What is mathematical thinking actually like?

[@benjaminkeep]

Here's the GiveInternet.org donation link: giveinternet.org/BenjaminKeep

[@SonOfSofaman]

"In some cases, misunderstanding the nature of what we are trying to learn is what impedes progress." -- very insightful. Thanks for sharing that!

[@rbpompeu1]

And sometimes, misunderstanding also may generate new approaches and ideas like Kojeve's (mis)interpretation of Hegel in Lacan's work.

[@scepticalchymist]

I once showed a student the graphical representation of (a+b)^2 and he got super excited about the geometrical insight hidden in plain sight of the algebraic formula. It was great to watch his joy about such an 'obvious' fact.

[@guillaumeleguludec8454]

My teacher used to say that you never understand math, you just get used to it. It's all about reshaping our intuition. When learning math it is often the case that a result seems miraculous (it contradicts our intuition in a way that makes us want to know more). Then we try to understand why it is the case until it becomes trivial and almost uninteresting, which is actually positive as it means we have succesfullly reshaped our intuition.

[@OmnivorousPancake]

btw, that is a famous quote by John von Neumann

[@user-li7gm7gv3v]

This thinking is called mathematical induction! Perfect for when you don’t know exactly what to do or which formula to invoke. :)

[@AlumniQuad]

Let me correct you: It's six plus minutes of lost-in-the-weeds introductory filler just to get to mathematical induction. At least this KZbinr will have maximized his $$$!

[@greger-u4y]

@@AlumniQuad !

[@jokelot5221]

This is similar to how programmers think while problem solving. Creating these mini examples to understand a larger whole. Like testing with something small, and then using that pattern to develop something big.

[@yavor05]

It's interesting how GiveInternet that this video advertised are doing somehow "divide and conquer" that programmers use to solve problems. Access to Internet is part of the problem of accessing education and as such, tackling it is tackling the access to education problem itself. :)

[@markkennedy9767]

8:30 is a very good description of learning maths. Going from struggling to understand a concept in maths to finding it trivial or obvious almost immediately and forgetting about your engagement with it. Its like the struggle is the thing that makes maths appealing, what makes maths worth it. I think this is a fairly common thing in university courses. Next year's maths is impossible. Last year's maths is trivial. The teaching of maths is arguably what is crucial. As you would think that anyone can understand maths since it is all just a tautology. 😂

[@varunv2584]

These kinds of mathematics stuff, is the most pleasurable thing to me.

[@griefingcloud4391]

Decent and straightforward representation of mathematical thinking. I think this was insightful.

[@Nothingx303]

So how can we increase our Mathematical thinking?or what an advanced Mathematical thinking looks like ??

[@pyepye-io4vu]

@Nothingx303 Some helpful books "How to Solve It" by George Polya (1971), "Mathematical Discovery" by Polya again (1981) "Mathematics and Plausible Reasoning" by Polya again (1954, two volumes) "Thinking Recursively" by Eric S. Roberts (1986).

[@julioaurelio]

​@Nothingx303 Read Polya, he's the master.

[@Genedide]

Mathematics can improve its approachability if produced non-genius underdog stories. Someone who indeed sucked at math durning their K-12 years, but someone came to love it later, got good at it, and then became a legend!

[@pyepye-io4vu]

"The Art and Craft of Problem Solving" by Paul Zeitz is a great book that shows mathematical thinking and strategies in detail. "Solving Mathematical Problems: A Personal Perspective" by Terence Tao is also excellent, and very short (about 100 pages).

[@benjaminkeep]

Thanks for the recommendations! I have The Art and Craft of Problem Solving. Really love it. I'll check out the other one when I get a chance.

[@Voltarez]

very cool, wish they had taught more angles like this at school.

[@pyepye-io4vu]

Most teachers don't know this stuff, and it would take a lot of time and effort to teach the teachers, and then teach the students this way. Sad but true.

