5:55 I made the comment, "the number of proofs is uncountably infinite." Many people have addressed this in the comments and I want to apologize for making this misleading remark. While researching for the video I came across this statement and accepted it as fact without checking it myself. While the statement as given is not technically inaccurate since we would first need to define a rigorous logical system and without this we can't comment on the validity of any statements (is this confusing or what?), under a few simple assumptions such as: 1) Languages have a finite number of characters; and 2) Proofs are finite, we can show that the number of proofs is a countable union of finite sets, which is countable. Sorry for the misleading statement. It will happen again! I appreciate all the comments point this out 😊
I really like that you are openly discussing the problems of how math is taught. I feel like I sound like a broken record talking to my friends about this stuff. This video makes me think I might be good as a math major, because I was learning proofs this whole time. Learning proofs has just been the way I connect new processes to my prior mathematical understanding. It just didn't feel right memorizing a formula if I couldn't verify for myself why it works.
This sounds like an area Australia excels at. We learn basic basic proof idea in High School senior math (Advanced, Extension 1 Math) (From NSW). In Extension 2 math we even learn formal proof techniques like proof by contradiction, induction, proof by contraposition, inequality proofs, number proofs, geometric proofs involving vectors. I think High School math really prepares me for university math in the future due to the focus on higher order thinking earlier on in my mathematics journey (topics learned by Year 12 Math if you take the top level courses for context: Sequences and Series, Trigonometry, Functions, Differentiation, Integration, Vectors, motion, differential equations, statistical graphs, binomial statistics, probability distributions, formal proof techniques, complex numbers (goes to De moivre’s theorem, roots of unity and exponential form), integral techniques (integration by substitution, integration by parts, general integration rules for repeated integration, integration by partial fraction decomposition), mechanics, 3D vectors. This is also reinforced in first year university math courses (like in Discrete Mathematics which is often mandatory for math majors has a part of the course look into formal mathematical proofs).
I am currently taking AP Calculus AB and will be taking BC next semester. I REALLY want to go into a math major in college cause I feel like that would be the only major I would enjoy. We did proofs in Geometry, it was a struggle at first because I had no idea on how specific I am supposed to be, but now I think I have gotten a lot better, but only for the geometry level.
I unfortunately completed my math major about 8 years before you created this video :P But I stuck through it even though I ran into the same wall a lot of these kids went through because I knew I had enough progress to just barely finish it, and I wanted to challenge my perseverance on a real life goal - by bringing it back from the brink of failure - for the first time in my life. Also, real analysis was easier than complex analysis for me :)
Getting my B.S. in Math at Georgia Tech this December :) My high school didn’t have classes past Algebra II but I was able to dual enroll at a California community college to take Pre-Calc, Differential Calculus, Integral Calculus, Multivariable Calc, Differential Equations, Linear Algebra, and a Math Proofs Seminar. The Math Proofs Seminar (and the Linear Algebra class being proof-based) is what made me fall in love with math. Now I have my dream job in Big Tech lined up to start post graduation 🥰
Also if you have the ability to take math classes at a community college instead of the AP equivalents: GO TO THE COMMUNITY COLLEGE. The teachers at community college honestly rivaled the best that I encountered at Georgia Tech because they were 100% focused on teaching (not research) and if you are a motivated high schooler who asks questions and comes to office hours, they will love you since they probably deal with a constant flow of mildly disinterested (“I’m only taking this because it’s a requirement”) college students.
As someone who has a backround in international math competitions, I personally think that the lack of exposure to proofs is the single biggest problem in a lot of school systems. It's just a lot of fun to do a wide range of math exercises, especially doing little variations on things you have seen a lot.
Really awesome video! I’m a senior in high school right now wanting to major in math so I’m probably going to go to the library and borrow a book on basic proofs now 😂
Intro to fundamental analysis, real analysis, and modern abstract algebra made me understand what many students feel in a math class. Glad i did experience and knew that going further in math was not my jam, though makes stuff easier in masters of CS
Well since high school geometry on khan academy has a structure of 2 column proof, I had to adapt fast since it was rigorous as hell. From algebra 1 to high school geometry felt like getting wooped by Euclid and his book of elements.
I was first introduced to proofs from a textbook called discrete math and its applications by Rosen, and honestly I’ve never felt so humbled. Growing up I’ve never made below a high A on a math test. Calc 1 2 and the highschool algebra and trig were easy… simply learn an algorithm, follow the steps and you get the solution. However when our textbook started introducing simple proofs I’ve never felt anymore stupid, sometimes the idea would be simple, for example proving the sqrt(2) is irrational. However I had no idea how to formulate my ideas, and I felt so humbled. Needless to say I passed the course, and made me consider switching from computer science to mathematics. Currently I’m still majoring in computer science, but I really want to give a course like real analysis or abstract algebra a shot….
I am contemplating the idea of studying mathematics because I have an interest in the theme, but I don't know if it's best to be self-taught for now. I started to really like how it can solve real life problems and improve my knowledge. I studied some logic and statistics in an English teaching degree and afterwards started a computer science related degree.
6:10 what might also be helpful here is recognizing how the statement relies on the fact that the real numbers form a field and how this property fails in more general rings like Z/12Z. If you never saw those examples it's harder to guess what properties you need to use to prove the statement.
This is the whole point. A proof like this one will be shown on the first day of an introductory Analysis course. None of the students know what a group or ring or field is since Abstract Algebra comes later. It's possible to simplify the proof further by saying, "A field has no zero-divisors," but this is circular. You have to start learning about proofs somewhere and that's a big part of the challenge.
I have a bachelor's degree in math, and proofs is definitely where I started to struggle. Abstract algebra basically almost killed me, but that was partially because the teacher I had for it made it a lot more difficult than it should have been. Even other math professors at my university agreed that the way she taught it was just way too hard for undergrad. Most math programs have an intro to proofs type course that helps prepare people for proof classes. Overall, Real Analysis actually wasn't that bad for me mainly because every proof has a specific process in Real Analysis. If you know the process for a proof in real analysis, it's not super difficult. Abstract algebra is a completely different story. I had a few failed courses throughout my undergrad, but I still graduated with my math degree. I am very proud of myself for doing what I did.
Makes me so glad I didn't major in math or engineering. But seriously, I'm very pleased to see how many non-US students are watching your new vlog. I'll bet they got taught bridge in their schools.
I feel like I'm going through the opposite process right now: I started as a comp-sci major, but I am trying more math classes this semester and I'm currently loving number systems (another proof based class, before real analysis). This video has given me the confidence that I would be a great math major.
The thing that made proofs kind of click for me is realising that I should just see proofs as an explaination for why something is true based on prior knowledge. Usually if I am stuck on a proof, it's a sign I need to look over or organize my notes again, and once I have all of my axioms, theorums, and everything in one place, then I can piece them together like a puzzle.
