@@UrielCopy That's of course because of Computational Irreducibility! (which makes understanding why "Why is understanding ''Why is understanding why understanding why maths is hard, hard, hard'', hard", a bit harder!)
It may sound petty but that’s one reason I didn’t go deep into math even though I was good at it. I couldn’t bear to be around that shirt all day every day
My high school math teacher, teaching math extension 2 for HSC (a thing in NSW, Australia, tough as f*ck), looking like Kim K, and she wearing t-shirts like this
You can throw a jacket on and look like a pro yet take it off and not have any sleeves to smudge the chalkboard! Lol i never thought about this before but as someone whose been wearing these shirts sence basically puberty it's so true 😂. Also, when staring down a couple blackboards of gibberish it helps to be dressed nice to stay focused, so T-shirts are a no go.
I know, it's like they're speaking English, and I know they're speaking English, so I can't even say that it sounds like gibberish, but I just can't understand wtf they're saying! This must be what English sounds like to non-English speakers.
@@HerveyShmervy based on that this isn't a novel. Life has no meaning, the world is a sandbox your life is yours and you can do whatever you want with it.
@@youssefchaoui2940 that isn't so, you can't do anything you want because of our conscience, and how does a novel relate to any of this. Btw I was looking for a more objective answer not subjective
Can’t wait to finish college to understand this. I literally tried to use my brain to the fullest and I understood maybe 20% of the conversation. I feel so so dumb rn. I have to study more 💔💔
Big shout out to Georgory Chaitin who was one of the founders of algorithmic information theory. He also proved via a LISP program a computer-theoretic result equivalent to Gödel's incompleteness theorem. Find out more in his "The Unknowable" book (1999).
Remembering the hundreds of algerethems into a specific field is why I'm struggling with math. Instead of years to master, you often times have few weeks to acknowledge every concept. Math in a educational level , moves to fast. In my feeble brain anyways.
@Johannes Terzis That's a somewhat disingenuous answer. Memorizing things is not *required* in math, but it certainly helps. The more you know, the less you need to calculate/derive, and therefore the faster you can work. Any good mathematician has a ton of stuff memorized.
I came to say exactly this. I've been studying tertiary mathematics for a few years now and some teachers (the ones who teach only rules) can make learning quite difficult; others on the other hand (ones who can form a relationship between seemingly abstract equations and real world applications, as well as derive the equations before you) can teach typically difficult content with relative ease. Knowing and understanding mathematics is quite different from being able to explain it effectively, although both share elements of one another.
I mean i do understand the teachers myself. Explaining complex math to be easily understood is hard. I got good mathematic grades in high school, so my classmates would ask me to tutor them. But I found myself in the position of not being able to explain the mathematics easily to be understood most of the time they probably think I'm just rambling on with sophisticated words. Makes me think, Math is just kinda understood intuitively.
When he says that math is hard because it is computationally irreducible, I think he is saying that it is just that, irreducible. It can't be simplified in your head, nor on paper, or else you wouldn't be able to say/think what you were trying to say/think (there is a certain amount of syntax necessary to maintain the semantics). i.e., there is no shortcut to thinking the necessary ideas to get through a mathematical thought process. Also - it is bizarre that after working with a mathematical idea for a bit it suddenly makes sense and we say that it "clicked". With what do we know it clicked? It is not to be taken for granted that we are aware when we finaly get something. Wolfram speaks of this order-2 space which contains the paths that connect the proof paths.. I wonder if we don't have similar levels like this within our own thinking, and that this "clicking" phenomenon is one order of processing confirming another order.
I think effectively the infinity of representation within a domain of mathematics is not wrangled in with effective communication. For example triple integral calculus as Length*Width*Height. It's best to have conversations with people who do not understand so you can grow, more than it is to want to stand out in a group of people who think exactly like you do and want to be delineated from others.
Learning math just takes a lot of time in my experience at least i am beginning algebra and it takes myself a lot of time for me to understand a concept.
jesus saquin well I would but I gotta learn algebra first calculus takes time to trust me man I’ve spent 6 hours on one concept just in algebra so I understand your frustration.
@@garretthamilton1929 keep up the hard work. It's not always those who understand something the fastest that excel. I found this to be true in my higher level math courses. (High relative to my own experience ,ie multi variable calc, diffyQ etc). I was one who in algebra and pre calculus required lots of time to grasp a concept. However the work ethic I built during that phase allowed me to crush my later math courses. Now most math concepts come easier to me since I know how to learn (since I struggled so much earlier ). Best of luck !
i am a beginner but it seems like he is talking about why super advanced maths is hard not maths in general..like theorem proving to prove a theorem you need axioms(other solved theorems) as a foundation..if there isnt already available axioms to build from then the problem becomes much more difficult...thats the only part i can really understand here
What he talks about when he mentions paths almost sounds like a neural net, where the weight between nodes corresponds to the distance between elements.
The space of possibilities for a “possible” solution granted it exists is the hardest part of math. There is rarely any vantage point towards the end of any proof except one is very experienced in that specific field. The only thing that persists is human persistence to continue building on mathematics despite it’s difficulty. For 1000s of failed attempts, one has got to work, and that one successful attempt nudges us forward just a slight bit.
What most people understand and use as rational tought is heuristics bound, not actually rational. When you get to math, physics, chemistry, there is only so far you can get following heuristics before you need a complete and actual rational systematic understanding not only in general but also case specific.
@@EkosPlatinum Dependent on evolutionary processing load reduction tools. It appears the person is thinking or being rational, but it is actually a very fine tuned instinct. Even apes operate addiction and subtraction this way, but anything more complex than naturals multiplication can't be operated in this manner by humans, although computers do it well by means rational modelled programs.
Anyone answers 2+2=4 in decimals without thinking (and they will assume it is in decimals). But if you change it to 2√4+2√4=? in decimals people will break, even if it is very basic math (4th grade fundamental year). The first operation is answered by means of conventional thinking (heuristic bound), which is instinctive by all interpretations. The second is actual rational thought.
