of our understanding of physics and computation. For example, the "AI" you changed your position on and have recently been examining more deeply is all math. Furthermore, I contest that the ability to construct literal intelligence out of silicon and electrons will fundamentally rely on all of contemporary mathematics and very possibly more than that. You cannot have an "AGI" system that equals human intelligence if you cannot embed mathematics, a human creation. That is not even mentioning the functorial relationship between an ML model and its "neuromanifold", i.e parametric space/ descriptive dimension. But it won't be Terence's or Salvijus's models that do that. Is math "incomplete" in the imprecise sense - of course. But that doesn't mean there isn't value in examining its dimensions. This is in fact a realization I had a year ago on 300 ug of LSD that pushed me deeper into math - that it in fact occurs in consciousness and that the mathematical realms are no less "real" than physical reality. (I was remembering Ramanujan's biography - an almost spiritual access to these realms without needing "proofs" - the formal logicist's ontology). In a similar fashion to how you explore consciousness, I examine these realms because of the clear value they have in building more refined understanding of our current dream - UFOs, intradimensional travel and communication, artifcial inteligence, synthetic life, genetic engineering - I contest that all these God-like technologies will only be possible through some future version of mathematics - post-rational, holistic and complex in ways we yet don't fathom. But that will not happen without rigour and technically-inclined individuals. I have watched all your videos on deconstructing science and paradoxes before I even started to study math more deeply. Nowhere was I being inflammatory, combative, or "playing gotcha". I was simply communicating to you that while people without a math background may not find issue with the less precise, you are possibly missing the opportunity to communicate your ideas to people that are more mathematically educated (I am again stressing the fact that I do not have resistance to your ideas due to you not phrasing them technically, as I said earlier). Similar to the pre- and post-fallacy, I believe that integrating the correct terminology and being careful about making precise statements when discussing these topics only would benefit you and the world by making your ideas more accessible, . It is creating unnecessary resistance to your ideas that I believe would benefit greatly the precise people that may conflate you with a woo-woo podcast armchair philosopher, the likes of Terence Howard. That is all. I respect the work you are doing and believe it is important for more people to listen to it.

@Leo Gura I am not disagreeing with any of your last statements, they are in fact non-controversial when it comes to people who study the foundations of math. Which is why I said earlier some future form of math will be about modelling the taxonomy of the various models themselves. You can for example choose which side of the paradox you want to examine (similar to Euclid’s 5th axiom depending on which you can switch between Euclidian, Spherical and Hyperbolic Geometry) in a yellow-stage approach to Math. The thing is, some of your statements were indeed imprecise, such as “all finite systems are inconsistent”, which you later refined to the more precise statement “any sufficiently complex system will have paradoxes” which is generally true. Again, my only pushback is not about your epistemological and metaphysical positions, with which I agree (logical formalism is not in fact the contemporary base of mathematics precisely because of Gödel), but rather that at times you state your position ambiguously when referring to terms with precise definitions. That’s all. I know you don’t like to get hanged up on stuff like that, which is why I removed my original post about this.

I think he is referring to Russel’s Paradox. It is true that in ZFC there is a distinction between “pure sets” and the type used in Russel’s paradox, which leads to inconsistency.

@zurew I actually wrote a statement that started as: " Leo, I don’t think zurew is being dismissive here. While from what I can tell you do have a deep understanding of the epistemological and philosophical limitations of math, you are indeed making imprecise statements that are generally not true. " (I referenced a few statements with corrections) "I think locking down on the terms and making sure you are more precise with your statements would help when speaking about math, because whilst I can overlook the ambiguity and understand the bigger point you are making, other people who have a stronger math background may get hanged up, which is something you can generally evade by just using the appropriate terms" But I decided to hide it, because tbh I am slightly tired of engaging on math topics in this forum, since it requires a certain level of care about the math itself which I don't think Leo himself is really interested in or many of the other people that engage in this discussion.

@BojackHorseman You are passing through the stage of nihilism (called also "dark night of the soul"). It occurs when your previous belief system is disbanded and you lose that sense of purpose. I don't know how long it has been for you, but it took about 6 months for me to get out of it when I was in a similar situation. I was in a state of "grey-ness", where I had no emotions and nothing seemed interesting. The way I got out of it is by starting to slowly engage with life in order to find a new purpose. Of course life is objectively "meaningless", because otherwise you wouldn't have the ability to choose what is meaningful for you. Two routines that helped me greatly were walks in nature and running/working out - both provably elevate your internal state. Through continuous contemplation and experience I was able to find a new intermediate purpose that later revealed itself as a life-long one. P.S - A lot of people still feel like zombies, hah. You are clearly higher consciousness than most, so it will inevitably feel this way. But you will eventually find people closer to you to match with.

