The main reason this is hard, is that learning to write proofs that are both rigorous and human-readable is extremely difficult without a teacher+grader.
This process is not taught in high school, most or all textbooks implicitly assume that you know how to do it, and if you don’t know how to do it then it isn’t obvious what’s wrong until you realize that you can’t do the exercises anymore.
Math is largely proof-writing. Proofs are an interactive process between writer and reader. Without a reader there is no feedback loop and you don’t learn to write understandable proofs. If you can’t explain math than you don’t truly understand it.
Once you know how to write proofs, it becomes possible to learn more through books. But you really need a class setting for that first part.
I've thought of this before, and it makes a lot of sense! Sometimes one can write a seemingly "correct" proof, but there may be gaps in the argument that isn't obvious until someone else looks at it. Heck, even professional mathematicians get it wrong sometimes.
Another thought I've had to help solve this issue is to supplement learning mathematics with formal methods. Using something like Lean, one may make a mathematical argument that is truly airtight and the student may feel at ease knowing their understanding of a proof is complete. This could be the feedback loop that you mentioned.
why is that the case? in the many years since maths has existed - have educators...or just...people in general not thought that folks may want to learn this for themselves? it boggles my mind how much of this stuff is locked away and gate kept.
Learning to write proofs is really learning a new kind of thinking, and that is not easy to do alone. For sure you can learn from books or other resources alone, but as with everything, when you are stuck in some way, good guidance can be very beneficial. I’ve thought proofs and logic at multiple levels and even though people struggle with similar issues, there is almost never a one-size-fits-all way to help an individual progress. It all depends on your context, way of learning, temperament etc etc.
It isn’t that it’s locked away. It’s just that it’s much, much easier to learn some skills with a teacher than on your own.
Imagine, e.g. trying to learn to play piano without a teacher. You could technically maybe do it but it would be 100x harder and you’d end up with a bunch of bad habits. But once you have the basics down you can learn new music on your own.
Proof-based mathematics is similar. There’s just a lot of technical, non-obvious stuff that you have to learn before you go off on your own.
Comments
The main reason this is hard, is that learning to write proofs that are both rigorous and human-readable is extremely difficult without a teacher+grader.
This process is not taught in high school, most or all textbooks implicitly assume that you know how to do it, and if you don’t know how to do it then it isn’t obvious what’s wrong until you realize that you can’t do the exercises anymore.
Math is largely proof-writing. Proofs are an interactive process between writer and reader. Without a reader there is no feedback loop and you don’t learn to write understandable proofs. If you can’t explain math than you don’t truly understand it.
Once you know how to write proofs, it becomes possible to learn more through books. But you really need a class setting for that first part.
I've thought of this before, and it makes a lot of sense! Sometimes one can write a seemingly "correct" proof, but there may be gaps in the argument that isn't obvious until someone else looks at it. Heck, even professional mathematicians get it wrong sometimes.
Another thought I've had to help solve this issue is to supplement learning mathematics with formal methods. Using something like Lean, one may make a mathematical argument that is truly airtight and the student may feel at ease knowing their understanding of a proof is complete. This could be the feedback loop that you mentioned.
why is that the case? in the many years since maths has existed - have educators...or just...people in general not thought that folks may want to learn this for themselves? it boggles my mind how much of this stuff is locked away and gate kept.
Learning to write proofs is really learning a new kind of thinking, and that is not easy to do alone. For sure you can learn from books or other resources alone, but as with everything, when you are stuck in some way, good guidance can be very beneficial. I’ve thought proofs and logic at multiple levels and even though people struggle with similar issues, there is almost never a one-size-fits-all way to help an individual progress. It all depends on your context, way of learning, temperament etc etc.
It isn’t that it’s locked away. It’s just that it’s much, much easier to learn some skills with a teacher than on your own.
Imagine, e.g. trying to learn to play piano without a teacher. You could technically maybe do it but it would be 100x harder and you’d end up with a bunch of bad habits. But once you have the basics down you can learn new music on your own.
Proof-based mathematics is similar. There’s just a lot of technical, non-obvious stuff that you have to learn before you go off on your own.