I don’t know how people can mention Eric Weinstein and Wolfram in one sentence. Wolfram has done great things in the past and gets very concrete In his explanation, Eric has neither. Thanks wolfram, this is Just gold!
Brian Beetle it’s obvious, that Wolfram always explains his ideas, Eric just says, that it’s useless to explain anything, because you don’t understand Dirac (see lex interview). Eric is a very shady character in that regard. I have learnt nothing by watching Eric, in contrast to watching wolfram.
@greenapplepear sure, there was just no enough time, sure. But there was enough time Wolfram’s theory is wrong, which Wolfram never said about Eric, because he can’t, because he does not know, I suppose and we know it is not easy to contradict a well thought out theory. Wolfram in the other hand said in another interview, that Eric’s work is important, which makes me wonder.
Here's the real reason. Math is a language, but it's not taught as a language. The way teachers tend to understand math and education in general is backwards. They know that people who comprehend things tend to memorize them, but instead of focusing on memorization by comprehension, they attempt comprehension by memorization. It's like being taught a language in sentences. Imagine you only learn how to translate two sentences: "I would like a coffee" and "your car looks nice". You spend 3 hours memorizing those two sentences with no idea what they really mean, and on the test, your teacher asks you to translate this sentence: "I think you left his coffee in the rental car." You think you didn't learn this in class because you didn't memorize this exact sentence, but your teacher expects you to know how to do it because you memorized things from the language, which (somehow) to them means you know the language. Instead, you have to speak math, and this is what word problems try to teach you to do, but they also suck because once again, you're taught a formula devoid of context. You're given the answer and told to just mindlessly drill it into your head and accept it. If you want math to get easier, give yourself a challenge to work through, something you can't quite solve immediately just by looking at it, but one that can be represented by a physical analog. Once you've found that problem, talk yourself through it. For example, you drill equal sized holes in a plank of wood that are all the same size. The holes are equally spaced from each other and the ends of the wood. Measuring edge to edge (or the material left between the holes, either phrasing works), what is the distance between each hole? Without reading ahead, think about how you'd solve this. Start with what you know- the length of the plank, the diameter of the holes, and the number of holes. If you take the length of the plank and subtract the number of holes times the diameter, you have the total remaining width of material between the holes, but that's all of the little wedge spaces. You're looking for the width of one of them. So, you divide by the number of those spaces, which is always one more than the number of holes. That's your answer. Let's make it a little tougher. You drill holes in a plank of wood, but there's a region of wood at one end that must remain unscathed, and you drill incrementally until your last hole is tangent to the edge of that region. What is the space between your holes now?
The part before 1:07 means: Math could be infinite, and it's made up by humans. We could measure really complex things sure but that doesn't mean that everything is solvable by simply logic. Logic is limited basically. There might be too many maths to be fully understood by one person. The foundations don't solve everything.
These men are clearly far beyond my skills and experience. But as a math/science tutor for 10 years now, I can say that math just isn’t for everyone. Those who get it will always get it with little to no external help-and vice Versa. But I will say, the one thing that my students who struggle HATE when I say, it’s “well, there’s not one sure way to do these problems every time, each one is different.” Some people just know how to do problem solving and utilize ABC 123 logic. People who speak that “language” to them it’s like “well it’s not that hard, you wouldn’t solve it any other way.” And I suppose some people just don’t “speak” that language-which is ok, everyone is just different. Admittedly I’m far from the best mathematician/engineer (BS in Civil), but I know enough to vouch for my experience and work.
Did u know logirithms have a sister issue. So say your base is 3 then your reverse log is the relation between which root dimension you need when your trying to find your base divided by your rooted base. The questions are 1 is this log as easy to deduce as standard logs and do they add any knowledge/ability to maths and number theory.
Stephen Wolfram is such a great guy, I love his enthusiasm for math. I'd very much like to talk to him (meaning sit & stare with blank expression on my face :D)
Lex Fridman having to podcast Wolfram from a living space thats crappy enough to surround him with painted concrete block says something about our world
This all makes me wonder if wolfram physics is just an alternative expression of mathematics, a different but equivalent format for writing down mathematical ideas... so it's not really about physics specifically, it is just able to express the math that we already use to model physics (so it could work just as well for economics or any other highly mathematical field)
1:42 actually if one can prove that the Riemann hypothesis is undecidable under the peano axioms, then it must be true because if it was false it would be provable as such using the peano axioms. Fun stuff.
this is actually not true. it is not possible to prove (using peano) that riemann hypothesis is undecidable under peano axoims because as you point out, that would make peano axioms inconsistent. you could prove the hypothesis is undecidable under peano using a different axiom set (such as ZF, for example).
@@multimoron11 Wait how would that make PA inconsistent? It would be incomplete but not inconsistent. But you're probably correct that you would need to prove that Riemann is undecidable under Peano axioms using something stronger, I'm not sure about that tho, I just remember hearing it in a Numberphile video.
@@federicovolpe3389 you are correct that proving RH is undecidable under PA means that RH is true. I'm pointing out this proof of undecidability under PA can't be done using PA, as that is an obvious inconsistency. you would need a different axiom set, if the proof is even possible.
The more words you add, the more tedious it gets. Math, if studied systematically from foundation to complexity is pure fun. Doing math whilst listening to music is so much fun and one doesnt get distracted by it. Music and word-heavy subjects dont always work for me.
While I think we can get further in mathematics using computers, I don't think that our brains can keep up. I think I'm quite intelligent, at least I'm good at abstracting and sort of compressing and unifying concepts. However, I think that the specific and the general are at odds, so that I'm merely deleting that which matters. I also think that there's a minimum amount of space needed for complex things which goes beyond my working memory, essentially locking me out of understanding. Similarities, symmetries, morphisms, etc. in mathematics are fairly simple, so it's easy to see that e.g. X relates to Y as Z relates to W, or to stack concepts to make new ones (if math and meta-math exists, then there's an infinite hierarchy of metas). So any person who is sufficiently intelligent should have an intuition for things like group theory before hearing about them, simply because they've already noticed that level of similarity in real life. But what about a concept which forms a more complex structure than just a square, and which doesn't fit in our working memoy? What use is a 200-IQ pattern recognition if the pattern doesn't fit in your mind? And what if math isn't more compressible than this? Space is stronger than time for a reason (complexity theory). In short, aren't we try to navigate a space which is far too large?
