how 2 develop mathematical maturity as a programmer

Added: 2026-06-01

[00:00:00]Mathematical maturity is a concept that describes how literate someone is in mathematical arguments and how well-versed they are when it comes to certain practices or ways of thinking in math. If you're coming from a computer science or a programming background and you're wanting to learn more about math and wanting to build this sort of mathematical maturity, then you're actually playing at a sort of advantage since there are a lot of things in programming that are analogous to math.

[00:00:23]Even if you have no actual interest in math for its own sake, training your mind to better assess logical arguments and think more abstractly is obviously going to benefit you when it comes to programming where you're doing that all the time. Also, it's just more interesting whatever the [ __ ] been going on in software for the past few years. If you've done a computer science program in any college or university within the United States, you've probably had to take a discrete math class in your freshman or sophomore year of college. Right now, this class is very important for computer science people since it's probably your first introduction to formal logic and proofs that or you might have done some basic proofs in a linear algebra course, right? So, regardless, you you have had at least some exposure to proof-based mathematics. In your junior and senior years of university, you're mostly taking major specific courses, right?

[00:01:10]Which might mean a few like theoretical CS courses. And you know, that's the thing about CS people is that they're probably more well-versed in pure math than engineers, right? That's that's the one singular uh thing that CS people are better at than engineers is that they're more experienced in pure math. Like I have a friend in electrical engineering and he thought that a a definition like just just some basic equation defining like some some function in like his engineering stats course was a proof.

[00:01:38]So, you know, GG's, bro. How do you actually build up this this maturity as a programmer? Well, the main thing that you have to do is that you have to learn math, right? Surprising, right? You have to read math textbooks. But the good thing is is that pretty much every math textbook is very similar in structure to software. In fact, it's a lot neater than software since instead of having to look through a bunch of different files for definitions of things, it's all laid out in a very linear fashion with everything clearly labeled with clear English explanations and justifications for them. The essential structure of most textbooks goes as follows. Right?

[00:02:13]Chapter 1 usually introduces the object of study that you're going to be working in for the rest of the book. Chapter 2 will usually introduce some sort of mapping or relation between objects, right? And then chapters 3 through 37 are just a bunch of nonsense essentially, right? I mean they're basically just exploring like all the different types of mappings and objects we can have, what constraints we could put on them, seeing like if if these things satisfy certain conditions, then certain properties of interest or usefulness then arise, right? And there are essentially only three components to a textbook that you're just going to be seeing over and over and over again for the entire book, right? And those are definitions, theorems, and exercises.

[00:02:54]Obviously, the author will try to let they'll have like paragraphs introducing certain things that lead into other things and try to string everything together in a way that's, you know, nice and cohesive like a book. But, I mean, these are essentially the three types of of things you're going to be seeing in the book. And they usually follow very predictable patterns. Pretty much every section starts off with a definition, right? And this definition introduces terminology and notation that you're going to be using for the rest of the section, if not for the rest of the book. And definitions aren't always super simple. Like often times you might like have to construct some object and then it gets like formally defined, right? But these these definitions are sort of analogous to like strrus or classes in programming, right? Like these sorts of objects or these things that exist that then act upon other things or get acted upon or are capable of doing certain actions or exhibiting certain behavior throughout the rest of the program. And usually you know these things are formally defined with certain variables and certain maybe methods or what have you, right? The same way definitions define certain objects with its own specific behavior and conditions that it has to satisfy. But you know definitions and declarations in programming of certain types of objects or classes or what have you, right? Are not necessarily what the part that makes your brain hurt, right? That's not really the hard part. Essentially the hard part, the part that's like equivalent to actual code or functions in software would be the theorems, right? The theorems pretty much every definition is followed by multiple theorems, right? And the and these theorems give you certain known true properties for a definition, right? For some, I guess, you know, object that that has been previously defined. And pretty much every theorem is followed by a proof for the theorem, right? or if not they tell you to do the proof yourself and if you're not used to studying proofbased mathematics then you know for some people this might come across as sort of like DLC like oh if you really want to and like you want to understand why this works and you you could read it but you know you could just read the definitions and read how you know these properties and how it works and just go on with your day which is not true at all right because reading and understanding the proofs of the textbook are the actual meat and potatoes of the textbook right it's the entire basically the entire point aside from you know having the prerequisite knowledge or just do the exercises but it's also really important to learn not only why these things are true but how to prove something is true you know in this sort of in this realm sort of of these specific objects and relations right you're basically training your mind on the common strategies and tactics employed to formally study these objects and it's also very likely that you're using definitions of objects and the properties of those objects and theorems to then prove things later on down the road about newer objects built upon the foundation formed from earlier parts of a textbook. And the reason why I say that theorems and their proofs are analogous to like functions and snippets of code, you know, like methods and stuff in programming is because that it's it's essentially describing the behavior of an object, right? And providing the logical reasoning for that behavior. Not only that, but it is very, you know, often that, you know, some theorem gets labeled with like a number and maybe even get it's given a special name, right? [music] Named after some like dead European guy and you will later invoke that proof in that property later on down the road to prove something else, right? You're essentially reusing something already proven, reusing the logic from something that we already discussed to then solve another problem or or built upon the logic of something else, right? which is very similar to how a function in some program could be defined to you know do some specific thing. But if that thing needs to be done multiple times throughout the program, uh you're not going to rewrite the same logic of the function multiple times, right? You're going to just reuse the logic you already [music] built, right? You're going to just reuse that code. Same way we reuse the logic from these previous proofs, sometimes even previous exercises to then build this subject further and further up, right? [music] A very important part of developing mathematical maturity is being able to read proofs and being able to understand them. [music] You know, you don't want to just like read a proof and just handwave it away is like, okay, yeah, I mean that that seems like it's true. Uh, you want to make sure I mean, if it's something very simple, then yeah, sure.