[@JatPhenshllem]

​@@pyepye-io4vu Doesn't mean you wont find really good examples like this in the textbook if you follow them

[@tarhelytarhely5662]

The problem is, that this process is not teachable to groups. The power of this process comes from the individual thought. Someone else's idea does not have this appeal. The textbook is full of these thought experiments, as ​ @JatPhenshllem said, but nobody cares for a reason. One day a student came to me with the question, what if the coordinate system is not right angled... The only thing I could say to him, it is an interesting question, think it through, write it down or make a video, I will read and comment on it. If I had the opportunity to sit with him for a couple of hours, I am sure, he would have learned a lot. Even if he asked this out of the will to disrupt the teaching... Most people do not find this way of thinking interesting or amusing. They feel they are thinking about a made up solution for a made up problem. They still come up with ideas, but they stop right there... They are watching this video and asking: "Why are you doing this to yourself? Don't you have anything more fun to do?" That is the reason most teacher do not even know this stuff, as​ @pyepye-io4vu said. It is unproductive. Teachers are the living example, that you can learn math without this process. If you want to be exposed to this way of thinking you have to get somebody who does it with you. School is a good place to arrange this, but the teachers will be your fellow students and their parents/relatives. Smart people, who use math, instead of teaching it, care about the integrity of their knowledge. So to fill the gaps between referential knowledge bits, they come up with analogies and though maps as years go by....

[@notgate2624]

Great video! I did very well at math in school but never once did anyone describe factoring this way. It immediately makes intuitive sense and I feel a bit silly for never seeing it. Many of the best insights in life are the obvious ones. Anything complicated is just composed of obvious observations if you frame it properly, but people have a weird tendency to underappreciated them or pretend they always knew them.

[@jonafrica6957]

Fascinating video, makes me wanna look up more mental models for math use in real life.

[@vimalneha]

Demand of earning, i stopped learning MATHs after honours, that i still mourn, but your video made me revisit my time utilization. Very good video! Subscribed.

[@Cossentine]

You were working your mathematical muscles, it's good to consider basics like this, that's where all the skill comes from

[@theimmux3034]

as someone else pointed out, this is mathematical induction. in general mathematicians think im terms of the relevant structure.

[@pyepye-io4vu]

Yes this is only scratching the surface, Benjamin is a bit out of his depth on this one (which is OK). Some good books on mathematical thinking: "How to Solve It" by George Polya (1971), "Mathematical Discovery" by Polya again (1981) "Thinking Recursively" by Eric S. Roberts (1986). Honorable mentions: "A Mathematician's Apology" by G. H. Hardy (1940) (this is more about beauty but still relevant to how a mathematician thinks) Mathematics Without Apologies by Michael Harris (2015) (partial response to above)

[@spgoo1]

The interesting thought I had was that this is somewhat cyclisation back the geometric origins of early algebra and the story of the origin of imaginary numbers. Very good video, as always, Ben!

[@lumosity2500]

You are on fire recently 🔥 Love that!

[@elunedssong8909]

Yes, Difference between squares is 2n+1, n being the base Now what's the formula for cubes? 1->2 = 7 2->3 = 27-8 = 19 3->4 =64-27 = 37 4->5 = 125-64 = 61 5->6 = 216-125 = 91 ... ? 19-7 = 12 37-19 = 18 61-37 = 24 91-61 = 30 Or, the formula for the formula is + 6n (61->91, at n=5) Or, the formula is, (1+2+3... up to n) *6 +1 We can check: 1*6 +1 =7 1+2 *6 + 1 = 19 .... We can always repeat this kind of process to figure out any object's formula, but as you can see we now have to use these head to wrap our mind around objects of (Sum from x=0->x=n) of x Maybe you can present this truth geometrically? I challenge you! :)

[@jfox8888]

Oddly, when you were explaining the math and patterns behind , I assumed it’s due to the go board that caused it . And that I’m and glad (luck would have it ) that I was exposed to get better at anything book , from Scott young Plucking and noting from existing problems and playing around with without expectations but noting down the patterns and sequences leads to more insights than traditional “ meta learning “ Perhaps we needed the AI models along to further accelerate the individual curiosity (side topic ) in learning depth and width , which Your video introduces, not by random , but by nature of nurture

[@alvarferrer592]

amazing video, super cool and super graphic! thanks a lot! greetings from spain

[@LucasOliveira-dw4zh]

I recently read Mathematica: A Secret World of Intuition and Curiosity by David Bessis and it had some insights related to your video. On the point of different representations, one of Einstein's proofs was achieved after changing the representation of one of the mathematical objects.