This is the type of problem we had to solve when I was a kid in elementary school in South Africa back in the 1950s. One cricket bat costs £14-18-9½ . How many cricket bats can you buy for £100 and how much change will you get? £14-18-9 means 14 pounds, 18 shillings and 9.5 pennies. £1 = 20 shillings and 1 shilling = 12 pennies. There were no handy pocket calculators back then. Answer: £100 = 48,000 halfpennies Once cricket bat costs 7,171 halfpennies divide 7,171 into 48,000 gives 6 remainder 4,974. So you can buy 6 bats and have 4,974 halfpennies in change. 4,974 halfpennies works out to £10-7-3 in change. And if you couldn't do that you were a wuss. The smallest coin was a farthing. One farthing was a quarter of a penny.
im sure its been said many times but i have a sneaky suspicion these apparent connections are more of a reflection of the nature of thought and its structural order.
Mathematics is hard because it requires abstract thinking which humans suck at. Also, like any language, it takes work to learn it and, frankly, most people's struggle with it is not due to Mathematics, itself, but their own laziness.
Good luck to all learning the ‘new science’ I haven’t read his foundational text but since the recent academic history of string theory, described in much the same way, an infinite potential space that maps to knowledge we know, let’s not lose ourselves sticking to one tribe. Always be cautious and celebrate new ideas that attempt to further our model of reality.
It's clear that math is simply the application of logic to numbers. That being said, I didn't understand much else about the latter half of this video. This is one of those videos where I feel smarter for having listened to it, but barely understood it.
There's probably something wrong with my understanding that someone even more informed could point out, but math isn't just the application of logic to numbers. It's the application of logic to axioms. You can use a set of axioms (i.e. ZFC) which are all you need to do anything we've thought of with numbers, with none of the axioms being explicitly about the notion of numbers. Additionally, there are many topics reachable via these axioms that axioms of numbers would be insufficient to reach. Mathematics is much more than just numbers in increasingly complex logical constructions.
Think what you are saying is, this is one of those videos that stimulates your curiosity. Most people stop there, but if you follow your curiosity, you will most certainly be smarter.
Notice that just because something is mathematically consistent, doesn't mean it works in the real world, this guy is on the very edges of what we need to understand to create AI
Math isn’t hard. It is badly taught though at the higher levels. Also math is easier to start with so there are more people doing it and therefore more competition. This competition just increases the amount and sophistication of today’s mathematics to incredible levels. Back in the 1600s one person could know most about all fields of mathematics. Not even the best mathematicians can know more than 1% of the whole field today.
Jeez, I came hoping to get an answer and left with none XD. Honestly though,as someone stated in the comments,maths is difficult because most professors charged with passing the knowledge to us suck at doing it. They might know maths but not able to properly transmit it to a brain that doesn't understand it the same way.
I find it rich that they are talking metaphysics this time, when a real metaphysical analysis of all the AI, Alien, futurist paradigm they hold so dearly would be utterly destroyed.
I don't understand his explanation of why mathematics being doable is relevant. He seems to just go off on a tangent saying something along the lines of, "There's an analogue of causal invariance, and there's this thing called homotopic type theory which came out of category theory and its an abstraction of the abstraction of mathematics and there's a thing called the univalence axiom, this axiom is equivalent to causal invariance." And I don't see how that tangent did anything to explain what he was saying.
Me, a writer but also shit with math: “Shit this is the perfect video, I can finally make my dreams come true of learning math!” Alien Mathematician: 0:36 “Computational Irreducibility” Also me: “What the fuck does that even mean?!?!?”
Maybe there would come a time in which we will rather build Mathematical Mindsets and Approaches rather than just mindlessly shoving Maths as an Subject which everyone hates.
There is nothing particularily hard about mathematics. Some people have a talent for it, others don't. Like with music, sports and everything else that depends on talent.
to summarise he finds patterns across mathematical fields and generalise thats what he says category theory and set theory both used for abstracting many fields in math
I suppose, that if you can reduce a problem to a single viable solution, you can then solve for an optimum solution utilising Quantum Computer for exactly what designed for - optimisation problems?
to summarise he finds patterns across mathematical fields and generalise thats what he says category theory and set theory both used for abstracting many fields in math
He is saying stuff like "math is hard" because of "computational irreducibility". When normal people say "math is hard" what they really mean is that it's difficult to practice enough to become fluent in doing calculations and proofs-- like in high school algebra or geometry. This guy is saying math is hard because of stuff like Godel's Incompleteness theorem and the most formal abstract research-grade problems. That's basically saying everything else is trivial except for extreme edge of mathematical knowledge which is accessible only to something like a few hundred people in the world. That's pretty smug, I think.
Spurius Tadius To be fair, I think the struggles that a normal person has with math can still be viewed through the lens of computational irreducibility. How “difficult” math is to a person depends on their experience/exposure to the underlying subjects, but complexity is still important. Irrespective of the subject of level of maths, people will tend to struggle less with one-step solutions, and struggle more with multi-step ones - especially if the most optimal solution _requires_ several steps. This mirrors what Wolfram is saying, I think. He could probably use a broader set of examples though.
@@skellington2000 Fair enough, Wolfram is "an expert", and us mere mortals are left to scratch our heads and wonder what he is on to. At the end of the day, however, he just leaves us with grandiose claims and not much has come of his "New Kind of Science". Except for the people who work for him, there does not seem to be others actively pursuing his ideas about cellular automata nor these newer hyper-graph concepts. In other words, he talks a big game but doesn't have anything to show for it. I think he would have more success if he were more willing to communicate and collaborate with others. Finally, remember that Lex Fridman's youtube channel is effectively for "the general public". For Wolfram to rapidly toss out concept after concept each of which require (at a minimum) a practicing PHD level understanding of theoretical physics and/or math is bad judgement-- I would say that's smug and posturing behavior. Sadly I think that there's an audience for that. Many people appreciate obfuscation. I had thought it was limited to the humanities (if you remember the Sokal affair), but it seems to afflict some STEM audiences as well-- especially with these "theory of everything" people like Wolfram and Eric Weinstein.
