Null Simplex

@Inliytened1 The US had the strongest post-pandemic economy in the world. The entire world went through an economic down turn and to blame that on the presidential administration is foolish, especially when the US economy faired far better than every other developed nation. What else was the Biden administration supposed to do specifically? In addition, the CHIPs act and the Inflation Reduction Act will greatly help the US economy in the long run, but plans like this take years before the effects can be felt, but American’s are incapable of thinking long term.

@Inliytened1 The right voted for a man who actively lies to his base and attempted to change the results of the 2020 election because people have to “put food on the table and make sure we have money at the end of the month”, but seem physically incapable of explaining how he’s going to do that more efficiently than the democrats. The lack of self-reflection is astounding.

Perhaps you are still in the process of teaching yourself how fo best utilize AI, but I think a video demonstrating how you use AI as a learning tool could be valuable. To you these things may be obvious, but for many of your students these tips could get help us started with using AI for inquiry rather than just shallow conversation and fun artwork. Probably not as interesting as some of your other videos, but it may be useful. Things like specific techniques and types of questions that people should be asking would be appreciated.

Even if this video does not convince you of its argument, I think it brings up some interesting ideas and made me change how I think about numbers. While not mentioned in the video, I think the way commas are used should be changed to better reflect the naming convention they use in the video. Specifically, the commas should themselves be represented with an alternative binary notation and be given the names shown at the 59:00 minute mark. Since binary is so long, this convention I’m proposing would help keep track of how large the number is.

@Ero I concede to your and Leo’s point. University is better for a maths education than no university. Your arguments have persuaded me, and going forward I will soften my rhetoric on this topic. Still, professors should put more emphasis on concept to motivate the theory.

@Ero I haven’t learned much past those topics as of now. I am working on a paper for how to share resources more fairly using bachelor’s level mathematics, so I consider myself an amateur mathematician. You may have a point, but I’m not sure what you could teach on a black board that you couldn’t also teach in an online format. I don’t see how lectures on elliptic functions or singular cohomology couldn’t be taught online in the same way that arithmetic and topology are taught online. My guess is that the number of people who teach graduate level mathematics is low, and people who teach online and who specialize in higher level mathematics is incredibly rare, not that these topics couldn’t be taught in an online format in the future. You are correct about proof writing though, as that requires practice and someone to correct the work. My broader point is that math education should focus on what the math is even about first, then discuss the theorems and the proofs. For example, I had a weak understanding of linear algebra in uni. I knew that determinants could calculate the hypervolumes of parallelepipeds and that a matrix is invertibile iff its determinant was not 0, but I had no idea what these disjoint concepts had to do with one another. It wasn’t until 3B1B’s series on linear algebra where it became clear to me how matrices transform vector spaces in a way which is “linear”, and how the determinant tells you how much the matrix transforms the hypervolumes of unit hypercubes. So when the determinant is 0, that means the hypercubes are flattened, meaning the linear transformation is not one-to-one, meaning the matrix is not invertible. It would have taken my professor half an hour at most to explain this and it would have made every subsequent theorem and proof related to determinants make more sense. There is one class I took which I don’t consider a waste of time and money. It was a topology course I took in graduate school which utilized the Moore method. Essentially, the professor gave no lectures, we were not allowed to read any text books, or use online sources. All we had was a pamphlet which started with two axioms for topology, some definitions based on those axioms, and then statements based on those axioms and definitions. We were not told which statements were true and which were false, and it was up to us to either prove or disprove the statement. As the pamphlet progressed, more definitions and statements were added which were based on the axioms, definitions, and statements prior in the pamphlet. Lectures were essentially students presenting there results to the class. My only gripe with this class is that I would teach it backwards. It started with very abstract concepts and from them it built more tangible structures. For example, it started with topologies, then discussed the separation axioms 1-4, then it went to metric spaces, and finally got to the real number line. I would start with the more intuitive structures, and then chisel away at them until all that remained were the axioms of topology, as this is closer to how topology was created in the first place. This class encouraged critical, independent, and creative thinking, as well as strengthened my proof writing more than any lecture-homework-study-test-repeat style class, and its not even close. If more math classes were like this, I wouldn’t consider bachelor level’s mathematics courses a waste of time and money.