Wait till you hear about the General Linear Group and its representation embeddings

I think Leo is just conflating the two terms (complete and consistent) which have precise meanings. From everything I have watched and read from him, I think he understands pretty well the limitations of math, precisely because of this very real existence of self-loops and references (the proof of Gödel is itself one)

You keep saying this but are yet to show us how would you readjust the axioms. If you are really that genuine about a civil debate, here you carry the burden of proof (have you not here the hebrew proverb “prove you don’t have a sister”?) Multiple people provided you with variety of demonstrations why you are wrong. I will remove myself from the discussion with this last example. From what I can tell your understanding of mathematics is only as far as arithmetic. So let me give you an example in geometry that you are clearly overlooking. Consider the unit circle S^1 defined in the Cartesian coordinate system through the equation x^2+y^2=1. You are aware of trigonometry, yes? You can redefine the xy coordinate systems into polar, whereas since the radius is one, you just have x =cos a and y=sin a. Now, have you heard of complex numbers? You can define that same circle using Euler’s identity e^(i pi a) = cos a + i sin a (this is the geometric meaning, there is also an algebraic one as the continuous Lie group U(1) that rests in the axioms mentioned earlier). By refusing to accept the existence of e^0=1 you are essentially saying that the circle is missing a point at (0,1). Yet i can clearly draw a circle “physically” without lifting the pen. How do you reconcile that?

There is a trade-off between consistency and decidability/ completeness. Gödel’s incompleteness theorem states that any formal axiomatic system strong enough to represent arithmetic is either inconsistent (i.e one can derive incorrect statements) or incomplete (there exist statements that are true but cannot be proven). The proof is in fact a genius self -loop. By labelling every symbol in math with a number, you can construct a number for each statement by raising primes in the corresponding order - say the number for 1 is 6, for = is 4, then 1=1 will be represented as 2^6* 3^4*5^6. Gödel showed that by this method, one can construct a statement that literally states “I have no proof”. Either the statement is true and there is no proof (incomplete) or it is false and that means there is a proof to a false system. A more real example is something called the Continuum hypothesis, shown to be undecidable in our current system. Here is the mindfuck - there is only a countable number of statements that are provable and an uncountable amount of unprovable statements. This in fact means in some far off future mathematics will actually be about modelling the taxonomy of different axiomatic systems/models. So yes, there are indeed an uncountable amount of different models.

I already showed you that your model is inconsistent. You cannot have a functional relationship that matches for the same value two different results.

@Salvijus The reason I asked what would the problem be was not to be a jackass about it, but because if you want to develop new math and be taken seriously, you want to be able to communicate what are the problems you are trying to fix and why we would want it. You have to understand that math did not become what it is because someone said so or because the “annunaki” gave us multiplication tables or whatnot. Mathematics is a millenia-old endeavour (the oldest of all sciences) that has been rooted in all the basic abstractions we would want to have as humans, such as arithmetic, geometric objects and so on (most of which were formulated independently across cultures - Hellenic, Vedic, Arabic, Chinese, etc.). Many of the current fields, such as Complex Analysis, Algebraic Geometry, Representation Theory, etc. actually emerged from wanting to answer basic questions about arithmetic (such a highlight is Fermat’s Last Theorem - look it up, fascinating stuff). Furthermore, some of the smartest humans to have ever lived have spent their entire lives on it. Consider child prodigies that never stopped working on math, such as Terence Tao and Noam Elkies, who is one of my professors. The likelihood that you have found an inconsistency in arithmetic, crowned the “queen of mathematics” by Gauss is close to none. The “palace of mathematics” is one of the most solid “structures” in the collective intellectual pantheon. That said, that doesn’t mean there isn’t place for revolutionary work. If you are familiar with Kuhn’s “The Structure of Scientific Revolutions”, you may recognize that math is not immune to paradigmic thinking, in fact the opposite - due to its size and complexity, majority of the subfields require specialization and an epistemic “trust” that the theorems proved in some adjacent field to yours are indeed correct. Novel work occurs at the interface between different fields, by demonstrating that different concepts are in fact one and the same (isomorphic) - Grothendieck’s Algebraic Geometry and the Langlands program are examples of such revolutionary work. To conclude, due to the fundamentally social nature of mathematics, you have to understand that unless your model allows for all of current mathematics, whilst very clearly solving an inconsistency/paradox within it or very fundamentally reformulates a field, the likelihood of it being accepted as the new canon is non-existent. Whilst people like Terence Howard may be able to sway people who need to brush up on middle school math, he is never going to be accepted as revolutionary, because this type of “rewriting” of some of the most basic arithmetic concepts has already been evolutionarily discarded as part of the cycle of paradigm formation in mathematics as an epistemic endeavour.