@@ElmerGLue Machine learning is an umbrella that covers a wide range of topics pulled from statistics, mathematics and computer science. There is nothing esoteric about it at all. It came to be in it's current form because some decidedly non-fringe computer scientists realized that, finally, we have enough speed and memory to crunch vast amounts of data and implement non-trivial neural nets (something which was first sketched out in the 60's by Marvin Minsky amongst others). Today, anyone with a modest technical background can fire up Keras (or other tools) and have a shot at using machine learning in their domain. But more to the point here, I think you'll find that world class scientists who make the effort to speak to the general public are able to adjust their language and express the essence of their ideas with clarity to the general public. There's a good example right here on Lex's channel (th-cam.com/video/IUHkhB366tE/w-d-xo.html) with Jim Gates who works in one of the most arcane of subjects, Super Symmetry Theory. He is able to edit himself and cogently discuss his work without using a firehose of specialist jargon. I am just saying that Wolfram should do the same and I judge him negatively because he does not (and this is separate from my own skepticism of what he proposes).
Even with simple problems, a friend is puzzled on how I know to add something or assume something to progress. When he look at it, it's like I just know. Reality is that at some point, I stop and ask: "What would be convenient ?" or "What am I missing here ?". Often time, it means I need to add/drop a constraint or add something I know. That part is the hidden path and most people can't see it. It's a close cousin of decomposition where you assume some part of the problem is already solved. This works very well in computing/programming. It also means that you should gather your thoughts at the same abstraction level. Changing the level of abstraction has a cost! As an example, I had to deal with real time software and high concurrency. I used high level primitives then implemented them. This lead on a rock solid solution that was FAST. The already existing solution was "optimized" (i.e. avoiding unnecessary abstractions), but slower and plenty of bugs (almost each corner case was a reason). On top of that, the first solution took 3 years, the second only 8 days spread across 2 months. Mathematicians (and good software programmers) can go very far by using proper abstraction and refrain to unnecessarily switch to another level of abstraction. It's like doing all your calculations in symbolic form before evaluating. Lastly, there is some political correctness, but mathematics (and programming) require a brain; It's not the forte of everyone and that's fine.
He looks like he's in jail for being a mathematical criminal.
He's the first man to successfully divide by 0. Very dangerous
@@ogbrosephstalin That made me laugh more than it should've.
@@ogbrosephstalin is this a reference to something Wolfram said?
@@xybersurfer no, it's just a math joke
which one? Fridman? or Wolfram?
Why is understanding why maths is hard, hard?
Why is understanding why understanding why maths is hard, hard, hard?
@@FXK23 Why is understanding ''Why is understanding why understanding why maths is hard, hard, hard'', hard?
@@UrielCopy That's of course because of Computational Irreducibility!
(which makes understanding why "Why is understanding ''Why is understanding why understanding why maths is hard, hard, hard'', hard", a bit harder!)
Lol
Let have a common knowledge that "Math is hard". Problem solved.
Why does literarily every 50+ mathematician wear a shirt like that? I'm starting to sense it holds some hidden mathematical super powers
See 'Academic Tribes & Territories' by Boettcher
It may sound petty but that’s one reason I didn’t go deep into math even though I was good at it. I couldn’t bear to be around that shirt all day every day
My high school math teacher, teaching math extension 2 for HSC (a thing in NSW, Australia, tough as f*ck), looking like Kim K, and she wearing t-shirts like this
You can throw a jacket on and look like a pro yet take it off and not have any sleeves to smudge the chalkboard! Lol i never thought about this before but as someone whose been wearing these shirts sence basically puberty it's so true 😂. Also, when staring down a couple blackboards of gibberish it helps to be dressed nice to stay focused, so T-shirts are a no go.
Same with my teacher, except his is a tropical tuxedo with the palm trees lol 🤣👌
This prison has the smartest inmates
super villain prison
"Mathematics is the exploration of the world through a proof trajectory".
What a brilliant way to define math.
I love when you go deep and I can't understand some of what's being said. Gives me more to go learn about. Please never back down.
That’s one of the little secrets of life. 👍
I too can go deep, u know...
I know, it's like they're speaking English, and I know they're speaking English, so I can't even say that it sounds like gibberish, but I just can't understand wtf they're saying! This must be what English sounds like to non-English speakers.
Rhett Melton dammit !!!
Keep learning. Keep on repeating. Things will make sense. If you love it.
he uses the word “human mathematics” alot. Alien confirmed. They’re amongst us. He’s trying to computationally irreduce us
I think your on to something. In fact the reason why he looks like he is prison is cause he’s a alien in Area 51?! •_•
man I can not be more grateful for your interviews. it's like having a direct conversation with brilliant minds. THANKS A LOT , THIS CHEERS ME UP
Totally agree I love this stuff!!
I just realised why mathematicians never get invited on podcasts
If he wore a robe and replaced "mathematics" with "life" you'd think he just solved existence
Wot? Wdym?
Existence is already solved. There's no meaning to any of it 😭😭😭
@@vishalsorout based on what?
@@HerveyShmervy based on that this isn't a novel. Life has no meaning, the world is a sandbox your life is yours and you can do whatever you want with it.
@@youssefchaoui2940 that isn't so, you can't do anything you want because of our conscience, and how does a novel relate to any of this. Btw I was looking for a more objective answer not subjective
You know you're leaving the biosphere when Lex says "is that something that is accessible to someone like me?"
Wolfram is one of those mathematicians that says "is" instead of "modeled by"
His answer to "why is math hard?" is itself hard to follow.
can confirm lol
0:36 Computational Irreducibility
Came up the motivation : Finally this would give me some concrete ideas.
Leaving : Why the f**k I clicked. I am dumb.
Think like this instead: Oh, I'm not as smart as Wolfram is.
@@Guztav1337 Yeah this works too.
