One of the highly generous mentors who dragged me kicking and screaming into the world of even making an attempt told me: “There are no bad math students. There are only bad math teachers who themselves had bad math teachers.”
Agreed with the above. Almost everyone can probably expand their mathematical thinking abilities with deliberate practice.
> But I do not think this is innate, even though it often manifests in early childhood. Genius is not an essence. It’s a state. It’s a state that you build by doing a certain job.
Though his opinion on mathematical geniuses above, I somewhat disagree with. IMO everyone has a ceiling when it comes to math.
Leibniz made that claim centuries ago in his critical remarks on John Locke's Essay on Human Understanding. Leibniz specifically said that Locke's lack of mathematical knowledge led him to (per Leibniz) his philosophical errors regarding the nature of 'substance'.
https://www.earlymoderntexts.com/assets/pdfs/leibniz1705book...
Whether that youthful immersion in math in fact benefitted me in later life and whether that kind of thinking is actually desirable for everyone as he seems to suggest—I don't know. But it is a thought-provoking interview.
At the very minimum, I ask people to always think of the distribution of whatever figure they are given.
Just that is far more than so many are willing to do.
Yes, but it's higher than you think: https://www.justinmath.com/your-mathematical-potential-has-a...
As info explodes and specialists dive deeper into their niches, info asymmetry between ppl increases. There are thousands of specialists running in different directions at different speeds. Leaders can't keep up.
Their job is to try to get all these "vectors" aligned toward common goals, prevent fragmentation and division.
And while most specialists think this "sync" process happens through "education" and getting everyone to understand a complex ever changing universe, the truth is large diverse groups are kept in sync via status signalling, carrot/stick etc. This is why leaders will pay attention when you talk in terms of what increases clout/status/wealth/security/followers etc. Cause thats their biggest tool to prevent schisms and collapse.
Any healthy/able individual could learn to deadlift twice their bodyweight with sufficient training, but the vast majority of people never reach this basic fitness milestone, because they don't put any time into achieving it. There's a very large gap between what people are capable of theoretically and what they achieve in practice.
I've been working a lot on my math skills lately (as an adult). A mindset I've had in the past is that "if it's hard, then that means you've hit your ceiling and you're wasting your time." But really, the opposite is true. If it's easy, then it means you already know this material, and you're wasting your time.
Run into too many people like that, who I daresay are common in the field, and it's easy to see how people become dispirited and give up.
He was right on the nature of the universe, he was wrong on making a better world. I for one forgive him on the basis of time served.
I’ve tried a bunch of courses (MIT linalg, Coursera ICL Maths for ML, Khan etc etc) but what I eventually realised is my foundations were so, so weak being mid 30s and having essentially stopped learning in HS (apart from a business stats paper at Uni).
Enter a post on reddit about Mathacademy (https://www.mathacademy.com/). It’s truly incredible. I’m doing around 60-90 minutes a day and properly understanding and developing an intuition for things. They’ve got 3 pre-uni courses and I’ve now nearly finished the first one. It’s truly a revelation to be able to intuit and solve even simple problems and, having skipped ahead so far in my previous study, see fuzzy links to what’s coming.
Cannot recommend it enough. I’m serious about enrolling in a Dip Grad once I’ve finished the Uni level stuff. Maybe even into an MA eventually.
I strongly believe that the average human being can be exceptional in any niche topic given enough time, dedication and focus.
The author of the book has picked out mathematics because that was what he was interested in. The reality is that this rule applies to everything.
The belief that some people have an innate skill that they are born with is deeply unhelpful. Whilst some people (mostly spectrum) do seem have an innate talent, I would argue that it is more an inbuilt ability to hyper focus on a topic, whether that topic be mathematics, Star Trek, dinosaurs or legacy console games from the 1980’s.
I think we do our children a disservice by convincing them that some of their peers are just “born with it”, because it discourages them from continuing to try.
What we should be teaching children is HOW to learn. At the moment it’s a by-product of learning about some topic. If we look at the old adage “feed a man a fish”, the same is true of learning.
“Teach someone mathematics and they will learn mathematics. Teach someone to learn and they will learn anything”.
I've long thought that almost all have the capability to learn roughly high school level math, though it will take more effort for some than for others. And a key factor to keep up a sustained effort is motivation. A lot of people who end up hating math or think they're terrible at it just haven't had the right motivation. Once they do, and they feel things start to make sense and they're able to solve problems, things get a lot easier.
Personally I also feel that learning math, especially a bit higher-level stuff where you go into derivations and low-level proofs, has helped me a lot in many non-math areas. It changed the way I thought about other stuff, to the better.