Eh the mathematics education at university was lack luster. It was mostly brute memorization of theorems and proofs with little emphasis on conceptual understanding or intuition for the math being taught. It became more about computation than knowledge. For example, I received a better understanding of linear algebra from 3Blue1Brown’s youtube series on the subject than I did from 2 quarters of uni. I learned more in uni for sure, but I was able to understand the core concepts of linear algebra much better from 3B1B’s free youtube series, which really shouldn’t be the case given how much time and money university classes cost in the US. As another example, I took multiple classes in group theory without understanding that group theory is the mathematical study of symmetry. Group theory felt like this very abstract, rudimentary version of algebra with no solid basis in reality. Had it been explained to me that everything in the class was related to the intuitive idea of symmetry, I would have been more motivated to learn the subject and the concepts would have made more sense since they would be grounded in something I understood rather than it feeling like I’m using logic and algebra to manipulate symbols to acquire some desired, yet ultimately meaningless, result. The modern education system just feels outdated. Before the internet, smartphones, and AI/LLM, it made sense to memorize an enormous amount of facts related to your field of specialization. But in this day and age where anyone can look up anything from anywhere at anytime, memorizing specific facts seems less important. Instead, education should put more emphasis on concepts and intuition as these things are easier to remember and do more to change how one thinks about the world. If your goal is to get a good job which requires a math education, then by all means, go to uni. However, if you want to learn math for math’s sake, there are cheaper, faster, and better resources online and great text books as well. Just my opinion, of course.

@OBEler I wouldn’t use the content of the video personally. I haven’t watched one of Leo’s videos in years, as I already have a pretty good idea of what his thoughts are and his videos are drawn out (always needed x2 speed while watching an Actualized.org video). However, his youtube audience is larger than his audience on this website. My main point is if he endorses the use of AI for higher purposes beyond images of unique pizza toppings, it could inspire others. The use of AI may be obvious to you and me, but a video on how to utilize it properly could be an incredibly important topic for a younger person just starting on their actualization journey.

You claim to be one of the few people on this planet who understands time thoroughly so I’d be interested in a metaphysical breakdown into the nature of time (and perhaps it’s relationship to space). My guess is that it would be too advance a video for you to put it on your Youtube channel and it would instead be something you would put in one of your paid programs, but I’d still like to see it on your YT channel if possible.

Existence.

Even fruit juice is mildly alcoholic. Hell, nonalcoholic wine is still 0.5% alcohol. Try not to worry about it. Tiny quantities of alcohol won't do anything to you. It's moderate to heavy consumption that you should be weary of.

By far the simplest explanation I’ve encountered for the intuition and fundamentals for what a neural network really is. If, like me, you’ve found yourself wondering how a neural network works but would get lost when the person explaining represents the neural network as a graph with nodes and edges connecting the nodes, then this is the video for you! It succinctly illustrates how an AI can take massive amounts of data and output something useful. PS: Thanks for the videos and Youtuber @LastThursday. Very interesting.

Primes have always been computable. The issue is they cannot be computed quickly.

Here are some videos on game theory, one of the most practical branches of mathematics. I believe that all children should learn some form of the Prisoner's Dilemma by the time they are in high school, if not sooner. What's important about the Prisoner's Dilemma is that it shows why, mathematically, selfishness emerges even when selflessness yields a more optimal situation for everyone, which feels like a relevant concept for this forum. What I found most interesting about this video is that even in this crude, overly simplified mathematical model about a game involving two entities which choose to either compete or cooperate, the most successful entities had 4 qualities: They were nice, forgiving, retaliatory, and predictable. This mimics the qualities of someone in the real world who would be good to cooperate with without being a pushover. Specifically, tit-for-tat, the most natural and obvious strategy, was the most effective overall. I absolutely love when simple mathematical models are able to capture intuitive aspects of physical reality. Counter intuitively, despite the fact that Tit-for-Tat can never outscore it's opponent in an individual match, it outscores everyone when all of the scores in the tournament are summed together. On the opposite end of the spectrum, the most selfish strategy can never lose an individual round, however it scores very few points throughout the entirety of the tournament, mimicking how selfishness may be a good short term strategy but a poor long term strategy. Here are some game theory videos by Primer showing how selfishness and selflessness could have evolved through natural selection. I was already planning on sharing Primer's videos on game theory, but as soon as I saw Veritasium posted a video on the topic, I realized now would be the perfect time to share the game theory videos.

Everyone knows about circular portals. Here's a generalization of that idea using knot portals. In the first video a trefoil knot, the second simplest knot after a circle, is used to connect six different universes depending on which angle you view the knot from. Here's a lecture on knot portals from the mathematician who created the concept.

Beautiful video demonstrating a simple editing technique which isolates motion in footage. It demonstrates how a shift in perspective can drastically change the context of something while adding another layer of depth to it. This technique seems both practical and fun. Here’s a second video demonstrating how similar techniques have medical and industrial applications.