Hated math in high school, but after taking a good break and now doing math in my late 20's, math is somewhat ok
Whoa. I'm researching this topic for an essay I'm writing and this is so complex. I love it
I have no idea whats going on but the fact that logic structures can manifest geometrically is really cool, would like to see modeling of that
he doesn;t mean quite the same thing by "geometrically" as the average person would
Can’t wait to finish college to understand this. I literally tried to use my brain to the fullest and I understood maybe 20% of the conversation. I feel so so dumb rn. I have to study more 💔💔
Wolfram is talking serious post-grad maths here...
@@kumoyuki Perhaps some of it is even post-doc maths
This isnt just “graduate college” level....
Bless this man for creating wolfram alpha, you have made my college life exponentially less painful 😌
This is refreshingly clear stuff from Wolfram up to 4:00. Great perspective.
Big shout out to Georgory Chaitin who was one of the founders of algorithmic information theory. He also proved via a LISP program a computer-theoretic result equivalent to Gödel's incompleteness theorem. Find out more in his "The Unknowable" book (1999).
Remembering the hundreds of algerethems into a specific field is why I'm struggling with math. Instead of years to master, you often times have few weeks to acknowledge every concept. Math in a educational level , moves to fast. In my feeble brain anyways.
@Johannes Terzis That's a somewhat disingenuous answer. Memorizing things is not *required* in math, but it certainly helps. The more you know, the less you need to calculate/derive, and therefore the faster you can work. Any good mathematician has a ton of stuff memorized.
Why is math hard? Because teachers can't teach it. I've found math teachers on TH-cam, that can teach me advanced math.
I came to say exactly this. I've been studying tertiary mathematics for a few years now and some teachers (the ones who teach only rules) can make learning quite difficult; others on the other hand (ones who can form a relationship between seemingly abstract equations and real world applications, as well as derive the equations before you) can teach typically difficult content with relative ease. Knowing and understanding mathematics is quite different from being able to explain it effectively, although both share elements of one another.
@@seanmaclean706 well said 👍
I mean i do understand the teachers myself. Explaining complex math to be easily understood is hard.
I got good mathematic grades in high school, so my classmates would ask me to tutor them. But I found myself in the position of not being able to explain the mathematics easily to be understood most of the time they probably think I'm just rambling on with sophisticated words. Makes me think, Math is just kinda understood intuitively.
The older I get the more I value people who can communicate clearly and teach new concepts well
@@mjfabian86 they generally have a well placed Jupiter or lots of sagitarius in their charts
mathematics surpass every other form of knowledge or field for the simply facts that it has and will last time and seem to have no bound
I sort of agree. Philosophy as a discipline is more general though .
Historic ground breaking interview... A good clip but the full video is amazing!
When he says that math is hard because it is computationally irreducible, I think he is saying that it is just that, irreducible. It can't be simplified in your head, nor on paper, or else you wouldn't be able to say/think what you were trying to say/think (there is a certain amount of syntax necessary to maintain the semantics). i.e., there is no shortcut to thinking the necessary ideas to get through a mathematical thought process.
Also - it is bizarre that after working with a mathematical idea for a bit it suddenly makes sense and we say that it "clicked". With what do we know it clicked? It is not to be taken for granted that we are aware when we finaly get something. Wolfram speaks of this order-2 space which contains the paths that connect the proof paths.. I wonder if we don't have similar levels like this within our own thinking, and that this "clicking" phenomenon is one order of processing confirming another order.
I think effectively the infinity of representation within a domain of mathematics is not wrangled in with effective communication. For example triple integral calculus as Length*Width*Height. It's best to have conversations with people who do not understand so you can grow, more than it is to want to stand out in a group of people who think exactly like you do and want to be delineated from others.
Learning math just takes a lot of time in my experience at least i am beginning algebra and it takes myself a lot of time for me to understand a concept.
Try taking calculus 2 now which is what I'm taking
jesus saquin well I would but I gotta learn algebra first calculus takes time to trust me man I’ve spent 6 hours on one concept just in algebra so I understand your frustration.
@@garretthamilton1929 keep up the hard work. It's not always those who understand something the fastest that excel. I found this to be true in my higher level math courses. (High relative to my own experience ,ie multi variable calc, diffyQ etc). I was one who in algebra and pre calculus required lots of time to grasp a concept. However the work ethic I built during that phase allowed me to crush my later math courses. Now most math concepts come easier to me since I know how to learn (since I struggled so much earlier ). Best of luck !
@@danielwatts3718 yea I agree good job for math is not an easy subject. He will get it
@@danielwatts3718 what if i don't want it to take long or struggle..i want to be a whiz like Ramanujan or Einstein..i dont want to be normal.
I'm always fascinated by Stephen's explanations, despite the fact the I never understand more than 2% of what he's saying.
God i am so fascinated by what theyre talking about and I cant even understand 2% of it
i am a beginner but it seems like he is talking about why super advanced maths is hard not maths in general..like theorem proving to prove a theorem you need axioms(other solved theorems) as a foundation..if there isnt already available axioms to build from then the problem becomes much more difficult...thats the only part i can really understand here
What he talks about when he mentions paths almost sounds like a neural net, where the weight between nodes corresponds to the distance between elements.
The space of possibilities for a “possible” solution granted it exists is the hardest part of math. There is rarely any vantage point towards the end of any proof except one is very experienced in that specific field. The only thing that persists is human persistence to continue building on mathematics despite it’s difficulty. For 1000s of failed attempts, one has got to work, and that one successful attempt nudges us forward just a slight bit.
It's clear that you're intelligent, but I have no idea what you're talking about.
What most people understand and use as rational tought is heuristics bound, not actually rational.
When you get to math, physics, chemistry, there is only so far you can get following heuristics before you need a complete and actual rational systematic understanding not only in general but also case specific.
What?
Really interesting, what is an heuristic bound? Some kind of logic limit?
@@EkosPlatinum Dependent on evolutionary processing load reduction tools.
It appears the person is thinking or being rational, but it is actually a very fine tuned instinct. Even apes operate addiction and subtraction this way, but anything more complex than naturals multiplication can't be operated in this manner by humans, although computers do it well by means rational modelled programs.