What may this problem be? I would appreciate if you can formulate it similarly to how I showed you that without this “rule” you get inconsistent results from what should be the same operation (i.e it is not well-defined)

As long as positive exponents are defined as the repeated multiplication of the base (which is how it has been defined in the first place) the need for well-definedeness (homomorphism) necessitates that negative exponents are reciprocals. To see this just use the example from above, namely, let 2^(-1) = x be the unknown and let us solve for it: For a homomorphism, we need h2(a + b) = h2(a)*h2(b). Replacing h2(-1) = x, we get: 2^3 * 2^-1 = 8 * x = 2^(3-1) = 2^2 = 4, hence x = 4/8 = 1/2.

@Salvijus So, what you are constructing is a different definition of exponentiation as a binary operation. When doing this, you always have to establish that is it is well-defined, i.e it is a homomorphism. Without drowning you in technicalities, consider that a binary operation is defined as a map , whereas when I give you two element, you return one based on the rule you have established (mapping two elements to one). In math, you can note this as a function, i.e 2^3 can be written as h(2, 3). Now, let us simplify by considering only when using base 2, as in all of your examples. Let this function be h2(x) = 2^x. Now, a necessary property when defining a binary operation, is to have that whether you perform the operation first within the parenthesis or after, it should be the same result (this is the meaning of a homomorphism - otherwise your operations is not well-defined), i.e h2(a + b) = h2(a)*h2(b). Using your current examples, let us consider a = 3 and b = -1. By what you wrote, h2(3) = 2^3 = 8 and h2(-1) = 2^-1 = 2. Then h2( 3- 1) = h2(2) = 2^2 = 4 but h2(3)*h2(-1) = 8*2 = 16, which is clearly not equal to 4. Hence, the operation you created is not well-defined.

Ah, I see what you mean. Yes, you are correct, no disagreement here. The epistemological position of empiricism applies to physics, i.e providing experimental demonstration to a statement is sufficient to establish its truth. In mathematics, the epistemological approach is different, whereas you need to provide a "proof" - a sequence of logical deductions based on collectively-established truths, namely axioms. These axioms can change and you can even construct something called "mathematical universes" where the same "proofs" lead to different statements. It gets really loopy. To conclude with an example of why you are right, consider String Theory - mathematically correct, but physically not even close (since we are not in 11 dimensions and the cosmological constant is positive not negative). Which is why people have won Fields medals and not Nobel Prizes, hah.

It follows from the fundamental properties of exponentiation, namely the additive property. When you have the same base, i.e a^b * a^ c, you can add the exponents : a^ b * a^ c = a^ (b+c). When you define negative powers as the reciprocals, i.e a^ (-b) = 1/ a^ b, you can convince yourself that a^0 = 1: a^ b * a^(-b) = a^(b-b) = a^0 = a^b* (1/a^b) = 1 due to cancellation.

I think Leo is referring more to the fact that the origin of mathematics was grounded in everyday reality (i.e empiricism), which is kinda true. Multiplication and addition had a physical meaning before they were abstracted as binary operations, whereas with the current axioms you could theoretically define any arbitrary set with a multiplication table. For example, in the binary set Z/2, i.e the set of {0, 1}, you do indeed have 1 + 1 = 0. The same was true for Calculus when developed by Newton and Leibniz - they almost entirely used their physical intuition. It was developed into a rigorous mathematical discipline (Real Analysis) much later when inconsistencies, such as Weierestrass' s function, appeared as pathological. That said, current mathematics operates in realms that are qualitatively different than reality. That doesn't make them less real or interesting, so you are generally right to question the idea that ALL of mathematics is empirical. The philosophical position that only mathematical objects that can be realized in reality is called finitism and has a host of problems on its own. Traditionally finitist reject the existence of infinity, imaginary numbers and a host of other very useful concepts which have allowed us to formulate Quantum Mechanics and General Relativity, which are clearly very successful (you having a screen to read this is further evidence).