[00:07:23]But if it's something a little bit more complex, you want to make sure you actually work your way through the logic and understand why one thing implies the next thing and why the next thing implies the next thing and why that might lead to a contradiction or might lead to something else that implies something else. Right? You have to understand like general like the structure of general proofs like how you prove an inequality or how you prove two sets are equal to one another or how you prove an if and only if statement, right? In fact, you know, it's probably good practice to actually read the theorem first and then sort of work out a proof in your mind or maybe on paper on what what the proof in the textbook might look like, right? It might not always be correct. You might it's not like like a required exercise obviously in like the textbook but it is good practice to be able to predict proofs to be able to use the logic previously established throughout the book or just general mathematical uh uh reasoning skills and and being able to prove things before the textbook tells you why they're true. You know, some textbooks might have like really concise and clever and creative proofs that you just weren't able to predict. you know, those are usually the the proofs that you're going to have to read over and over and over again to understand like how exactly they got from one point to another, right? That's really like the the hard sort of work that that that's like the core like labor that goes into actually building up your reasoning skills to see why some argument is valid or invalid. And hopefully this is something that you're already well verssed in when it comes to programming and being able to reason logically, you know, throughout different steps of of a function why things happen the way they happen, right? Like if a certain bug if a function is not acting appropriately and it has like some buggy behavior, then usually you'll read the function, you read the code and then you'll logically work out, okay, this variable is at is at this value at this point and then it goes to this value and then this variable is introduced and then we call this function and then we do this and we do that. you're working out the logical steps through your head to deduce the possible reasons why a certain outcome is given when when that logic is called, right? And this is sort of working out the same muscles that you do when you're reading a proof, right? It's rigorous sanity checks against your own logical arguments or other people's logical arguments to see why something might be true or false or to see why something might happen that isn't intended to happen or why something is correct like why why code is correct why a proof is correct and that there is no possible logical outcome for why unwanted behavior or consequences can be exhibited. Usually at the end of each section or chapter of a math textbook, there will be a series of exercises for you to do for you to test these skills and this knowledge that you've learned to really see like how well you understand the concepts. Now, it's pretty important that if it's a subject that you really want to learn and that you're not fairly familiar with, then you should probably at least attempt every single exercise, right? Doing these exercises are really the best practice or or workout you can do for your brain to really cement what these concepts are about, right? To to really build intuition for how these objects behave and what the properties are and what these certain things imply.

[00:10:34]Intuition is a very big component of mathematical maturity as well. And it's mainly built through doing exercises.

[00:10:40]It's like the entire textbook is like a software, some large system of programs that is slowly building upon each and every single previous stage to become more and more complex and become more and more expressive and functional. And these exercises are sort of a way for you to contribute your own sort of uh ideas to them. Often times in certain sections in the future, they will invoke certain proofs that you they expected you to do from certain previous exercises to further strengthen a concept or illustrate some underlying theme of of another concept, right? Or even very commonly like other exercises will call upon previous exercises as a way to sort of like recall this idea.

[00:11:23]Well, okay, do this, use this idea to then prove this, right? Basically, right? So it it's sort of like a partnership between the author and the reader to build up the system of logic.

[00:11:33]Now I would say there's definitely a priority when it comes to exercises.

[00:11:36]Like if you can't do all of them, you should do the ones that are the hardest for you. The ones of lowest priority would be the ones that are, you know, so easy you can just do them in your head.

[00:11:45]Some people will skip computational exercises, but I think they're kind of important to at least do a few. You don't have to do like all of them, but at least like a few, you know, to to really understand how to compute something. and to understand like how it works in application. But you really want to be prioritizing the ones that are the hardest, the ones that seem like very weird cuz usually those require more creative solutions that require you to think more deeply about the subject.