@@TheR971 What most people take as rational thinking is actually a very finely tuned instinct in use.
Anyone answers 2+2=4 in decimals without thinking (and they will assume it is in decimals). But if you change it to 2√4+2√4=? in decimals people will break, even if it is very basic math (4th grade fundamental year).
The first operation is answered by means of conventional thinking (heuristic bound), which is instinctive by all interpretations.
The second is actual rational thought.
typo, do another pull request :)
Not the most accessible conversation but somewhat interesting.
Stephen freezes in time for a couple seconds at 2:00 exactly.
I don’t know how people can mention Eric Weinstein and Wolfram in one sentence. Wolfram has done great things in the past and gets very concrete In his explanation, Eric has neither.
Thanks wolfram, this is
Just gold!
Brian Beetle it’s obvious, that Wolfram always explains his ideas, Eric just says, that it’s useless to explain anything, because you don’t understand Dirac (see lex interview). Eric is a very shady character in that regard. I have learnt nothing by watching Eric, in contrast to watching wolfram.
@greenapplepear sure, there was just no enough time, sure. But there was enough time Wolfram’s theory is wrong, which Wolfram never said about Eric, because he can’t, because he does not know, I suppose and we know it is not easy to contradict a well thought out theory.
Wolfram in the other hand said in another interview, that Eric’s work is important, which makes me wonder.
Computashskkkdhhaggd disability
Needed to activate captions to know that that word is irreducibility
lmao
Me too! 😂
Our lord google Ai has bestowed upon us his blessed knowledge.
I’m bad at math! It must be my computational disability!
@@l429930 in other words adhd
Here's the real reason. Math is a language, but it's not taught as a language. The way teachers tend to understand math and education in general is backwards. They know that people who comprehend things tend to memorize them, but instead of focusing on memorization by comprehension, they attempt comprehension by memorization. It's like being taught a language in sentences. Imagine you only learn how to translate two sentences: "I would like a coffee" and "your car looks nice". You spend 3 hours memorizing those two sentences with no idea what they really mean, and on the test, your teacher asks you to translate this sentence: "I think you left his coffee in the rental car." You think you didn't learn this in class because you didn't memorize this exact sentence, but your teacher expects you to know how to do it because you memorized things from the language, which (somehow) to them means you know the language.
Instead, you have to speak math, and this is what word problems try to teach you to do, but they also suck because once again, you're taught a formula devoid of context. You're given the answer and told to just mindlessly drill it into your head and accept it.
If you want math to get easier, give yourself a challenge to work through, something you can't quite solve immediately just by looking at it, but one that can be represented by a physical analog. Once you've found that problem, talk yourself through it. For example, you drill equal sized holes in a plank of wood that are all the same size. The holes are equally spaced from each other and the ends of the wood. Measuring edge to edge (or the material left between the holes, either phrasing works), what is the distance between each hole? Without reading ahead, think about how you'd solve this.
Start with what you know- the length of the plank, the diameter of the holes, and the number of holes. If you take the length of the plank and subtract the number of holes times the diameter, you have the total remaining width of material between the holes, but that's all of the little wedge spaces. You're looking for the width of one of them. So, you divide by the number of those spaces, which is always one more than the number of holes. That's your answer.
Let's make it a little tougher. You drill holes in a plank of wood, but there's a region of wood at one end that must remain unscathed, and you drill incrementally until your last hole is tangent to the edge of that region. What is the space between your holes now?
The part before 1:07 means: Math could be infinite, and it's made up by humans. We could measure really complex things sure but that doesn't mean that everything is solvable by simply logic. Logic is limited basically. There might be too many maths to be fully understood by one person. The foundations don't solve everything.
Kierkegaard's leap of faith
These men are clearly far beyond my skills and experience. But as a math/science tutor for 10 years now, I can say that math just isn’t for everyone. Those who get it will always get it with little to no external help-and vice Versa. But I will say, the one thing that my students who struggle HATE when I say, it’s “well, there’s not one sure way to do these problems every time, each one is different.” Some people just know how to do problem solving and utilize ABC 123 logic. People who speak that “language” to them it’s like “well it’s not that hard, you wouldn’t solve it any other way.” And I suppose some people just don’t “speak” that language-which is ok, everyone is just different. Admittedly I’m far from the best mathematician/engineer (BS in Civil), but I know enough to vouch for my experience and work.
I realize how hard it is to understand him if you've never tried to construct math from square one yourself
I came for an answer, left more confused!
Did u know logirithms have a sister issue. So say your base is 3 then your reverse log is the relation between which root dimension you need when your trying to find your base divided by your rooted base. The questions are 1 is this log as easy to deduce as standard logs and do they add any knowledge/ability to maths and number theory.
LOVE IT. Great work Lex, you have the most interesting guests. Keep it up y que viva tu audiencia de México! :D
Stephen Wolfram is such a great guy, I love his enthusiasm for math. I'd very much like to talk to him (meaning sit & stare with blank expression on my face :D)
The lighting in this room is superb.
Most confusing thing I've watched this year
Lex Fridman having to podcast Wolfram from a living space thats crappy enough to surround him with painted concrete block says something about our world
This all makes me wonder if wolfram physics is just an alternative expression of mathematics, a different but equivalent format for writing down mathematical ideas... so it's not really about physics specifically, it is just able to express the math that we already use to model physics (so it could work just as well for economics or any other highly mathematical field)
yes the relationship between all these abstractions is unclear
I understand each and every single word individually, but when he puts those word in that order, i get lost
1:42 actually if one can prove that the Riemann hypothesis is undecidable under the peano axioms, then it must be true because if it was false it would be provable as such using the peano axioms.
Fun stuff.
this is actually not true. it is not possible to prove (using peano) that riemann hypothesis is undecidable under peano axoims because as you point out, that would make peano axioms inconsistent. you could prove the hypothesis is undecidable under peano using a different axiom set (such as ZF, for example).
@@multimoron11 Wait how would that make PA inconsistent? It would be incomplete but not inconsistent.
But you're probably correct that you would need to prove that Riemann is undecidable under Peano axioms using something stronger, I'm not sure about that tho, I just remember hearing it in a Numberphile video.