TLDR: It is called the axiom of multiplicative identity for rings in Abstract Algebra when applied to the set of Whole numbers (Z) Whole numbers as a set defined with an operation of addition, constitute a group. The axioms of a group are: 1. Associativity, i.e the order doesn't matter : (2+ 3) + 4 =9= 2+ (3 + 4) 2. Identity, i.e an element (0) with which the operation by definition leaves the other element the same: 0 + 2 = 2 = 2 + 0 3. Inverse, i.e an element that negates the other, s.t the result is the identity : 2 + (-2) = 0 = (-2) + 2 In this case the whole numbers are abelian (commutative), i.e the order of summation doesn't matter: 2+ 3 = 3 + 2 When you add multiplication, the commutative group now becomes a ring when you have two extra axioms fulfilled: associativity of multiplication (equivalent to (1), just switch + with *) and distributive law, i.e (a + b) * c = a* c + b*c A ring can have a multiplicative identity, which in the case of the whole numbers is 1, because by definition when you multiply 1 with anything, you still get the same element: 1x2 = 2, 1x3 = 3, 1xa = a, so just replace "a" with 1 and you have 1x1 = 1.

As someone who studies graduate mathematics at a top university, I actually agree with this. The axiomatic approach was created for the sake of consistency and it is already leading to problems. My intuition is it represents a Blue-stage conception of mathematics as a formal logical system, something Gödel already disproved. That said, the braindead stuff Terence Howard espouses is Purple-level mythology that is entirely disconnected from that precise naturalistic origin of mathematics - as an instrument of abstraction to reason about reality. P.S. The dichotomy of “realness” of mathematics vs physics is one I have found very good to contemplate about in general. Hint: both occur in Consciousness.

I think everyone should approach this issue individually and access the personal toll from psychedelics. There are examples, such as Sasha Shulgin who experimented with hundreds of different psychedelic substances and lived until 88. I also remember reading about cases when people took 10-100x the dose of LSD and were fine. Neurogenesis and -plasticity are some of the documented effects of psychedelics, but so have been psychoses and cardiac arrests. In the end, you have to make a responsible decision. Opening yourself up to consciousness is like increasing the voltage for an appliance and depending on what "alloy" you are made of, the effects may be different. For. example, I personally have consistent stomach irritation from all psychedelics and almost always vomit, whereas my brothers have had none of that from the same batch and dose.

@Leo Gura This precise exploration is one of the driving motivations behind my life purpose, that of the "Consciousness Engineer", which requires deep understanding of its innumerable facets. Non-dual and other simplistic philosophies do not properly appreciate the sheer complexity of Consciousness. Taming the Chaos is the natural evolution of my current human dream (I take greater inspiration from Tesla, Ramanujan, Grothendieck and Shulgin moreso than from Ramana, Ralston or Spira) The thing is, I don’t believe there are tools even remotely close to what psychedelics have to offer in this exploration, a realization you yourself share. Having experienced hellish realms and their physical toll, I nonetheless believe that is a cost I am willing to pay. Considering what you yourself had recently shared with respect to psychedelics and insanity, I wonder whether you think the latter is problematic. I believe I am uniquely grounded and resilient, considering my and my people's life experience, and moreso than that I see this as the only way of creating abundance for without abuse or exploitation. But of course, the Chaos can bite my arm off in the process.

A few years back I remember Leo posting this video about the hyperbolic geometry of DMT experiences that had a profound impact on me: Fast forward to now, as a math student at the same university I have the unique opportunity of continuing some of this work. I would be interested to hear about people's experiences with various and possibly more obscure psychedelics and what "form" the different trips took. I personally have about 10 trips on mushrooms and 20 on LSD with varying intensity which I am currently using as a reference. Although they have been very significant in my spiritual journey, they are a small "sample size" from the molecular variety accessible. Feel free to share any experiences, insights or suggestions you may have. Post-rational strange-loops and paradoxes are more than welcome, this is an epistemically open discussion.

This illuminating NY article tells the story of "Christiania", a 50-year old Danish anarchist commune that is currently shutting down due to failure of handling organized crime groups that have lead to surge in violence. “The mafia doesn’t listen to anybody” is a quote of one of the long-time residents. It is a perfect example of a clash between the green and red spiral stages - the former approaches issues with "acceptance" and "kindness" that is then perceived as weakness and exploited by the ruthless conduct of an organized crime group. As someone who comes from a country riddled with violence from "mafia"-style organized crime groups, I have grown directly aware of the non-trivial and highly complex nature of ensuring safety and freedom for a group of people. It is not a given, nor a guarantee. Libertarian and anarcho-utopian philosophies/ form of governance will fundamentally fail due to its naive model of people and social systems. What is to stop non-state actors, such as drug cartels and neo-liberal corporations from extracting all value at the expense of your and your family's life and well-being? A strong government. Something that I have realized in the last 2.5 years of living in the US is taken for granted.

Cute U clearly have a lot of free time