I personally always wonder if you don't see ANY proofs in America during high school I grew up and went to school to Belgium and we already saw quite a lot I think We saw the definitions of limits, derrivatives, integrals... and derived their properties trough the definitions Because I follewed a math-heavy program I even saw the basics of linear algebra, number theory and projective geometry, of which we also learned part of the rigour
It's certainly not everyone's experience that proofs first come about in university level courses, but I think it is for a large majority. Geometry, taught in high school, does cover the fundamentals of proofs, but the core material is new so students see the proofs as a new technique. Contrast that with proofs in Analysis where even in the comments of this video I've seen plenty of arguments (some since deleted) trying to prove a concept as simple as fields having no zero divisors which lack understanding and rigor. The difference is students believe they understand the properties of numbers (the real numbers) since they have worked with them for over a decade, but they never learned the underlying axioms or what rigid principles must be obeyed and why. As for Calculus proofs, I do believe many schools go over the derivation of the definition of limits, derivatives, and (Riemann) integrals. The issue is that the teachers then don't ask for understanding of these concepts on the exams and instead just ask students to apply the power rule to a function, thinking that this constitutes understanding.
5:58 the number of proofs/theorems in general is actually countably infinite not uncountably infinite. Since all proofs/theorems are of finite length, the set of theorems is the infinite union over all possible lengths, and for each length k there is a finite amount of theorems as words/mathematical signs are a finite set. So it's an countably infinite union of finite sets which ends up as just countably infinite
I think the distinction isn't proofs vs. non-proofs, but rather problems that can be solved using a standard series of steps vs. problems that require more out-of-the-box thinking. Proofs can be just as formulaic and simple if there's no variation in the questions.. For example, in high school we had a proofs unit where we (for the most part) just had to prove some properties such as divisibility and rationality. Once our teacher showed us the steps to do this, we could pretty much just use these same steps for any question we got on the test, plugging in different numbers/expressions. I'd argue that there really isn't much of a difference between proofs and questions without proofs. Proofs are just regular questions where they give you the answer beforehand, and the question would be more difficult in most cases if they didn't give the answer. "Prove that if 2x = 4, then x = 2" counts as a proof, but it isn't any different from "find x if 2x = 4". You can turn any proof into a question just by phrasing it differently: instead of "prove that x is true", you can ask "is x true?" You definitely are right about students not being taught how to come up with unique solutions to problems themselves though. Most of high school math is just memorizing a few procedures that are taught by your teacher, and just plugging in different numbers. There is contest math however, which has questions that are not so straightforward, and I believe most students thinking of majoring math have tried it out.
I’m a senior in an arts degree who’s recently become obsessed with how math is developed and what constitutes rigorous proof… I don’t know if there’s a way to switch to math this late, but this video just makes me more passionate about that possibility.
Wow... As a person taking an undergraduate degree in Mathematics I can relate. When I first started Real Analysis, it was my weakest topic. I couldn't understand what half of the lecturers were stating. Besides, the topic is almost completely unintuitive for me. Meanwhile, I can't say I'm good at discrete mathematics or linear algebra, though I am definitely more confident in those topics than analysis (real OR complex). This video deserves more views.
My geometry classes in high school went over proofs a bit, but I didn't really learn proofs until linear algebra. I think that helped with Real Analysis. I made it through, but that's when I decided grad school wasn't for me. I just didn't have the talent to go further. I'm glad I took it though since it actually proved the Fundamental Theorem of Calculus for me, which I found fascinating. Although I always had a problem with some of the stuff they taught about Cantor. Yes the reals are bigger than the naturals. But I never accepted that the interval from 0 to 1 had the exact same number of elements from 0 to 2. You can put them in a 1 to 1 mapping of course, but you can also put them in a 2 to 1 mapping. And probably other maps as well.
Im a stats and data science major and was guessing this would be about real analysis and was right. Been avoiding this vid cuz it seemed clickbaity but since im procrastinating, i clicked anyway. Im kinda tempted to take real analysis at some point. Its not necessary at all to take it, but my proofs class has been neat enough to where im curious.
Weird , i recently started to study math in collegue (well , the equivalent of collegue here , in Spain) and we are already doing some kinds of proofs , and even we have a subject called "sets and numbers" which give us basic notions in proofs and mathematical logic.
for anyone doubting about mathematics: remember that math is creative. if you love math and enjoy finding creative and original solutions to problems, you are on the right track. if you have the patience and discipline to work hard at it every day, not only are you in the right field, but mathematics will be a BREEZE. dont let anyone scare you away! best of luck
Going into my 4th year math major at a top university in Canada, and I would say 100 percent real analysis was a bit of a hurdle. It was worsened for me because I went from doing my first and like half my second year at a smaller college where things were taught in a very procedural way. Then I went into an honors math program when I transferred to the university. Man, what a way to just be thrown into proofs. First proof class is an honors real analysis 😢 I've made it through all of that now though.
As a 15 year old high school student with a strong foundation in Algebra, Trigonometry, Mathematical proof writing, Real Analysis Requires A level of Abstract and deductive thinking,l i have currently not adapted to. Limits, Convergence ,Ratio tests,Alternating series test etc. were and still are super hard to apply but due to my strong foundation in proof writing and prior training in abstraction. It is fairly easy! but maybe i am just still woefully ignorant. My English is very bad so please don't roast me 😅 (
Wait, what? I'm not from the US so I don't know the school system. Do you really not get to do proofs for 2 years even at university? At my uni in Germany you have to work on some kind of proof in the first week. What is taught in Calculus and what's Real Analysis? I think at my uni those are grouped into one series of lectures. But maybe I still don't get what Real Analysis is actually about.
I don't have a wholistic perspective on what happens outside the US, my only understanding of other school systems comes from my interaction with graduate students from other countries. What is taught in Calc vs. Real Analysis? Calculus focuses on problem solving techniques like the power rule for derivatives or formulas for arclength or surface area integration. Real Analysis focuses on proving formulas from Calculus, like using the delta-epsilon definition for a limit. I think the school system's rationale behind not teaching the proof techniques earlier than the 3rd year is that many engineering students need calculus for their jobs, but they don't need to know what a proof is. It's expected that only students who study pure math will take real analysis and therefore proofs are delayed to weed out the rest of the population.
When I was at school, in France, we started to do proofs (on parallelograms, I think) at what is the French equivalent of US eighth Grade (i.e "4ème") . It may unfortunately have changed since these happy days (1979), though.
Oh, interesting. Here we have different courses for math, physics, computer science students. I don't think one can get an engineering degree at my uni. The focus on proofs is different among those courses. The math students definitely learn Calculus tools while proving why they work (or more commonly the professor proving why they work). This way most students drop out in the first or second semester if they don't wanna do that kind of math. Even for comuter science students the most common dropout reason are math courses unfortunately. They might still be too proof focused.