[@tofutherizzler]

some days when im in school or when im just bored i have mathematical debates in my head and i really thought that was normal

[@AuthenticBranding]

I am a big believer that mathematics is actually a language. Now to be clear, language isn't about the words we speak, its about communication, which uses words, but also music, metaphor, mechanics, and so much more. We can look at the same thing and get different opinions on it: the famous example of the movie with the squares and triangles. This is because we take communication into our mind to best understand it. So that really is a questions is what is math: I believe it is the language of the universe, or whatever you believe like a proof for God or the rules of evolution. In any case, it is clearly a structure, the same structure we use in language, but also in story, which is where I come in. I am a narratologist, a good one, which is good cause I am not good at anything else. I am kinda an idiot, and no where is that more true than science. However, I have been stumbling down a rabbit hole of quantum mechanics lately, and to be clear I do not understand it well, but there are parallels that are allowing me to make connections that I previously would not have been able to. Now this isn't an example of lets take the English moron and dumb math down as far as we can, which has been my process, but wondering if there are set methods that allow for people to be good at one thing over another and analogy aside how to translate that information between the processes.

[@santerisatama5409]

When we remember that silence is inherent and necessary part of language, and can be also very meaningful, I agree. Mathematical intuitions are very often "prelinguistic" when language is defined more narrowly as a formal mathematical language which the art of mathematical poetry constructs in order to express and communicate intuitions among sentient beings like us. I'm a translator and a poet, and my teen age interest in QM starting from learning about David Bohm and Jiddu Krishnamurti has gradually evolved into hobby of study of foundations of mathematics as an amateur simpleton. I remember that when reading Plato's Sophist and how the discussion of Great Kinds is dramaturgically narrated there, I got an aha moment and went: Plato is already talking QM here! Much more recently, I started to read from Proclus what Greeks actually thought about ontology etc foundations of mathematics, and I was flabbergasted: What Proclus is saying corresponds extremely closely with what Bohm is saying about explicate orders unfolding from implicate order, active information and Holomovement. Also for Proclus the term "unfolding" is the key term for describling the movement of dianoia/intuition from inclusive organic order of Cosmic Nous to parts, and then as backfeed to Holomevement. Narratology of epic cycles of oral poetry is also a movement of unfolding, just like any live performance of a composed piece or theme of music. On this path I've also stumbled on a foundational formal language of mathematics, which is very simple and IMHO beautiful and can be for starters narrated as geneologies of ancestors, parents and children. The language forms a natural quantum metric in the form that everybody can learn without hard and obfuscating jargons behind which current bubblegum and wire math of mathematical physics hides behind. If you are interested, I'd be happy to talk more.

[@AuthenticBranding]

@@santerisatama5409 Ma, as they Japanese put it. I am a narratologist who wasn't satisfied with the current crop of narrative structures. Really, it came down to a realization that all structures were a part of the same system. I was always good at math, especially geometry, but my expertise in science stopped the moment physics left the geometric area. My mind just doesn't comprehend it the same way. However, like math, narrative structure is also a prelinguistic system. It is the order and shape by which stories are able to maximize their abilities, especially as defined by Ethos, Logos, and Pathos. As I dived deeper in, utilizing a lot of psychological principles, my only other real area of expertise outside of narrative, I was able to connect different narrative paths together through a concept of how we are driven to make certain decisions. This drove me into the concept of game theory. From game theory, of which I still have so much to learn, I was pulled into a plethora of concepts that basically revolved around entropy, information theory, energy, and the idea of how the world unfolds. I do not have the ability to fully grasp quantum mechanics, nor to I have the expertise to where any thoughts of mine go above the idea of "belief", but in my limited understanding I really am grasping the idea of: Quantum mechanics basically being a look into the multiple possibilities of the future, energy being placed on those possibilities which creates the future, creating information, which then essentially creates time. This is a very poorly put together idea, but I am currently just grasping at straws by adding what narratological concepts I can put together. I am always down to chat more, but I can't promise you that you won't get unbelievably frustrated by my lack of, not just knowledge, but my struggle to even understand. Which I think goes back to the point of this video. It is easy to point out that we all have biases, not only for information like trying to rationalize Newtonian Physics with General Relativity, but a deeper system where if we can explore the deeper concepts below language like math, narrative, and perhaps even one for learning, it would allow for an environment where learning becomes effortless once we are able to clearly translate that information. That too is a narrative concept I am working on, but again my experience has a certain stopping point that will need to be aided by others when that time comes.

