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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

@melodydanielluna If I go off the title, the balance is really nuanced and you build it dynamically with experience. You are uniquely different, so there's no formula. It's evident you can't only break your back working, cause you'll burn out and live life without any enjoyment and if you only relax, you'll never be fulfilled. When, how exactly, what frequency and so on - that's for you to figure out. There's no inherently bad way to relax, you just have to be brutally honest with yourself whether you are doing this in the healthy manner or not. And from you asking the question, I'm pretty sure you are not aware. I myself was playing a lot of games a few years back. I stopped them altogether. For this stage of your life, it's best to redirect that energy for more constructive relaxation practices. But of course I can be wrong. That's why it's always about where exactly you are at.

@StarStruck That's gonna happen a lot. Women are just flakey until they get attached. Just move on. Analyse where maybe you weren't on point with your game, improve and move on. There could be a thousand different reasons, some of which may not even concern you. So don't overthink it.

@AlwaysJoggin What @Preety_India is pointing to is worth considering about your attitude. When a man is needy and sexually frustrated his perspective is greatly distorted. She's right about your framing and wording. At some point when you wish to have a worthwhile GF, she wouldn't tolerate this attitude. But for you to reach this point you have to build yourself up and learn how to attract girls. For that what @Leo Gura refers to is the most effective way of building yourself up. But first drop all of this bitter rhetoric, because that is only slowing you down. And that is coming from someone who had immense frustration with women and feelings of incompetence.Believe me, I had the same exact story like yours at the beginning of 2020. What you were espousing is literally cancer for your development as a man. So it's best to stop victimising yourself and take ownership. You shouldn't feel like that she owes you, because that's just not true. If you want badass women to be attracted, you gotta be worth it. And that's not gonna happen if you get stuck in this debate.

It definitely takes time. There is a reason you have buried those those things in your psyche. Exposing yourself to them at once will break and traumatise you. That said, the more consistent you are, the more momentum you build over time. In the beginning everything is fuzzy, confusing and overwhelming. But if you keep on, you'll eventually push through.

I had a similar disillusionment. But the fact is, finding your vision is not going to magically make you a disciplined and without resistance. In fact, even the opposite. So you are on the right path. You have to work up gradually. Keep reminding yourself your vision, start breaking it down in intermediate steps, and start acting. Make things digestible. Otherwise it will be overwhelming and that's when the greatest resistance comes. You have to start acting, because you are frozen from overthinking. That's the only way out of procrastination.