@@eagelwizard290in america u learn way more math before U start with our Proof based uni system, but because of that u ll do a speedrun of what is tought in analysis 1,2,3 in real analysis
@@eagelwizard290 That lack of proofs until real analysis would be atypical for my experience too. At my math major program in Canada, we did proofs from semester one, because that's what made it a math program and not an engineering program.
Is it really normal to not have taken a proof based class before real analysis? My uni requires you to take an introduction to proofs, and proof based linear algebra/calc 1 + 2 in first year. Second year you do more proof based linear algebra, so by the time you get to real analysis you have a good amount of experience with proofs.
I would add that many of these students are probably better served studying applied maths, whether has part of a different degree (physics, CompSci, engineering) or as part of a dedicated applied maths degree, which does exist. A lot of the thinking and type of problem structures in that are closer to the high school maths you talk about.
What you said about real analysis being the first course with proofs isn’t accurate in my experience. We did them in high school geometry and then I had to take an intro course for math majors which was linear algebra but proofs based and it was a good bridge between calculus and pure math. Also discrete math for my cs major. If you’re school does this they have a shit math department
It is indeed a tricky one but you're on the right track! Since the integration interval is centered around 0, think about f(-x) in relation to f(x) where f is the integrand. Can you divide the integral into two equal parts that may allow for algebraic cancellations?
@@LilBiteOEverything ah, I see. It would utilise f(x)=1/2 (f(x)+f(-x)) + 1/2 (f(x)-f(-x)) It all reduces to the integral on [-1,1] of 1/2 cos(x) dx = integral on [0,1] of cos(x) dx = sin(1)
I was struggling with very basic proofs in high school, and that made me confused because im usually very good at math, i asked my teacher about it and then he told me about how in his analysis class only 4 of the 30 students finished the class, and they had to reduce the minimal grade so theyd pass, else only 2 had enough grade to pass the class
Who knows, you may be surprised. I thought I was going to flunk out at first, but I ended up joining one of the rec sports teams on campus and that time away from studying actually gave me some help and encouragement.
Any proof-based course will have many of the same ingredients, so Linear Algebra with proofs will be a good introduction to the concepts. But Real Analysis is very focused on producing rigorous proofs from day 1 while Linear Algebra will still spend at least half of the time covering problem solving techniques like the Gram-Schmidt process. My guess is that Real Analysis is still a substantial bit more difficult than other introductory proof courses.
The answer does depend on how you define language and a proof. Your idea of finite-length proofs in a finite-sized language resulting in countable proofs is certainly more popular than the one I presented.
5:55 "... the number of possible proofs in uncountably infinite..." It certainly is not! It's infinite, but not uncountably infinite. This guy needs to take some math classes.
I really want to major math but i still need to wait a few years (i'm 15), however, i would like to know a good place to find studying material both for entry level college maths or american HS algebra/calc Every help is welcome.
For high school math, I've heard Kahn Academy is good. For Calc or college math MIT OpenCourseWare is excellent. Hope this helps! Best of luck on your journey!
Computers do all the computations anyway. The proofs are where you get to get creative. Which is fun until test day. Then when you can't get creative on demand it's not quite as fun. But I still enjoy math as a hobby.
5:59 no it's not. Certainly not the numbers of proofs in languages that exist on earth. Don't use mathematical language wrong while talking about mathematical language
This is of course up for debate. To be rigorous in making a mathematical statement it's important to use rigorous notation and that would take several weeks of studying logic to do. I did no such thing. Neither did you. And it's well beyond the scope of this video anyway. I'll concede that a more popular belief would be that the number of proofs is countably infinite, but my statement is in no way 'wrong'. I could write down a formal set of axioms and notation for which the number of proofs in uncountably infinite.
@@LilBiteOEverything you're right of course - sorry I phrased my comment this harshly. It simply depends on how you define a mathematical proof and most importantly it has little to do with the point you are making. I just think that it is not useful to just throw in the word uncountable in a context where it makes little sense and is not needed. But of course that is just a detail and other than that I did like the video!
In Morocco we take real analysis 1 and 2 in the first year as well as a 2 proof based algebra courses and I'm not even a math major I'm finding it very diffuclt why the hell do we need to learn this as informatics students
Proof by contradiction If x&y are non-zero real numbers and their product is zero, then, since multiplication is repeated addition of a value, there must exist at least one non-zero real number which when added to itself a non-zero real amount of times must equal zero. However, we know that if you add any real number not equal to zero to itself any amount of times, it will never equal zero. Therefore their products must be zero, unless you’d like to say that anything other than zero added to itself n amount of times will equal any value not a multiple of themselves, which is ridiculous.
It's interesting how the proof presented here and in the video are kind of distinct. To use mathematical language, you are using the fact that the real numbers have characteristic zero while the video used the fact that the real numbers form a field. What makes them distinct is that there are rings with characteristic zero that are not fields and fields with positive characteristic (both cases fulfill the statement in the video). There are also rings like Z/pZ[x] that fulfill the statement but where both arguments fail which is pretty interesting.
To restate your proof. Proof by contradiction If (x,y) €ℝ and (x,y) =/= 0 and xy=0. Since xy = yΣn=1 x. Then there exists (yΣn=1 x )= 0 where (x,y) €ℝ and (x,y) =/= 0. However we know that there does not exist (yΣn=1 x)=0 that (x,y)€ℝ and x=/=0. Your proof lacks justification of the non existence of a non-zero real number that when added to itself a non-zero real number of times doesn't exist. That statement would also need proof.
I think this proof doesn't work: The statement 'However, we know that if you add any real number not equal to zero to itself any amount of times, it will never equal zero.' is wrong. This is stating that x + kx cannot equal zero unless x = 0, but if we take k = -1 it will equal zero (although I don't think this is what you meant). But if you see 'adding a number to itself any amount of times' just as the product xy, this statement is just stating that xy = 0 is impossible without either x or y being zero, but that was the thing we were trying to prove in the first place.
@@smikkelma3504 you're right! I didn't think about the case where you could add a value multiplied by a constant. I was thinking in terms of x*y is the same as x plus itself y amount of times. could you help me understand why the proof in the video works?
the deeper problem with this proof is that the statement "multiplication is repeated addition of a value" is not quite correct. it's basically true in the integers, but not in more complicated systems. for instance, what is (2/3)*(5/4)? there are certainly ways to sort of view this as repeated addition, but it uses a lot of machinery that needs to be proven first. and this is only in the case of fractions: something like e*pi is even harder to think of in this way. (it is still possible, just quite complicated.) the proof in the video sidesteps this issue by forgetting the notion of multiplication as repeated addition. it's much more common in math to think of multiplication as a binary associative operation that distributes over addition, and especially in the real numbers, we also require that every nonzero number x has an inverse (some number x^-1 such that xx^-1= 1). as such, this conceptualisation gives us another characterisation of zero-ness: a number is nonzero if and only if it is invertible. now, if x and y are invertible, then xy is also invertible (y^-1 x^-1 always works), so xy is also nonzero.