[@santerisatama5409]

@@AuthenticBranding In terms of narratology, we can state that the necessary creative opposition-distinction of narratology is narrator and listener. In math we like to generalize such distinctions as far as we can. I suggest that the narrator is mathematically defined as temporal movement outwards: < >, and the listener is defined as temporal movement inwards: > < also form a Boolean reversible inverse relation (the bitwise NOT operation, as it is officially called) between them. The pair can be seen with mathematicians gaze (that goes beyond the surface of the screen and can imagine what happens in the depth behind the surface) to form also a loop and/or rotation, which we can perhaps see more clearly when we reapeat them: String contains substring >< nested in the middle of it, and >< contains . The "hidden" side of the loop becomes revealed by repeating the "visible" side and and gluing together (aka "concatenating" in math jargon) the repetition of the string, so that the repetition forms it's inverse "Other" in the middle of the repetition also on this screen surface. To generalize narratology into quantum direction, we can at least suggest that mythic cycles exist primarily in 2-directional quantum time, and narratological performances of a myth are parts of such inclusive wholes that happen in 1-directional time. However, narratological events such as mathematical proof events etc. spread their waves towards both past and future, because that is the the temporal essence of the art of story telling when it is intended to create social cohesion and adaptive sustainable behavior in s concatenating touch between past and present of e.g. a tribe. When we accept that a mythic cycle exists in 2-directional quantum time, which in math we like to express as parsimonous symbolic poetry as we can and necessity requires, the next step is to generate more flesh into our mathematical narratology. To communicate with people who associate math with number theory, let's generate number theory into our story. We can do that by nesting algorithm that move from whole to parts in syntropic creative manner, which can be called "mereological decomposing" and "autopoiesis", like an organic order creates and maintains it's parts. Let's start from < > as the generator, and generate new words called "mediants" form our ancestor words < and >: < > < > < > < > etc. Mediants are children of their parent words, formed by parents making copies of themselves and joining the copies together in between them into new chid words. The ancestor words from from the first row stay with the family from row to row and keep on participating in procreation. Just like indigenous peoples of animistic cultures conceive and narrate, that our ancestors stay with us, they are are spiritually alive and active, if not alive in flesh. Speaking of Spirit, the word means breathing, and mathematical generalization of breathing is movement outwards < > and inwards >

[@AuthenticBranding]