[00:12:10]And you know, if you don't challenge yourself, you're not really ever going to get better, right? If if you avoid all the hard ones, like the same way if you [ __ ] like decide not to do past like 20 push-ups because anything past that hurts, you're not going to actually improve. if you're not actually going to make yourself stronger in any capacity, right? So, you need to you need to regularly challenge yourself and, you know, make yourself do hard things. But, I mean, I think I've talked about like the how of studying math long enough.

[00:12:39]Like, you know, if you're a CS major or a CS graduate, how do you actually progress in math? I mean, what direction do you go in? What subjects do you study? You know, what books do you read?

[00:12:48]There's pretty much two directions you can go into, right? you could study new math subjects that you're unfamiliar with. That's sort of like a continuation of of your university math education, which for the average CS person, that would be, you know, subjects like real analysis or abstract algebra. Or you can look at a new proof-based pure math perspective of a subject that you might already be familiar with or have some exposure to stuff that you might even be able to directly apply or that you use every day when writing software which is usually going to be subjects that were introduced in like your discrete math class like graph theory, combinotauric, stuff like that. But uh I mean the way I continued to pursue math on my own was through a sort of compromise between the two which is linear algebra. Right?

[00:13:32]Every CS person has taken linear algebra before, but it was probably matrixbased computational linear algebra, right?

[00:13:39]Since most of the people in a linear algebra course at a university are probably going to be engineers and CS people. The book I personally used was linear algebra done right by Sheldon Axler, which is a book I've read first of all front to cover and is also a very well-known book for you know specifically not introducing determinants until like the very end. So you're basically going through linear algebra through the perspective of vector spaces and linear transformations. And if you've only taken like a computational matrix space course in linear algebra before, it gives you a much clearer perspective on the subject. Like at first you're like, "Oh, wow. Matrices are like not even like what linear algebra is really about. But then you read further along in the book and you gain a a much greater appreciation for just how useful matrices are." And you know, the further along you get in the book, you go from like a proof-based perspective on linear algebra 1. And then like in the second half of the book, you're going well into like concepts that you don't learn until linear algebra 2, which is typically like a proof-based course ma mostly taken by math majors, right? And in especially in the later half of the book, you're introduced to even even earlier on though, like even in chapter 3, you're introduced to certain concepts that you won't really explore deeply until you read about other subjects like abstract algebra or even like functional analysis. So it's sort of like this nice continuity between different subjects of math where you see certain concepts introduced that then are reused later on down the line but maybe in a different context right or like like even in linear algebra done right there are certain exercises that tell you to treat the what was it was telling you to treat the norm of the difference between two linear maps as a metric in a metric space and to prove that it satisfies the requirements of a metric like the triangle inequality the D of P and Q equals D of Q and P etc. Right? And it's uh you know it it it's cool because I was first learning about like uh basic analysis at the time and it just seems like you know the more you understand about these foundational subjects the more like they you you see them cross over time and time again you know in like interesting ways. you know, linear algebra done right isn't like the hardest textbook in the world. But if you're not super strong and super familiar with proofs, like if you maybe took discrete math, but you forgot about all of it before, I think linear algebra done right will like really strengthen your sort of ability to read and write proofs. But if you want to learn uh analysis and you're not really comfortable with writing proofs yet, then you could probably start out with something like understanding analysis by Steven Abbott, which uh is like a really really popular introductory real analysis textbook. I mean, analysis is just really good to study because it's such like a foundational stepping stone in like higher math where it's probably going to be your first exposure to stuff like point set topology and you know your reintroduction to certain concepts in calculus but this time with much more rigorous definitions of what makes something continuous or you know what a limit is what makes a set countable or or finite or etc. Right? One thing I'll add before the end of the video is using AI when you're studying math, which is something that most people do not recommend because you're probably not going to learn much from it. I mean, using a chatbot to summarize a certain concept or like, you know, maybe give like some basic reasoning for why something might not be true if you're just like like treating it basically like a teacher when you're asking questions in a classroom or something.

[00:16:54]Sure. But like often times AI is going to be self-contradictory, right? You might ask it a question and it'll say, "No, that's false." Then it'll go through the logical steps of why it's true. Then at the end, it'll conclude, "Yes, it's true." And something like ridiculous like that. It's not really something you could inherently trust unless it's like pulling from the internet like known solutions to to known problems in like textbooks and stuff. So maybe you can use it to verify solutions to problems. Do not use it to like do exercises for you and then just read them as if they were like, you know, cuz you know, doing the actual exercises yourself is the entire point, right? That's how you actually build mathematical maturity. It's how you actually strengthen your mind to be able to understand math better. So, it's like, you know, don't offload don't offload your your thinking to uh to anthropic or [ __ ] Open AI or Google or whatever the [ __ ] you know?

[00:17:41]Especially don't use it as like a replacement for a textbook. That's just completely [ __ ] bro. Like, why?

[00:17:46]I've seen people do that and it's like, dude, oh, holy [ __ ] We're we're [ __ ] as a species, bro. But, um I mean, that's basically all I wanted to say on the topic. So, I guess I'll just leave it at that.