Since x & y are not zero let x = m+k and y = n+k ; k ≠ 0 and m, n belong to R xy = (m+k)(n+k) = mn + km + kn + k² => xy is non zero because k exists independently
So I'd like to point out that when distributing (m+k)(n+k) that k is actually k^2. Which messes with your final proof because if k(m+n) + mn = -(k^2). At (m,n)=0 => k^2 =0. Though we know k is not zero, we don't know k^2 is not zero because that is what we're trying to prove.
@@jangmaster3000 Oh yeah that is my mistake. But I think the point still stands, if k² = 0 then ultimately k has to be 0. And then contradiction. And we actually do know for sure k² ≠ 0, because otherwise it allows k = 0
@@abdoonyoutube7997 Why would k^2 = 0 show that k = 0. The problem is that if k^2 = k*k =0 where k =/=0 and a real number then isn't that what were tring to prove? We're just start back at the beginning but x and y are just replaced with k it's a circular proof.
Yep. Cost becomes a consideration when it's significantly longer than 4 years. Otherwise I'd probably still be in school too. Just studying a different major probably.
Tell me about it. I ended up going into accounting which is MUCH simpler than math if you don't go for the CPA. But I still respect my bachelors in math more than my masters in accounting even if potential employers do not.
@@theboombody yeah, is probably the hardest major out there, but is not that hard, I realized a long time ago that being an student is easy because is just solving already solved problems. Going into grad school would be hard tho... But with a math bachelor's you can't do much for real, and that's the real difficulty, because is not just a bachelor, is going to undergrad school+grinding a lot to get into a decent grad school+finishing a PhD to maybe land a position that is most likely low pay given the amount of effort you put in.
@@gabrielbarrantes6946 Right. No reason to go for it unless you're passionate about the material. Might as well go into engineering if you're just looking for a secure job. Some math I get bored to tears by (mostly computational math) and other math I get absolutely mesmerized by. So I just pursue the math I find interesting and do it as a hobby now. I do accounting for the money but I do math for the love. I wish I had the talent of Euler or Gauss, but I don't even have the talent of a normal PHD. So it makes no sense to pursue that road when I have so little to offer and the chances of making a living doing it are not good. My mediocre accounting talent serves me much better financially than my less than mediocre mathematical talent.
I've not seen a single mathematician talk about why we use just the 2 order logic system. As if it's superior?? And you are talking about circular reasoning, the whole premise of real analysis depends on saying that the least upper bound exists (yes I know you'll say it's an axiom) but it's just a joke. The deeper you go into analysis the more you see what a scam it is. Not abstract algebra which is the best.
what is your math education? i don't mean formal training, i mean how much serious math have you done? im asking bc you sound like me two years ago before i switched majors from computer science to math. ive learned (and suffered) and ive come out the other side realizing some important things: 1. (most) mathematicians dont care about logic. one of the reasons behind this is bc for all intents and purposes, ZFC or NBG+grothendieck universes does the trick. remember that math is not about finding the best way to arrive at truth, math is about arriving at truth given a set of axioms. our job as mathematicians is not always to question these axioms but moreso to discover how far they take us. this is important bc babylonians were solving quadratic equations 2000 years ago and i can guarantee you that they did not have a formal axiomatic system. yet, we teach those same formulas today... 2. logic is rly only talked about in math exposition. dont get me wrong, there is A LOT of logic during undergraduate mathematics (think point set/algebraic topology, homologies, categorical group/ring/field theory etc.), but it always serves as an approach, and not as a theory in and of itself. it's kind of like graph theory: logic is a concept that requires no training to grasp and thus videos about logic (or graph theory) can get a lot of views. but mathematics extends far beyond logic and graph theory and those elementary concepts that capture the attention of millions. do not fall for the math exposition trap: do the math. you will realize that your comment-- although well meaning-- is very obviously incorrect. math is not a semantic of second-order logic (if it were mathematicians wouldnt ever be able to prove anything of value) and there is nothing wrong the supremum axiom (its just an axiom). the further you go in math, the more you realize that the boundaries between analysis, geometry, (abstract) algebra, and probability are blurred. you also realize that math today is still very much like the ones championed by the babylonians: it is pragmatic. if you want to remain in your safe bubble of "logic," be my guest. but logic is a mess, and frankly you'll have a better time doing theoretical computer science. but dont take my word for it. venture out and DO REAL MATH. there are plenty of very serious logicists out there (bless them) who are probably giddy at the idea of sharing their work with interested and curious people like yourself. just please, dont go around commenting this kind of stuff. keep working hard and in a year or two you'll come back to this and look at your comment with puzzled disbelief, if not cringe. its also ok to struggle with analysis, but work at it, become virtuosic: it will free you. anyway enough rambling from me, hopefully you actually see this... have a nice day.
5:55 I made the comment, "the number of proofs is uncountably infinite." Many people have addressed this in the comments and I want to apologize for making this misleading remark. While researching for the video I came across this statement and accepted it as fact without checking it myself.
While the statement as given is not technically inaccurate since we would first need to define a rigorous logical system and without this we can't comment on the validity of any statements (is this confusing or what?), under a few simple assumptions such as:
1) Languages have a finite number of characters; and
2) Proofs are finite,
we can show that the number of proofs is a countable union of finite sets, which is countable.
Sorry for the misleading statement. It will happen again! I appreciate all the comments point this out 😊
I found this video 7 years late, now I’m in a math phd
Nerd
@@skomants2997 zionist
What a nerd
?
@@skomants2997
*Hrmmmmmm Intensifies*
I really like that you are openly discussing the problems of how math is taught. I feel like I sound like a broken record talking to my friends about this stuff. This video makes me think I might be good as a math major, because I was learning proofs this whole time. Learning proofs has just been the way I connect new processes to my prior mathematical understanding. It just didn't feel right memorizing a formula if I couldn't verify for myself why it works.
As someone considering a math major this actually encouraged me, not that I think I'm any good at this, but rather it looks like fun!
It's fun, but it's not lucrative. The interesting math doesn't pay. The practical math pays.
@@theboombodyif you learn how to code or get a minor in comsci you can get a job
This sounds like an area Australia excels at. We learn basic basic proof idea in High School senior math (Advanced, Extension 1 Math) (From NSW). In Extension 2 math we even learn formal proof techniques like proof by contradiction, induction, proof by contraposition, inequality proofs, number proofs, geometric proofs involving vectors. I think High School math really prepares me for university math in the future due to the focus on higher order thinking earlier on in my mathematics journey (topics learned by Year 12 Math if you take the top level courses for context: Sequences and Series, Trigonometry, Functions, Differentiation, Integration, Vectors, motion, differential equations, statistical graphs, binomial statistics, probability distributions, formal proof techniques, complex numbers (goes to De moivre’s theorem, roots of unity and exponential form), integral techniques (integration by substitution, integration by parts, general integration rules for repeated integration, integration by partial fraction decomposition), mechanics, 3D vectors. This is also reinforced in first year university math courses (like in Discrete Mathematics which is often mandatory for math majors has a part of the course look into formal mathematical proofs).