@@santerisatama5409 I have broken up my tools into two sections: structure and framework. Both operate the same way, but structure I think is more applicable to this idea. I believe that narratology is more than creativity, but a structure of communication, not language, but communication, that allows us to use creativity to share experiences, but also can be used in direct communication, mental health, education, and so on. As such, the structure can adapt to each communication method, at least I have designed it. Now on a narratological side it can be broken down into three parts: immersion, relatability, and engagement. Immersion is the world of the communication and what this does is it primes the listener for every possibility that can happen in this space. Relatability is what is happening in the space and the emotions that are connecting the communication to the audience. The engagement is essentially does of emotion that create higher levels of engagement as they start to wear off. The reason narratology work over any form of communication is because communication is explaining what you know to someone who doesn't know. Narratology is explaining what you know to someone who does know, but from your perspective. However, yeah it can be read L2R and R2L. The author creates a world that is a dressed up one like where they live, where anything could happen, but then something did, and they moved through it as it shifted. They wrote about it from their perspective and others from their perspective understood. It isn't as clean as math I assume, though perhaps PEMDAS says otherwise, but the concept is the same. Same concept works whether using the creator of the narrator. In defining it in terms of time however, is where it gets interesting. Time itself exists and moves forward for everyone. Just in the world around us our bodies experience time. However, trauma disrupts the mental flow of time in our minds. Narrative is a flow of time measured through experiences, which is why it doesn't hit every possible event. Now reading these experiences the same L2R as R2L depends on your frame of reference. A story can be divided into 7 segments that create 5 scenes: the universal scene types being a need, conflict, resonance, connection, and value. So there is an entropy, an arrow of time that doesn't create the same symmetry within the system. However, I am also continuously working on a theory of another 7 segments that may indeed create this symmetry, but I am not there yet. There is an overall idea of a union of opposites within deeper psychological concepts that reflect narrative, but as time has passed in the field, more complex characters that operate beyond just that line of symmetry. This doesn't take away the movement being bilateral though. We have positive and negative arcs for many characters over many lines of development, however, overall at least, we do try to steer a character forward. The ability to do that does directly correlate between who they are and the world that has been created for them. This is another question of how to measure time in narrative. Trauma easily sticks people in a single moment until they can come from it. Negative arcs can also regress individuals, not pulling them back in time exactly, not to an exact spot they are in, but at least regressing from the overall goal they would wish to reach. But moving onto breathing I think illustrates an interesting point. The wave returns to the source in another way. Breathing in is the reception of oxygen required to work, while breathing out is the rest and release. We use pacing as the same kind of idea. I didn't start in quantum physics, but regular physics under the idea that emotion is a type of energy. Every moment in a narrative is an emotion, behavior, belief, and value. A character has something happen to them, they have an emotional reaction which causes them to act because they believe it will get them closer to what means the most to them. Emotion is the energy of the action. The frequency of the story is the number of actions that happen in quick succession, each one with its own breath, its own wave, which the amplitude is determined by the power of the emotion. To answer your question: it depends. Narrative structure is the skeleton that holds fantasy to the real experiences that line the mythical events. When the fantasy gets closer to the events of a particular listener, they feel it deeper, but with narrative structure, it should affect most people at least. We see this from C.S. Lewis who said a Children's story meant only for children isn't a good children's story in the slightest. A story made for kids should relate to kids, but somewhere deep down we all remember how that felt for us all. I will say I would be lying if I understood all of this, but I doubt clarification would help. To be honest, there are some days where I wonder if I am truly insane in trying to create these connections. Puzzle solving is where I shine and sometimes that can lead to putting together Lego pieces into a new form that Lego themselves would never made: the danger of building with instructions. I am still so new to all of this, but a lot resonates with me. My path started with trying to understand game theory. If you have two people who value the same thing: what is the mathematics that will dictate the percentages of what someone will pick. What if you have multiple values? That led to motivation and how emotions insert energy into actions, like adrenaline. Along side this I was wondering about how energy spreads out. Drive theory dictates that we start with one drive for the first two actions, 2 for the next 2, and 3 for the next 2, ending up with 4 drives for the last. This creates a matrix of emotions that can be felt depending on the situation and character. This is essentially the spreading out of that emotional energy which requires engagement to keep moving forward. The way we track randomness through percentage helps in development of a story and information theory and how it measures entropy creates this correlation between energy and information. This is where I started. I am currently trying to remember why I went quantum with what I was searching for, but I do know what caught my eye was the idea of every possibility being seen before it happened. Before I get close enough to quantum mechanics to truly know through, I know I need to finish my game theory studies, and even then I still have my narratology research to finish. There is a lot to do and seemingly so little time to do it, but I am glad for something to do. But this is the fun stage where I can muse of possibilities before I dive to far into the application.

[@santerisatama5409]

@@AuthenticBranding Thanks for sharing more of your thoughts. Mathematical immersion really is game theory, picking up some collection of mathematical symbols and then starting to investigate the possible in interesting syntaxes of your game pieces with playful attitude. Tthat's how we can stumble also on something new, at least from our perspective, instead of just mechanically following rules. Immersion comes from putting your hands on the games of your symbols might suggest to play, if they happen to have the power to evoke immersive flow states in your case. The game pieces < and > have given me immense pleasure, as they have opened ways to define syntropic win-win games, which in QM associate with the technical term Creation operator. The Annihilation operator (in my language >

[@szymonbaranowski8184]

the problem with these abstract ideas and patterns is that you can't remember them as small reduced pieces or pictures and while you apply it you usually do it through set of remembered examples not applying the rule taking time to compute these for new problem or data - only geniuses with much faster processing power and enough mental ram to hold more in mind can do it... or optimising this process better than others not loosing track of thinking about it... that's why we do better with less distractions and going as slow as it's required for undisrupted process and only slowly forcing speed up as it becomes easily repeated habit

[@PranaySakpal-m3l]

The thing about factoring being about forming rectangles is shown in numberblocks (the kids show) when numberblock 19 is introduced

[@eg_tasin]

This observation is neither stupid nor obvious. In fact, I hadn’t considered it before, but it highlights an important point: visualizing the topics we need to learn in our minds is essential for thorough understanding.