@@federicovolpe3389 you are correct that proving RH is undecidable under PA means that RH is true. I'm pointing out this proof of undecidability under PA can't be done using PA, as that is an obvious inconsistency. you would need a different axiom set, if the proof is even possible.
I think u both r true tbh
The more words you add, the more tedious it gets. Math, if studied systematically from foundation to complexity is pure fun. Doing math whilst listening to music is so much fun and one doesnt get distracted by it. Music and word-heavy subjects dont always work for me.
But what can you use this abstract mathematics for ? Like what is it’s use
I would love to watch Mr. Wolfram make all the worlds supercomputers cry, and cry hard.
Wolfram on point here.
Why is he in a prison cell ?
So he could study more math.
lol
"Computational irreducibility" sees a small bump in searches
While I think we can get further in mathematics using computers, I don't think that our brains can keep up. I think I'm quite intelligent, at least I'm good at abstracting and sort of compressing and unifying concepts. However, I think that the specific and the general are at odds, so that I'm merely deleting that which matters. I also think that there's a minimum amount of space needed for complex things which goes beyond my working memory, essentially locking me out of understanding. Similarities, symmetries, morphisms, etc. in mathematics are fairly simple, so it's easy to see that e.g. X relates to Y as Z relates to W, or to stack concepts to make new ones (if math and meta-math exists, then there's an infinite hierarchy of metas). So any person who is sufficiently intelligent should have an intuition for things like group theory before hearing about them, simply because they've already noticed that level of similarity in real life.
But what about a concept which forms a more complex structure than just a square, and which doesn't fit in our working memoy? What use is a 200-IQ pattern recognition if the pattern doesn't fit in your mind? And what if math isn't more compressible than this? Space is stronger than time for a reason (complexity theory). In short, aren't we try to navigate a space which is far too large?
Everyone's gangsta until shit is computationally irreducible
Why did you have to interrupt him at 10:18? I was dying to hear what he was about to say about relating physics to all that.
This is the type of problem we had to solve when I was a kid in elementary school in South Africa back in the 1950s.
One cricket bat costs £14-18-9½ . How many cricket bats can you buy for £100 and how much change will you get?
£14-18-9 means 14 pounds, 18 shillings and 9.5 pennies. £1 = 20 shillings and 1 shilling = 12 pennies. There were no handy pocket calculators back then.
Answer:
£100 = 48,000 halfpennies
Once cricket bat costs 7,171 halfpennies
divide 7,171 into 48,000 gives 6 remainder 4,974. So you can buy 6 bats and have 4,974 halfpennies in change. 4,974 halfpennies works out to £10-7-3 in change.
And if you couldn't do that you were a wuss.
The smallest coin was a farthing. One farthing was a quarter of a penny.
42069 IQ podcast right here
I’d love to know Stephen Wolfram’s take on the Yoneda Lemma. Ever since I discovered it I’ve been convinced it is a key to the universe.
im sure its been said many times but i have a sneaky suspicion these apparent connections are more of a reflection of the nature of thought and its structural order.
I love your podcasts bro
Him: Computation irreducibility...
Interviewer: Riiiight.... Riiiight... **confused**
Mathematics is hard because it requires abstract thinking which humans suck at.
Also, like any language, it takes work to learn it and, frankly, most people's struggle with it is not due to Mathematics, itself, but their own laziness.
Math is a scam created by con artists
What did he say, "rulio multiway system"? Its hard to tell?
Sam S thats what I heard but i'm not sure
Vincent Candela i googled it but nothing came up, sometimes I think Lex tries to act like he knows more than he does
Rulial
Stop interrupting this man’s show, Lex.
But can you give change back from a $20???
I phrase I coined many years ago, "There is nothing hard, just things you haven't learned yet."
Good luck to all learning the ‘new science’
I haven’t read his foundational text but since the recent academic history of string theory, described in much the same way, an infinite potential space that maps to knowledge we know, let’s not lose ourselves sticking to one tribe.
Always be cautious and celebrate new ideas that attempt to further our model of reality.
I understand some of these words
It's clear that math is simply the application of logic to numbers. That being said, I didn't understand much else about the latter half of this video. This is one of those videos where I feel smarter for having listened to it, but barely understood it.
There's probably something wrong with my understanding that someone even more informed could point out, but math isn't just the application of logic to numbers. It's the application of logic to axioms. You can use a set of axioms (i.e. ZFC) which are all you need to do anything we've thought of with numbers, with none of the axioms being explicitly about the notion of numbers. Additionally, there are many topics reachable via these axioms that axioms of numbers would be insufficient to reach. Mathematics is much more than just numbers in increasingly complex logical constructions.
Think what you are saying is, this is one of those videos that stimulates your curiosity. Most people stop there, but if you follow your curiosity, you will most certainly be smarter.
I thought Maths was the patterns bound the universe
Most difficult undergrad mathematics course you took? Go. For me it was advanced calc or abstract algebra.
For me it’s transform theory 😭😭.
4th grade decimals.
Z transforms
Fourier Transforms
the path is a long road, there are no shortcuts, learn the language as you go
Notice that just because something is mathematically consistent, doesn't mean it works in the real world, this guy is on the very edges of what we need to understand to create AI
Back to school with conics, directrix, and iterations.
what is this "computational aerodusibility"?
This was amazing!!
Math isn’t hard. It is badly taught though at the higher levels.
Also math is easier to start with so there are more people doing it and therefore more competition. This competition just increases the amount and sophistication of today’s mathematics to incredible levels. Back in the 1600s one person could know most about all fields of mathematics. Not even the best mathematicians can know more than 1% of the whole field today.
Mathemagics
Jeez, I came hoping to get an answer and left with none XD.
Honestly though,as someone stated in the comments,maths is difficult because most professors charged with passing the knowledge to us suck at doing it. They might know maths but not able to properly transmit it to a brain that doesn't understand it the same way.
What a kid response, "its the teachers fault!"
I find it rich that they are talking metaphysics this time, when a real metaphysical analysis of all the AI, Alien, futurist paradigm they hold so dearly would be utterly destroyed.