I am currently taking AP Calculus AB and will be taking BC next semester. I REALLY want to go into a math major in college cause I feel like that would be the only major I would enjoy. We did proofs in Geometry, it was a struggle at first because I had no idea on how specific I am supposed to be, but now I think I have gotten a lot better, but only for the geometry level.
I unfortunately completed my math major about 8 years before you created this video :P
But I stuck through it even though I ran into the same wall a lot of these kids went through because I knew I had enough progress to just barely finish it, and I wanted to challenge my perseverance on a real life goal - by bringing it back from the brink of failure - for the first time in my life.
Also, real analysis was easier than complex analysis for me :)
Getting my B.S. in Math at Georgia Tech this December :)
My high school didn’t have classes past Algebra II but I was able to dual enroll at a California community college to take Pre-Calc, Differential Calculus, Integral Calculus, Multivariable Calc, Differential Equations, Linear Algebra, and a Math Proofs Seminar. The Math Proofs Seminar (and the Linear Algebra class being proof-based) is what made me fall in love with math.
Now I have my dream job in Big Tech lined up to start post graduation 🥰
Also if you have the ability to take math classes at a community college instead of the AP equivalents: GO TO THE COMMUNITY COLLEGE. The teachers at community college honestly rivaled the best that I encountered at Georgia Tech because they were 100% focused on teaching (not research) and if you are a motivated high schooler who asks questions and comes to office hours, they will love you since they probably deal with a constant flow of mildly disinterested (“I’m only taking this because it’s a requirement”) college students.
current junior math major. love math but hating my life atm
As someone who has a backround in international math competitions, I personally think that the lack of exposure to proofs is the single biggest problem in a lot of school systems.
It's just a lot of fun to do a wide range of math exercises, especially doing little variations on things you have seen a lot.
Really awesome video! I’m a senior in high school right now wanting to major in math so I’m probably going to go to the library and borrow a book on basic proofs now 😂
Intro to fundamental analysis, real analysis, and modern abstract algebra made me understand what many students feel in a math class. Glad i did experience and knew that going further in math was not my jam, though makes stuff easier in masters of CS
Well since high school geometry on khan academy has a structure of 2 column proof, I had to adapt fast since it was rigorous as hell. From algebra 1 to high school geometry felt like getting wooped by Euclid and his book of elements.
I was first introduced to proofs from a textbook called discrete math and its applications by Rosen, and honestly I’ve never felt so humbled. Growing up I’ve never made below a high A on a math test. Calc 1 2 and the highschool algebra and trig were easy… simply learn an algorithm, follow the steps and you get the solution. However when our textbook started introducing simple proofs I’ve never felt anymore stupid, sometimes the idea would be simple, for example proving the sqrt(2) is irrational. However I had no idea how to formulate my ideas, and I felt so humbled. Needless to say I passed the course, and made me consider switching from computer science to mathematics. Currently I’m still majoring in computer science, but I really want to give a course like real analysis or abstract algebra a shot….
I was gonna scroll past this vid but the thumbnail made me laugh, subbed.
I am contemplating the idea of studying mathematics because I have an interest in the theme, but I don't know if it's best to be self-taught for now. I started to really like how it can solve real life problems and improve my knowledge. I studied some logic and statistics in an English teaching degree and afterwards started a computer science related degree.
I major in Yu-Gi-Oh and I use probably and statistics to win! More math majors should play TCGs, especially Yu-Gi-Oh!
That's a real major?
Is there also pokemon major
*probability
@@ZDTF 😅Only on UA-cam, I have acquired a PHD in Dueling due to my studies in AI & The Hypergeometric Distribution!!!
For those that aren't aware: Richard Garfield (the inventor of the modern TCG) has a PHD in combinatorics
Yoo 😂
5:59 the number of possible proofs is *countably* infinite
Second year CompSci major planning on double majoring in math, and I'm taking real analysis next semester. Oh boy.
6:10 what might also be helpful here is recognizing how the statement relies on the fact that the real numbers form a field and how this property fails in more general rings like Z/12Z. If you never saw those examples it's harder to guess what properties you need to use to prove the statement.
This is the whole point. A proof like this one will be shown on the first day of an introductory Analysis course. None of the students know what a group or ring or field is since Abstract Algebra comes later. It's possible to simplify the proof further by saying, "A field has no zero-divisors," but this is circular.
You have to start learning about proofs somewhere and that's a big part of the challenge.
I have a bachelor's degree in math, and proofs is definitely where I started to struggle. Abstract algebra basically almost killed me, but that was partially because the teacher I had for it made it a lot more difficult than it should have been. Even other math professors at my university agreed that the way she taught it was just way too hard for undergrad. Most math programs have an intro to proofs type course that helps prepare people for proof classes. Overall, Real Analysis actually wasn't that bad for me mainly because every proof has a specific process in Real Analysis. If you know the process for a proof in real analysis, it's not super difficult. Abstract algebra is a completely different story. I had a few failed courses throughout my undergrad, but I still graduated with my math degree. I am very proud of myself for doing what I did.
Makes me so glad I didn't major in math or engineering. But seriously, I'm very pleased to see how many non-US students are watching your new vlog. I'll bet they got taught bridge in their schools.
I feel like I'm going through the opposite process right now: I started as a comp-sci major, but I am trying more math classes this semester and I'm currently loving number systems (another proof based class, before real analysis). This video has given me the confidence that I would be a great math major.
The thing that made proofs kind of click for me is realising that I should just see proofs as an explaination for why something is true based on prior knowledge. Usually if I am stuck on a proof, it's a sign I need to look over or organize my notes again, and once I have all of my axioms, theorums, and everything in one place, then I can piece them together like a puzzle.
Basically: a priori
I personally always wonder if you don't see ANY proofs in America during high school
I grew up and went to school to Belgium and we already saw quite a lot I think
We saw the definitions of limits, derrivatives, integrals... and derived their properties trough the definitions
Because I follewed a math-heavy program I even saw the basics of linear algebra, number theory and projective geometry, of which we also learned part of the rigour
It's certainly not everyone's experience that proofs first come about in university level courses, but I think it is for a large majority.
Geometry, taught in high school, does cover the fundamentals of proofs, but the core material is new so students see the proofs as a new technique. Contrast that with proofs in Analysis where even in the comments of this video I've seen plenty of arguments (some since deleted) trying to prove a concept as simple as fields having no zero divisors which lack understanding and rigor. The difference is students believe they understand the properties of numbers (the real numbers) since they have worked with them for over a decade, but they never learned the underlying axioms or what rigid principles must be obeyed and why.