[@LifeKiT-i]

Nice video!

[@vanessamgates6018]

I got lost at the first "n squared." Sorry. Maybe I'll come back and go through it over and over until I can get past that piece first. I was taught that squared means that number times itself, so how can it be just two? Confused. Anyway, thank you for speaking in a way at least that doesn't make me feel like an idiot. Of course, I had a brain tumor in the area responsible for words and math, apparently.

[@rambleswolf]

Are you talking about 1:58? I'm not very good at explaining things, but he's saying that if you take a square with sides of n length, then its area will be n squared (or n×n). Then the square of the next size up from n will be: n×n + 2×n + 1. Let's let n=3 for demonstration purposes. A square with sides of 3, will have the area of 3 squared, which is 9. OOO OOO OOO (There are 9 circles in this 3×3 array) To work out the area of the next square up (which, if n=3, would be a square with sides the length of 4) i.e. looking like this: OOOO OOOO OOOO OOOO You start with the original square (3^2) OOO OOO OOO and then you add 3 to two of the sides (2×3 (this is the "2n" in the video)) O O O OOO and then add 1. O So (3^2)+(2×3)+1 = 9+6+1 = 16 which is 4 squared. Hope this helped, sorry if it's a bit of a word salad 😅

[@vanessamgates6018]

@@rambleswolf That actually helped. Sometimes when two different people explain, the explanation begins to make sense. Still not quite understanding, but I'll return and keep reading and thinking. My hope is to improve my brain's functioning in this particular area. Thank you very much for taking the time to respond, especially for not being condescending with some typical glib online rudeness. This is when the internet has meaning.

[@rambleswolf]

@@vanessamgates6018 No problem! Happy I could help! Good luck with the recovery and brain (re)training! 😁 Something I've found that's helped me is if you take an actual pen and paper and try to figure out what's going on. Treating it sorta like any other puzzle people do for fun, like a sudoku or crossword. Something about physically writing/ drawing and actively thinking about the problem helps a lot. Sometimes you may still end up confused, but that's just what learning is like sometimes, haha.

[@gamedog9542]

​​@@vanessamgates6018the number above N is how many times you multiply it by itself. So like N² is N times N, the 2 means two N's that you multiply together. So like N⁵ is saying multiply 5 N's all together like this. N times N times N times N times N Hope this helps!

[@epphrisarose7494]

This video is really helpful to me. I have been pondering what differentiates mathematical thinking from the way I think. Like I know what you're talking about. I hear the words you're saying. I have learned these concepts. But I think math has always been a challenge for me because I don't use those tools to think, about pretty much anything at all, the way you described. I use mathematical tools in specific contexts and not at all otherwise. Rather I think in a more narrative, metaphorical, philosophical way... Is probably a good-ish way to describe it. So as obvious as it may seem to you. The idea that people use math to think is very strange to me. 😅😂

[@pyepye-io4vu]

You can train your mind to think in every different way, narratively, metaphorically, mathematically, etc. They are not mutually exclusive, and you are not "born" to think one way over the other. Mathematical thinking is actually not limited to what is shown in this video. Mathematicians have to use all kinds of different thinking, INCLUDING narrative, metaphorical, etc. The more kinds of thinking you can use, the better.