Though, helping my family members and friends taught me that different people might need quite different approaches to start to understand new material. Some have an easier time approaching things from a geometrical or graph perspective, others really thrive on digging into the formulas early on etc. One size does not fit all.
One positive effect of having studied pure mathematics when young might have been that I became comfortable with thinking in multiple layers of abstraction. In topology and analysis, for example, you have points, then you have sets of points, then you have properties of those sets of points (openness, compactness, discreteness, etc.), then you have functions defining the relations among those sets of points and their properties, then you have sets of functions and the properties of those sets, etc.
I never used mathematical abstraction hierarchies directly in my later life, but having thought in those terms when young might have helped me get my head around multilayered issues in other fields, like the humanities and social sciences.
But a possible negative effect of spending too much time thinking about mathematics when young was overexposure to issues with a limited set of truth values. In mainstream mathematics, if my understanding is correct, every well-formed statement is either true or false (or undecided or undecidable). Spending too much time focusing on true/false dichotomies in my youth might have made it harder for me to get used to the fuzziness of other human endeavors later. I think I eventually did, though.
I met the guitar guy a few years later outside his house. He always had just one guitar but now owned something like 20, something like a hundred books about music. Quite the composer. It looked and sounded highly sophisticated. The dumb guy didn't exist anymore.
I was initially celebrated for the mathematical talent.
But as life progressed, I my family started seeing me as an academic loser.
Basically, no girls would be interested in me because "mathemetical talent" doesn't help you with that.
And i seen handsome men had more respect from society than spending countless time on math.
So, i later gave up because my family kept pressuring me to attain real success, girls, money and car and i became a programmer.
Funny enough, I was still a loser in societal view doesn't matter I started clearly half a million a year.
So most people don't try hard at math because math is not rewarding, for most people.
It's much better to build physique, music talent, comedic talent, this helps you get girls and respect from peers.
I would argue something different. The "skill" angle is just thinly veiled ladder-pulling.
Sure, math is hard work, and there's a degree of prerequisites that need to be met to have things click, but to the mindset embodied by the cliche "X is left as an exercise for the reader" is just people rejoicing on the idea they can needlessly make life hard for the reader for no reason at all.
Everyone is familiar with the "Ivory tower" cliche, but what is not immediately obvious is how the tower aspect originates as a self-promotion and self-defense mechanism to sell the idea their particular role is critical and everyone who wishes to know something is obligated to go through them to reach their goals. This mindset trickles down from the top towards lower levels. And that's what ultimately makes math hard.
Case in point: linear algebra. The bulk of the material on the topic has been around for many decades, and the bulk of the course material,l used to teach that stuff, from beginner to advanced levels, is extraordinarily cryptic and mostly indecipherable. But then machine learning field started to take off and suddenly we started to see content addressing even advanced topics like dimensionality reduction using all kinds of subspace decomposition methods as someting clear and trivial. What changed? Only the type of people covering the topic.
Plato's Meno has Socrates showing that even a slave can reason mathematically.
It's not really math alone but modeling more generally that activates people's reasoning. Math and logic are just those models that are continuous+topological and discrete+logic-operation variants, both based in dimension/orthogonality. But all modeling is over attribution - facts, opinions, etc., and there's a lot of modeling with a healthy dose of salience - heuristics, emotions, practice, etc. Math by design is salience-free (though it incorporates goals and weights), so it's the perspective and practice that liberates people from bias and assumptions. In that respect it can be beautiful, and makes other more conditioned reasoning seem tainted (but it has to work harder to be relevant).
However, experts can project mathematical models onto reality. Hogwash about quantum observer effects and effervescent quantum fields stem from projecting the assumptions required to do the math (or adopt the simplifying forms). Yes, the model is great at predictions. No, it doesn't say what else is possible, or even what we're seeing (throwing baseballs at the barn, horses run out, so barns are made of horses...). Something similar happens with AI math: it can generate neat output, so it must be intelligent. The impulse is so strong that adherents declare that non-symbolic thinking is not thinking at all, and discount anything unquantifiable (in discourse at least). Assuming what you're trying to prove is rarely helpful, but very easy to do accidentally when tracking structured thinking.
As anti-intellectual as it sounds, you could imagine someone asking, is it worth devoting years of your life to study this subject which becomes increasingly esoteric and not obviously of specific benefit the further you go, at least prima facie? Many people wind up advocating for mathematics via aesthetics, saying: well it’s very beautiful out there in the weeds, you just have to spend dozens of years studying to see the view. That marketing pitch has never been the most persuasive for me.