As for Calculus proofs, I do believe many schools go over the derivation of the definition of limits, derivatives, and (Riemann) integrals. The issue is that the teachers then don't ask for understanding of these concepts on the exams and instead just ask students to apply the power rule to a function, thinking that this constitutes understanding.
5:58 the number of proofs/theorems in general is actually countably infinite not uncountably infinite.
Since all proofs/theorems are of finite length, the set of theorems is the infinite union over all possible lengths, and for each length k there is a finite amount of theorems as words/mathematical signs are a finite set.
So it's an countably infinite union of finite sets which ends up as just countably infinite
I think the distinction isn't proofs vs. non-proofs, but rather problems that can be solved using a standard series of steps vs. problems that require more out-of-the-box thinking. Proofs can be just as formulaic and simple if there's no variation in the questions.. For example, in high school we had a proofs unit where we (for the most part) just had to prove some properties such as divisibility and rationality. Once our teacher showed us the steps to do this, we could pretty much just use these same steps for any question we got on the test, plugging in different numbers/expressions.
I'd argue that there really isn't much of a difference between proofs and questions without proofs. Proofs are just regular questions where they give you the answer beforehand, and the question would be more difficult in most cases if they didn't give the answer. "Prove that if 2x = 4, then x = 2" counts as a proof, but it isn't any different from "find x if 2x = 4". You can turn any proof into a question just by phrasing it differently: instead of "prove that x is true", you can ask "is x true?"
You definitely are right about students not being taught how to come up with unique solutions to problems themselves though. Most of high school math is just memorizing a few procedures that are taught by your teacher, and just plugging in different numbers. There is contest math however, which has questions that are not so straightforward, and I believe most students thinking of majoring math have tried it out.
I’m a senior in an arts degree who’s recently become obsessed with how math is developed and what constitutes rigorous proof… I don’t know if there’s a way to switch to math this late, but this video just makes me more passionate about that possibility.
Analysis is hard.... But like.... Honestly things get harder afterwards that it seems easy now
Looks like learning to prove things like how opposite angles of a cyclic quadrilateral are supplementary wasn't a complete waste.
Exactly on 10k views rn
Wow...
As a person taking an undergraduate degree in Mathematics I can relate. When I first started Real Analysis, it was my weakest topic. I couldn't understand what half of the lecturers were stating. Besides, the topic is almost completely unintuitive for me. Meanwhile, I can't say I'm good at discrete mathematics or linear algebra, though I am definitely more confident in those topics than analysis (real OR complex).
This video deserves more views.
Too late for me, now I’m in my last semester of undergrad 🤕
My geometry classes in high school went over proofs a bit, but I didn't really learn proofs until linear algebra. I think that helped with Real Analysis. I made it through, but that's when I decided grad school wasn't for me. I just didn't have the talent to go further. I'm glad I took it though since it actually proved the Fundamental Theorem of Calculus for me, which I found fascinating. Although I always had a problem with some of the stuff they taught about Cantor. Yes the reals are bigger than the naturals. But I never accepted that the interval from 0 to 1 had the exact same number of elements from 0 to 2. You can put them in a 1 to 1 mapping of course, but you can also put them in a 2 to 1 mapping. And probably other maps as well.
Im a stats and data science major and was guessing this would be about real analysis and was right. Been avoiding this vid cuz it seemed clickbaity but since im procrastinating, i clicked anyway.
Im kinda tempted to take real analysis at some point. Its not necessary at all to take it, but my proofs class has been neat enough to where im curious.
Weird , i recently started to study math in collegue (well , the equivalent of collegue here , in Spain) and we are already doing some kinds of proofs , and even we have a subject called "sets and numbers" which give us basic notions in proofs and mathematical logic.
I’m glad Spain is teaching math better than we are!
ive just started studying maths in spain this year too and we also have that class (along with intro to analysis, which also dives into proofs)
Leaving a comment to boost the algorithm
Great video btw!
I will take real analysis next semester
for anyone doubting about mathematics: remember that math is creative. if you love math and enjoy finding creative and original solutions to problems, you are on the right track. if you have the patience and discipline to work hard at it every day, not only are you in the right field, but mathematics will be a BREEZE. dont let anyone scare you away! best of luck
I don’t know I’m watching this when I’ve already graduated with a math degree. Started as Physics though then switched to Math pretty late.
Going into my 4th year math major at a top university in Canada, and I would say 100 percent real analysis was a bit of a hurdle. It was worsened for me because I went from doing my first and like half my second year at a smaller college where things were taught in a very procedural way. Then I went into an honors math program when I transferred to the university. Man, what a way to just be thrown into proofs. First proof class is an honors real analysis 😢 I've made it through all of that now though.
As a 15 year old high school student with a strong foundation in Algebra, Trigonometry, Mathematical proof writing, Real Analysis Requires A level of Abstract and deductive thinking,l i have currently not adapted to.
Limits, Convergence ,Ratio tests,Alternating series test etc. were and still are super hard to apply but due to my strong foundation in proof writing and prior training in abstraction. It is fairly easy! but maybe i am just still woefully ignorant.
My English is very bad so please don't roast me 😅
(
Also it's pretty satisfying to write less than epsilon in a big rigourous proof and writing QED.
Wait, what?
I'm not from the US so I don't know the school system. Do you really not get to do proofs for 2 years even at university?
At my uni in Germany you have to work on some kind of proof in the first week.
What is taught in Calculus and what's Real Analysis? I think at my uni those are grouped into one series of lectures. But maybe I still don't get what Real Analysis is actually about.
I don't have a wholistic perspective on what happens outside the US, my only understanding of other school systems comes from my interaction with graduate students from other countries.
What is taught in Calc vs. Real Analysis? Calculus focuses on problem solving techniques like the power rule for derivatives or formulas for arclength or surface area integration. Real Analysis focuses on proving formulas from Calculus, like using the delta-epsilon definition for a limit. I think the school system's rationale behind not teaching the proof techniques earlier than the 3rd year is that many engineering students need calculus for their jobs, but they don't need to know what a proof is. It's expected that only students who study pure math will take real analysis and therefore proofs are delayed to weed out the rest of the population.
When I was at school, in France, we started to do proofs (on parallelograms, I think) at what is the French equivalent of US eighth Grade (i.e "4ème") .
It may unfortunately have changed since these happy days (1979), though.
Oh, interesting. Here we have different courses for math, physics, computer science students. I don't think one can get an engineering degree at my uni. The focus on proofs is different among those courses. The math students definitely learn Calculus tools while proving why they work (or more commonly the professor proving why they work).
This way most students drop out in the first or second semester if they don't wanna do that kind of math. Even for comuter science students the most common dropout reason are math courses unfortunately. They might still be too proof focused.