Usually when people think of noneuclidean geometry, they think of hyperbolic space or sometimes spherical. But for 3 dimensional spaces, their are 8 different types of geometries which are possible (if I understand the theorem correctly). This first video is a tour through these 8 geometries using balls to demonstrate how space and light curves in these different geometries. In at least one of these geometries, shapes which in our flat, Euclidean space are deemed paradoxical or merely optical illusions created by playing with perspective on a 2D image are in fact viable objects in these realities. Here is one such shape, the Penrose Triangle. Clearly this shape cannot work in our world. However, in Nil Geometry, the shape is perfectly valid due to the twisty nature of the space. Here is another such example of an impossible shape which is possible in Nil Geometry, the Penrose stairs. These stairs have you either constantly ascending or descending depending on which direction you circle them, and were featured prominently in the film Inception. This video shows what it would be like to walk up a set of Penrose stairs in Nil geometry where it is possible to constantly ascend or descend. When your grandparents talk about walking up hill both ways to and from school, they actually lived in a Nil reality.

A physical simulation of half of a hyperbola and waves originating and propagating from the focii of the hyperbola. The hyperbola reflects the waves, but due to the properties of a hyperbola, it appears that the waves go straight through the hyperbola, creating perfect circles centered at the focii despite the circles being reflected. This video is a similar concept but for the sister of the hyperbola, the ellipse. Visually this simulation is boring, but mathematically I find it interesting. Another interesting video on the reflective properties of an ellipse. This one is more visually stimulating than the previous ellipse video.

A video on a very strange sequence of numbers with large implications on various branches of mathematics, some of which are very abstract and completely counter intuitive. For example, if one could compute the 27th term of this sequence, than we would automatically know whether or not the Goldbach conjecture is true or false. Similarly, the Reimann Hypothesis can be proven to be true or false if the 744th term of this sequence can be computed. There are terms of this sequence whose value can NEVER be proven to be true or false (true but unprovable), meaning the term has a definite value, but there is no way to tell what that value is given the type of mathematics we use today. It's not a matter of using more advanced mathematical techniques within our current mathematical axioms, you would have to use different mathematical axioms to begin with. Here is a follow up video that I had not watched yet when I shared the first video. This video basically goes into more detail of some of the strange properties of the Busy Beaver's Sequence mentioned in the original video. This video is more interesting than the original video, since it demonstrates that when a finite system like mathematics goes meta, it runs into strange issues involving the nature of infinity. Here, this sequence looks at all computable algorithms with n states which halt when given an infinite string of all 0's. However, the sequence itself is not computable after a certain point. Very counter intuitive. In addition, it links the Busy Beaver's Sequence to the famous Collatz Conjecture.

Not sure if this has been posted before. Didn’t want to search all 100 pages.

This question feels self-referential to me. Though I prefer UnbornTao's answer.

I personally found online resources such as Youtube, Khan Academy, Math Stack Exchange, etc., to be a far more effective learning tool than university (I got my bachelors in pure mathematics and am working on a mathematical research paper currently). I preferred online resources over university mainly because you are able to go at your own pace and really absorb the information, where as in university you are forced to go along with the speed of the class. The other issue is that professors rarely go into the intuition about the math concepts and instead skip straight to the method used to solve a problem with little to no understanding as to how the method works in the first place. Statistics classes are an especially egregious example of this. I would stick with Khan academy and Youtube personally, but I have ADD and learning disabilities (including reading and note taking speed), and so I have my own biases against university style education. If you get stuck, you can search around the internet. There’s a good chance your question or a similar question has been asked already on the math stack exchange, and if you are unable to find it you can post the question there. The main issue with this sight is that they can be a bit rude, blunt, and anal with your question, but they will still likely answer it (assuming the question isn’t asked too frequently and can be easily found on the site). Lately, I have also been asking math questions to chatgpt. Currently it feels like a friendlier version of a google search, and is easier to just ask it a question and see what insights it has rather than rummaging through a bunch of ad-infested links in order to find the answer to your question (though you should be doing this as well). You have to be very careful with chatgpt when using it as a source of information though, even more so than with Wikipedia. Often times the responses I get are just flat out wrong or self-contradictory, and I may not have been able to catch it if I hadn’t already known so much in the field. It is probably best to just ask it a question and then look up it’s response on the internet to verify whether or not what the AI was telling you is true. Lastly, what was the question you got stuck on? I’d be happy to explain it to you.