Greetings to all of you. It was one year ago that I decided to finally drop all of the mental games and dive head first. To give you some context - I am an 18 year old, living in one of the most underdeveloped countries in Europe (and also in a really toxic family environment - Stage Red/Blue). Without going into much detail, there was serious ultra-conservative ideological pressure and many, many limitations. It became clear to me that I cannot develop myself without first addressing this fundamental issue of my environment that would keep stunting my growth. You see, I very much liked to think that I was spiritually gifted - I was a meticulous meditator and yoga practitioner. Whether through meditation, inquiry or psychedelics I had many realizations and states of consciousness of Love, Infinity and Oneness. But however developed I thought myself to be there were recurrent neurotic patterns, backsliding and frustration. It culminated with a series of trips all of which were clear in their message- "come back with focus". However painful at first, I accepted that unless I address these more pragmatic issues of my environment, I wouldn't be able to develop myself. I wanted to somehow feel "talented" and "gifted" by skipping those early developmental stages. Of course, in the long-term this would actually have been an extremely immature and unconscious decision missing the fundamental systemic nature of our lives. I cannot possibly describe all the things that changed over the last year, so I will list only the most impactful: 1) I sought out a mentor - an American businessman and educational activist who had a project at our school. (Stage Orange/ Green). He became a crucial pillar in my development by counteracting the impairing influence of my family. 2) I moved out from my parents' apartment and have been living independently for 7 months now. 3) Where previously I would struggle with relationships, I really got determined and eventually set this part of my life in motion. Over the last year I met many new friends and romantic partners that led to really valuable experiences and lessons learned. 4) I got my shit together and decided to do everything I could to study abroad. The only way for me to really charge my growth was to change my environment. You see, I am very aware of the systemic limitations and ideological framework of a university, but you have to understand that this really is the only option for someone of my background. So after almost a year of gruesome hard work (and many more prior) I am the only one from my country that got accepted at Harvard this year. 5) I started clearing up my life purpose through having many different experiences - conducting scientific research, working on NGOs and businesses. I became aware that I have a distinctly different path than any standardized path could give me, so I have to put in the extra time to figure it out and make it work. There were so many lessons and realizations, but since I chose this to be addressed to other people in my situation, these are a few things I would have told my myself one year ago: 1) Sorting out your problems on a pragmatic level is not in any way less spiritual - in fact you really have no other option if you want to be as conscious and as developed as possible. Otherwise those problems will haunt you as your shadow, manifesting in toxic habits, dissatisfactions and even health issues. 2) Have as many and as diverse new experiences as you can. One of the most important lessons for gaining clarity is to experience life as much as possible - meet new people, try out things that interest you, go to different places, read different books. This would not only boost your creativity and general satisfaction, but also give you a clear sense of what you want out of life. 3) Set clear boundaries for yourself of what you accept as friends, partners and acquaintances. This is the only way to start elevating from your toxic environment and relationships. Your satisfaction and fulfilment should be a priority, otherwise you will build up resentment. 4) Focus on you. Stop debating, conversing or trying to change anyone's opinion or view of things. If you cannot cut the toxic people off, at least minimize as much your confrontations as possible (for example I stopped debating or talking about anything with my family that I knew would lead to ideological conflicts) 5) Your environment is an extension of you. Be deeply aware of how it influences you and your development and constantly strive to improve it. 6) You should work on all aspects of your life systemically. There would be times for more focused work on a certain area/project, but in general you don't want to leave a part of your life unaddressed. I of course had many visions and spiritual realizations, the results of countless hours of contemplation, but I decided to make this post as pragmatic as possible. I hope this helps anyone who finds himself in a similar situation.

I appreciate very much the support. Same goes to you while figuring things out I understand. In my case the money I have came from merit scholarships for accomplishments, work from funded projects where there are operational costs, and some of the work I do as an intern for my mentor. It's way less than what I would make than if I worked, but in my case all my time had to be strategically directed for uni prep and exams. This is not sustainable long-term, because I eventually would run out, so it was a risk I had to take. But I am grateful things worked out - I would have to leave the apartment right around when I have to leave for the US. Your situation is probably distinctly different, so I don't know if this would help you, but what I would advise is to plan it strategically. Look for opportunities that would suit you and what you want to achieve in the long-term. You may live in a city where the rents are ridiculous, so that is something you also have to take into account. I am certain you can work things out. This is not good financial advice in my opinion. The bank loans you are referring to are the fastest way to chain yourself to wage slavery. I've set a rule for myself to not take out loans under any circumstances. As for looking at ways to make money on Google - this assumes there is a one-template solution which of course in our world is not the case.

Hey. I am an 18-year-old in a similar situation to yours. I wrote a post in this section concerning my experience in figuring things out. For starters, it's best to discard the notion of one "True Path". You are a distinctly different person with your own interests, passions and life situation. I would advise you to start figuring out what you want out of life. This comes from experiencing many different things and seeing what you resonate with. As younger people, we have to start setting the foundation for our life to come. Check @Leo Gura's videos titled "Advice for young people" part 1 and 2.They are greatly illuminating and I think you will take a lot from them.

IME you should do whatever feels right for you. In my case despite going all in, I suffered a lot because of unaddressed issues within my psyche and environment. So it meant I have to sort out that first. You may be different. One thing @Leo Gura has mentioned is on the issue of ambition - I know for sure fact that the hermit lifestyle isn't for me, because I would be miserable. I previously haven't had a problem with sharing my background on the forum, but I have given information on this post that could make it trace back to me and I prefer to evade this for now. I rented a small apartment. It was relatively inexpensive, because it's in a neighbourhood at the end of my city. I am affording it through scholarships and different work I have done. Thank you for your words.