@@eagelwizard290in america u learn way more math before U start with our Proof based uni system, but because of that u ll do a speedrun of what is tought in analysis 1,2,3 in real analysis
@@eagelwizard290 That lack of proofs until real analysis would be atypical for my experience too. At my math major program in Canada, we did proofs from semester one, because that's what made it a math program and not an engineering program.
Is it really normal to not have taken a proof based class before real analysis? My uni requires you to take an introduction to proofs, and proof based linear algebra/calc 1 + 2 in first year. Second year you do more proof based linear algebra, so by the time you get to real analysis you have a good amount of experience with proofs.
I would add that many of these students are probably better served studying applied maths, whether has part of a different degree (physics, CompSci, engineering) or as part of a dedicated applied maths degree, which does exist. A lot of the thinking and type of problem structures in that are closer to the high school maths you talk about.
Real analysis was easy and intuitive for me
Not me finishing my major watching this vid
What you said about real analysis being the first course with proofs isn’t accurate in my experience. We did them in high school geometry and then I had to take an intro course for math majors which was linear algebra but proofs based and it was a good bridge between calculus and pure math. Also discrete math for my cs major. If you’re school does this they have a shit math department
I was hoping for a look at that "easy" integral. I assume it would involve a parity trick.
It is indeed a tricky one but you're on the right track! Since the integration interval is centered around 0, think about f(-x) in relation to f(x) where f is the integrand. Can you divide the integral into two equal parts that may allow for algebraic cancellations?
@@LilBiteOEverything ah, I see. It would utilise
f(x)=1/2 (f(x)+f(-x)) + 1/2 (f(x)-f(-x))
It all reduces to the integral on [-1,1] of 1/2 cos(x) dx = integral on [0,1] of cos(x) dx = sin(1)
I hope the math major you resurrected from quitting will weigh in :)
And they teach us proofs in last year of high school in Iran.
I was struggling with very basic proofs in high school, and that made me confused because im usually very good at math, i asked my teacher about it and then he told me about how in his analysis class only 4 of the 30 students finished the class, and they had to reduce the minimal grade so theyd pass, else only 2 had enough grade to pass the class
1st semester math major here can confirm my ass is cooked
Who knows, you may be surprised. I thought I was going to flunk out at first, but I ended up joining one of the rec sports teams on campus and that time away from studying actually gave me some help and encouragement.
Very interesting video! How does proof-based linear algebra stack up to real analysis in terms of difficulty?
Any proof-based course will have many of the same ingredients, so Linear Algebra with proofs will be a good introduction to the concepts. But Real Analysis is very focused on producing rigorous proofs from day 1 while Linear Algebra will still spend at least half of the time covering problem solving techniques like the Gram-Schmidt process. My guess is that Real Analysis is still a substantial bit more difficult than other introductory proof courses.
@@LilBiteOEverything i appreciate the feedback👍
How do i major in math??
5:55 aren't proofs supposed to terminate? In that case wouldn't there only be countably many?
The answer does depend on how you define language and a proof. Your idea of finite-length proofs in a finite-sized language resulting in countable proofs is certainly more popular than the one I presented.
5:55 "... the number of possible proofs in uncountably infinite..." It certainly is not! It's infinite, but not uncountably infinite. This guy needs to take some math classes.
Am i dumb if i cant solve that calculus question about finding the area but was able to prove xy≠0?
Why can't xy≠0? Is there a rule that neither x or y can be 0?
What website are you showing us at 0:55 ?
www.princetonreview.com/majors/all
I really want to major math but i still need to wait a few years (i'm 15), however, i would like to know a good place to find studying material both for entry level college maths or american HS algebra/calc
Every help is welcome.
For high school math, I've heard Kahn Academy is good. For Calc or college math MIT OpenCourseWare is excellent. Hope this helps!
Best of luck on your journey!
One might also see this fact as something encouraging, especially the less computationally inclined among us (that's including me haha).
Computers do all the computations anyway. The proofs are where you get to get creative. Which is fun until test day. Then when you can't get creative on demand it's not quite as fun. But I still enjoy math as a hobby.
5:59 no it's not. Certainly not the numbers of proofs in languages that exist on earth. Don't use mathematical language wrong while talking about mathematical language
This is of course up for debate. To be rigorous in making a mathematical statement it's important to use rigorous notation and that would take several weeks of studying logic to do. I did no such thing. Neither did you. And it's well beyond the scope of this video anyway.
I'll concede that a more popular belief would be that the number of proofs is countably infinite, but my statement is in no way 'wrong'. I could write down a formal set of axioms and notation for which the number of proofs in uncountably infinite.
@@LilBiteOEverything you're right of course - sorry I phrased my comment this harshly. It simply depends on how you define a mathematical proof and most importantly it has little to do with the point you are making. I just think that it is not useful to just throw in the word uncountable in a context where it makes little sense and is not needed.
But of course that is just a detail and other than that I did like the video!
Clicked this as soon as it reached 10k views
In Morocco we take real analysis 1 and 2 in the first year as well as a 2 proof based algebra courses and I'm not even a math major I'm finding it very diffuclt why the hell do we need to learn this as informatics students
Bit late I applied last week😅
Proof by contradiction
If x&y are non-zero real numbers and their product is zero, then, since multiplication is repeated addition of a value, there must exist at least one non-zero real number which when added to itself a non-zero real amount of times must equal zero. However, we know that if you add any real number not equal to zero to itself any amount of times, it will never equal zero. Therefore their products must be zero, unless you’d like to say that anything other than zero added to itself n amount of times will equal any value not a multiple of themselves, which is ridiculous.
It's interesting how the proof presented here and in the video are kind of distinct. To use mathematical language, you are using the fact that the real numbers have characteristic zero while the video used the fact that the real numbers form a field. What makes them distinct is that there are rings with characteristic zero that are not fields and fields with positive characteristic (both cases fulfill the statement in the video). There are also rings like Z/pZ[x] that fulfill the statement but where both arguments fail which is pretty interesting.
To restate your proof.
Proof by contradiction If (x,y) €ℝ and (x,y) =/= 0 and xy=0. Since xy = yΣn=1 x. Then there exists (yΣn=1 x )= 0 where (x,y) €ℝ and (x,y) =/= 0. However we know that there does not exist (yΣn=1 x)=0 that (x,y)€ℝ and x=/=0.
Your proof lacks justification of the non existence of a non-zero real number that when added to itself a non-zero real number of times doesn't exist. That statement would also need proof.
I think this proof doesn't work: The statement 'However, we know that if you add any real number not equal to zero to itself any amount of times, it will never equal zero.' is wrong. This is stating that x + kx cannot equal zero unless x = 0, but if we take k = -1 it will equal zero (although I don't think this is what you meant). But if you see 'adding a number to itself any amount of times' just as the product xy, this statement is just stating that xy = 0 is impossible without either x or y being zero, but that was the thing we were trying to prove in the first place.