Care to elaborate? Since you seem to know it all..
some prisons do look like that,,.....,but aren't we all in a prison ???
thank you for helping us enlighten our minds..
I don't understand his explanation of why mathematics being doable is relevant. He seems to just go off on a tangent saying something along the lines of, "There's an analogue of causal invariance, and there's this thing called homotopic type theory which came out of category theory and its an abstraction of the abstraction of mathematics and there's a thing called the univalence axiom, this axiom is equivalent to causal invariance." And I don't see how that tangent did anything to explain what he was saying.
His definition of math is hard is on the "experimental" and departmental side of mathematics.
0:36 what does he say ?
Me, a writer but also shit with math: “Shit this is the perfect video, I can finally make my dreams come true of learning math!”
Alien Mathematician: 0:36 “Computational Irreducibility”
Also me: “What the fuck does that even mean?!?!?”
Maybe there would come a time in which we will rather build Mathematical Mindsets and Approaches rather than just mindlessly shoving Maths as an Subject which everyone hates.
There is nothing particularily hard about mathematics. Some people have a talent for it, others don't. Like with music, sports and everything else that depends on talent.
I love when jumping to the comments section and found, literally not only me think that is hard. I have no reason to hate myself after watching this
to summarise he finds patterns across mathematical fields and generalise thats what he says category theory and set theory both used for abstracting many fields in math
I suppose, that if you can reduce a problem to a single viable solution, you can then solve for an optimum solution utilising Quantum Computer for exactly what designed for - optimisation problems?
This did not make it any clearer why I was terrible at math in school.
to summarise he finds patterns across mathematical fields and generalise thats what he says category theory and set theory both used for abstracting many fields in math
I was listening for a good bit and then I was like wait who is this guy, then I looked at the title and was like :o
Hey buddy can you interview James Woodward and Steve Bassett?
Dunno why people call Wolfram arrogant. He comes across as really down to earth to me
He is saying stuff like "math is hard" because of "computational irreducibility". When normal people say "math is hard" what they really mean is that it's difficult to practice enough to become fluent in doing calculations and proofs-- like in high school algebra or geometry. This guy is saying math is hard because of stuff like Godel's Incompleteness theorem and the most formal abstract research-grade problems. That's basically saying everything else is trivial except for extreme edge of mathematical knowledge which is accessible only to something like a few hundred people in the world. That's pretty smug, I think.
Spurius Tadius To be fair, I think the struggles that a normal person has with math can still be viewed through the lens of computational irreducibility.
How “difficult” math is to a person depends on their experience/exposure to the underlying subjects, but complexity is still important. Irrespective of the subject of level of maths, people will tend to struggle less with one-step solutions, and struggle more with multi-step ones - especially if the most optimal solution _requires_ several steps.
This mirrors what Wolfram is saying, I think. He could probably use a broader set of examples though.
@@skellington2000 Fair enough, Wolfram is "an expert", and us mere mortals are left to scratch our heads and wonder what he is on to. At the end of the day, however, he just leaves us with grandiose claims and not much has come of his "New Kind of Science". Except for the people who work for him, there does not seem to be others actively pursuing his ideas about cellular automata nor these newer hyper-graph concepts. In other words, he talks a big game but doesn't have anything to show for it. I think he would have more success if he were more willing to communicate and collaborate with others.
Finally, remember that Lex Fridman's youtube channel is effectively for "the general public". For Wolfram to rapidly toss out concept after concept each of which require (at a minimum) a practicing PHD level understanding of theoretical physics and/or math is bad judgement-- I would say that's smug and posturing behavior. Sadly I think that there's an audience for that. Many people appreciate obfuscation. I had thought it was limited to the humanities (if you remember the Sokal affair), but it seems to afflict some STEM audiences as well-- especially with these "theory of everything" people like Wolfram and Eric Weinstein.
@@ElmerGLue Machine learning is an umbrella that covers a wide range of topics pulled from statistics, mathematics and computer science. There is nothing esoteric about it at all. It came to be in it's current form because some decidedly non-fringe computer scientists realized that, finally, we have enough speed and memory to crunch vast amounts of data and implement non-trivial neural nets (something which was first sketched out in the 60's by Marvin Minsky amongst others). Today, anyone with a modest technical background can fire up Keras (or other tools) and have a shot at using machine learning in their domain.
But more to the point here, I think you'll find that world class scientists who make the effort to speak to the general public are able to adjust their language and express the essence of their ideas with clarity to the general public. There's a good example right here on Lex's channel (th-cam.com/video/IUHkhB366tE/w-d-xo.html) with Jim Gates who works in one of the most arcane of subjects, Super Symmetry Theory. He is able to edit himself and cogently discuss his work without using a firehose of specialist jargon. I am just saying that Wolfram should do the same and I judge him negatively because he does not (and this is separate from my own skepticism of what he proposes).
my stat mech prof said he made fun of him after a confrence for having a "trivial" job as an experimental physicist.
Even with simple problems, a friend is puzzled on how I know to add something or assume something to progress. When he look at it, it's like I just know.
Reality is that at some point, I stop and ask: "What would be convenient ?" or "What am I missing here ?".
Often time, it means I need to add/drop a constraint or add something I know.
That part is the hidden path and most people can't see it.
It's a close cousin of decomposition where you assume some part of the problem is already solved. This works very well in computing/programming.
It also means that you should gather your thoughts at the same abstraction level. Changing the level of abstraction has a cost!
As an example, I had to deal with real time software and high concurrency. I used high level primitives then implemented them.
This lead on a rock solid solution that was FAST.
The already existing solution was "optimized" (i.e. avoiding unnecessary abstractions), but slower and plenty of bugs (almost each corner case was a reason).
On top of that, the first solution took 3 years, the second only 8 days spread across 2 months.
Mathematicians (and good software programmers) can go very far by using proper abstraction and refrain to unnecessarily switch to another level of abstraction.
It's like doing all your calculations in symbolic form before evaluating.
Lastly, there is some political correctness, but mathematics (and programming) require a brain; It's not the forte of everyone and that's fine.