[@santerisatama5409]

Representation of analogs of some basic mathematical idea in different contexts aka "landscapes" (e.g thinking self-referentially about the idea of "proportion", which is Latin translation (and not very good as such) of Euclid's term 'analogos', which is inherently associated with Euclid's term 'logos' (which has been badly translated as "ratio")... a mathematician with foundational interest may be inclined to start to think about the question: are all analogical representations relationally peers, or is there some landscape, some foundational context that is more fundamental than others, more complete in comparison to less complete landscapes? Foundational method is not so different from the first method demonstrated in this video. Staying still and letting math come to you intuitively. Greek word skhole (school) means originally leizure :) i liked the comparison to ecology very much. The deeply intuitive requirement/quest that our math needs to be coherently interconnected organic whole in order to be true can be found in Zeno and Euclid and Proclus and Whitehead. The deep intuition that ontology of mathematics is animistic, mathematics is not a dead object, a mere abstraction, but in some mysterious sense alive, an animated spirit. This representation of quadratics as "obvious" gave something novel to my obsession of thinking quadratics in the landscape of zig-zag paths of continued fractions along a binary tree, where quadratics are periodic constants in change... including even "transcendental" like e. When we include the new origami method in the old method of straight edge and compass, is e still "transcendental" in relation to our more complete method of constructive pure geometry that can now solve trisection of angle and bunch of other old problems with that? I don't know, the question just popped up.

[@gusty7153]

i like it when somehing is stupid and obvious. it makes the complicated things involved easier.

[@adam4759]

This is a really cool video with beautiful animations and a really cool philosophy behind it all in the end, but there's just one thing bothering me: the plots are incorrect, e.g. the blue line when it crosses x = 4 should cross y = 16 as 4² = 16 and the function for the plot is y = x² . Tiny nitpick and you can ignore it if you think it is not important to convey the message of your video. But as a mathematician you should always strive to be as factually correct as you can be in my opinion.

[@natanvance]

"Think Like a Mathematician"? I feel a remake of a song by The Bangles coming up.

[@dr.fabriciogerardoacevespl5390]

2:55 Regarding this,I have several questions, clarifying that they are respectful. They are in no way misleading. and they are born from a genuine intellectual concern on my part. They are these: 1 is that "extra section" (small square n1), which causes the period to change and there cannot be a "regular" iteration when factoring, due to Gallileo G.'s "square-cube law"? (Lilliputian laws) that states that, as a shape that grows in size, its volume grows faster than its surface area?. Is it related to the "nonlinear iterative transformation" of the Mendelbrot set?, and finally...is it related to the distribution of prime numbers (visualized as sets of "stacked" elements)? I would greatly appreciate your response, greetings!

[@paschalcharles6097]

This is what we call abstraction

[@openroomxyz]

Nice video thanks for the video was a joy

[@FONEternal]

I started teaching myself complex analysis recently. I wonder - would there be a rectangle to complete in 4-space if we are solving these quadratics over the complex numbers?

[@maryo1065]

What did you use to make the square models ? And How did you make the square models ?

[@Pedritox0953]

Great video! Peace out

[@user_08410]

As no.1 mathematics in my school, I m agree with him. For more, to point out mathematics people's and normal people mind, mathematics people will always think of numbers during eating, think of geometry things like angle, triangle when see special objects, think of graph when see rope, think of calculus when see things change and etc. Yea as he say is kinda like stupid, but that's where build our logic and analysis skill as we doing it will produce the system/structure to understand things, which is an important factor of being good at maths.

[@rodericwalter2862]

Thanks.

[@ucchuman]

so thinking mathematically is understanding the patterns and genralising them?

[@pyepye-io4vu]

That's a big part of it, there is quite a lot more to it too.

[@NoDoubt747]

Hi, what is the program that you use for those animations? great video

[@Rockyzach88]

Are you saying weed thoughts are actually legitimate?

[@user-op8ov1sm6t]

What are the cognitive effects of combining flashcards and free recall for strengthening long-term memory and enhancing deep understanding? Is there any empirical evidence suggesting a better approach to integrating these two techniques?

[@SimplyMaths-zu8zg]

well explained

[@shishirsanjeev4051]

Is that painting a portrait of somebody?

[@absiddi.7712]

What does "affordances" mean at 7:15?

[@darnelcantorfierro8911]

Different representations of mathematical objects have their own tools, intuitions and limitations. So, each representation ("afford") allows us to see some characteristic of the mathematical object in a specific way.