@@smikkelma3504 you're right! I didn't think about the case where you could add a value multiplied by a constant. I was thinking in terms of x*y is the same as x plus itself y amount of times.
could you help me understand why the proof in the video works?
the deeper problem with this proof is that the statement "multiplication is repeated addition of a value" is not quite correct. it's basically true in the integers, but not in more complicated systems. for instance, what is (2/3)*(5/4)? there are certainly ways to sort of view this as repeated addition, but it uses a lot of machinery that needs to be proven first. and this is only in the case of fractions: something like e*pi is even harder to think of in this way. (it is still possible, just quite complicated.)
the proof in the video sidesteps this issue by forgetting the notion of multiplication as repeated addition. it's much more common in math to think of multiplication as a binary associative operation that distributes over addition, and especially in the real numbers, we also require that every nonzero number x has an inverse (some number x^-1 such that xx^-1= 1). as such, this conceptualisation gives us another characterisation of zero-ness: a number is nonzero if and only if it is invertible. now, if x and y are invertible, then xy is also invertible (y^-1 x^-1 always works), so xy is also nonzero.
I am not a math major but am taking Discrete Math which covers many concepts in writing formal proofs, is that not normal for math majors?
Since x & y are not zero
let x = m+k and y = n+k ; k ≠ 0 and m, n belong to R
xy = (m+k)(n+k) = mn + km + kn + k²
=> xy is non zero because k exists independently
Wouldn’t you also have to prove that mn + km + kn =/= -k?
@@happypig8690 we can see if, k(m+n) + mn = -k². At (m,n) = 0 => k² = 0 => k = 0. This is a contradiction bc by def. k≠0
So I'd like to point out that when distributing (m+k)(n+k) that k is actually k^2. Which messes with your final proof because if k(m+n) + mn = -(k^2). At (m,n)=0 => k^2 =0. Though we know k is not zero, we don't know k^2 is not zero because that is what we're trying to prove.
@@jangmaster3000 Oh yeah that is my mistake. But I think the point still stands, if k² = 0 then ultimately k has to be 0. And then contradiction. And we actually do know for sure k² ≠ 0, because otherwise it allows k = 0
@@abdoonyoutube7997 Why would k^2 = 0 show that k = 0. The problem is that if k^2 = k*k =0 where k =/=0 and a real number then isn't that what were tring to prove? We're just start back at the beginning but x and y are just replaced with k it's a circular proof.
I think medical school and law have more dropouts, because of length.
Yep. Cost becomes a consideration when it's significantly longer than 4 years. Otherwise I'd probably still be in school too. Just studying a different major probably.
@@theboombody Oh yeah I'm not considering cost it's free here. But still more dropouts.
This video is about 14 months too late
YOU DID IT YOU GOT 10K VIEWS
It's at 10k rn
BRADYBOT IM A MATH MAJOR BUT WTF
Nah, difficulty is not the problem, the issue is that is oversaturated and has no jobs prospects.
Tell me about it. I ended up going into accounting which is MUCH simpler than math if you don't go for the CPA. But I still respect my bachelors in math more than my masters in accounting even if potential employers do not.
@@theboombody yeah, is probably the hardest major out there, but is not that hard, I realized a long time ago that being an student is easy because is just solving already solved problems. Going into grad school would be hard tho... But with a math bachelor's you can't do much for real, and that's the real difficulty, because is not just a bachelor, is going to undergrad school+grinding a lot to get into a decent grad school+finishing a PhD to maybe land a position that is most likely low pay given the amount of effort you put in.
@@gabrielbarrantes6946 Right. No reason to go for it unless you're passionate about the material. Might as well go into engineering if you're just looking for a secure job.
Some math I get bored to tears by (mostly computational math) and other math I get absolutely mesmerized by. So I just pursue the math I find interesting and do it as a hobby now. I do accounting for the money but I do math for the love. I wish I had the talent of Euler or Gauss, but I don't even have the talent of a normal PHD. So it makes no sense to pursue that road when I have so little to offer and the chances of making a living doing it are not good. My mediocre accounting talent serves me much better financially than my less than mediocre mathematical talent.
17k views, today, more than 10k. Proved. It is easy, next.😂
It's real analysis.
Is this bradybot?
Yes
Nope.
🙏🙏🙏
bump
I've not seen a single mathematician talk about why we use just the 2 order logic system. As if it's superior??
And you are talking about circular reasoning, the whole premise of real analysis depends on saying that the least upper bound exists (yes I know you'll say it's an axiom) but it's just a joke. The deeper you go into analysis the more you see what a scam it is. Not abstract algebra which is the best.
what is your math education? i don't mean formal training, i mean how much serious math have you done?
im asking bc you sound like me two years ago before i switched majors from computer science to math. ive learned (and suffered) and ive come out the other side realizing some important things:
1. (most) mathematicians dont care about logic. one of the reasons behind this is bc for all intents and purposes, ZFC or NBG+grothendieck universes does the trick. remember that math is not about finding the best way to arrive at truth, math is about arriving at truth given a set of axioms. our job as mathematicians is not always to question these axioms but moreso to discover how far they take us. this is important bc babylonians were solving quadratic equations 2000 years ago and i can guarantee you that they did not have a formal axiomatic system. yet, we teach those same formulas today...
2. logic is rly only talked about in math exposition. dont get me wrong, there is A LOT of logic during undergraduate mathematics (think point set/algebraic topology, homologies, categorical group/ring/field theory etc.), but it always serves as an approach, and not as a theory in and of itself. it's kind of like graph theory: logic is a concept that requires no training to grasp and thus videos about logic (or graph theory) can get a lot of views. but mathematics extends far beyond logic and graph theory and those elementary concepts that capture the attention of millions. do not fall for the math exposition trap: do the math. you will realize that your comment-- although well meaning-- is very obviously incorrect. math is not a semantic of second-order logic (if it were mathematicians wouldnt ever be able to prove anything of value) and there is nothing wrong the supremum axiom (its just an axiom).
the further you go in math, the more you realize that the boundaries between analysis, geometry, (abstract) algebra, and probability are blurred. you also realize that math today is still very much like the ones championed by the babylonians: it is pragmatic. if you want to remain in your safe bubble of "logic," be my guest. but logic is a mess, and frankly you'll have a better time doing theoretical computer science. but dont take my word for it. venture out and DO REAL MATH. there are plenty of very serious logicists out there (bless them) who are probably giddy at the idea of sharing their work with interested and curious people like yourself. just please, dont go around commenting this kind of stuff. keep working hard and in a year or two you'll come back to this and look at your comment with puzzled disbelief, if not cringe. its also ok to struggle with analysis, but work at it, become virtuosic: it will free you.
anyway enough rambling from me, hopefully you actually see this... have a nice day.
Oops too late