[@notgate2624]

Different representations have different "properties". They provide different perspectives and tools for analysis, visualization, etc.

[@averagetechnologyenojyer]

How can I effectively schedule revisions for topics I’ve studied? For instance, I’ve read Study Like a Champ, but I still feel confused about how to set my timetable up(but the book is super awesome!).

[@ingmat3741]

Hola, buena tarde, por favor como desactivo el auto doblaje?

[@87advil]

Haz click en la ruedita en la esquina del video y selecciona "pista de audio"

[@physjim]

thumbs up for the goban, how strong are you?

[@ruatapungawhare-pounamu4050]

Hi g, I have some questions if you would kindly answer in the future 😅

[@ConradMarbourg3635]

Somewhere, sometimes in the middle of the night Her : "What he is thinking about ? I bet he is cheating on me." Bnejamin : "how do you jump for one square to another ?" All jokes aside, I'm not a mathematician and the video was nice to watch.

[@madisonsherrie7793]

I guess I know I like math since I paused and 1:30 and went to go figure it out myself 😂

[@Safarti]

lol thats me whenever im bored

[@reversicle212]

Please make videos on IQ and related topics. Does being low IQ (average or just below average, while not having ASD), mean that you'll never succeed in a career involving STEM (CS, Math, Physics, Chemistry etc. majors)? Can a person with that IQ ever complete a PhD? If yes, then what's the benefit of having high IQ?

[@pyepye-io4vu]

(In the US) Average IQ of math PhDs is around 130, average IQ of physics PhDs is around 140. That should tell you something at least. In my PhD program, at a certain point I could feel a "wall" that I could not overcome. Not sure if it's due to IQ or not, but you'll get stuck at some point. IQ seems to be the bare minimum prerequisite, it takes a lot more than IQ to finish it. A certain special kind of personality / character, an obsession / drive with the subject, creativity, motivation, and even social skills (conferences, collaborating with others)

[@wernerviehhauser94]

IQ tells you about the probability of reaching an intellectually challenging goal, nothing more. Don't overrate this number. Someone with lower IQ will have a harder time trying to BSc/MSc/PhD in math and the lower the IQ the more will fail at it compared to a higher IQ group, otoh the top 25% of janitors have a higher IQ than the lower 25% of mathematicians.

[@hi-sj7cu]

I'd like to think that IQ as an amplifier that accelerates your learning on a subject. That being said, don't be encouraged just because you are born as an average or below average person. Always remember that: "While its true that biology gives everyone a different starting point, the race isn't only about where you begin - it's about how you run."

[@erockbrox8484]

Thats not math thinking, math people think in terms of theorems and lemmas and proofs.

[@paschalcharles6097]

What are those theorems

[@rashedulkabir6227]

This video was difficult for me to watch because English is not my first language.

[@__hannibaalbarca__]

To think like mathematician, First stay away of Internet, and very close to Books, Internet will teach you nothing, all my mathematicians skills is back 1999, and learn computer in 1992, but i see how social media, mobile phone, how distract a pure thinking, Second : pen and papers and excercise no matter is easy and difficult. Third : stay away from porn.

[@techguide57]

Obvious? Nobody says anything about it

[@papaskwatt-k4g]

this feels less like looking at mathematical thinking and more like you have adhd

[@jnhrtmn]

I never trusted math 100%. A child is a scientist with "The wheels on the bus go round and round." This song accurately and consistently describes everything the bus does exactly like math, but the child does NOT understand the bus, because wheels going round are not causal. The variables in gravity math are exactly like those wheels, they are NOT causal, so what do we actually understand using math? Relativity uses transform equations to CHANGE dimensions that are themselves theoretical to CREATE itself on your paper. The NON-transformed reality does not disappear by math on your paper. The NON-transformed reality IS STILL THERE ignored. Everyone thinks angular momentum and a conservation law cause the gyroscopic effect. This is physics in your face, not small or far away, and it's wrong. Look at my version that don't need a right-hand rule first. I understand it causally, and math didn't get me there.

[@fading_figure]

YAPPER

[@ugcomputerscience21]

Ur voice is like jordan peterson, just decrease pitch a bit.

[@rheymanda1074]

That's not very nice

[@akashpallepati61]

What bs