I do appreciate the growing trend of presenting material in a more down-to-earth way, maybe with less-formal language and showing the reader it's not as scary as they might've thought previously. Kudos to the author, this is cool.
It seems great on the surface. More people might read texts about mathematics, but if the trend were taken past some threshold, then there might be a global consequence as well.
Obviously this is just speculation. Another possability is it will simply result in a change in the personality type of those ammeture mathemeticians whom make contributions to science. I suspect there will be some sort of net effect, but it might not be what we expect.
Is there anyone here that was inspired by layman articles on mathematical theory and later went on to learn rigorous formal notation?
In a world with solid coverage of mathematical theory presented informally to appeal to a larger laymen audience, will the world produce more amateur mathematicians that make great contributions to science or less? Will would-be amateur mathematicians be less motivated to learn formal notation given that they can more easily read and understand the same material presented in an informal manner? If they did pick up a base level of formal notation, would they be less motivated to learn how to understand or write rigorous proofs using formal notation for the same reason?
I don't know what the answer is. Maybe there are studies out there that attempt to answer these questions.
The number of potentially great mathematicians who stop learning because they're satisfied with the informal coverage has to be absolutely dwarfed by the number of potentially great mathematicians who are scared away from mathematics entirely at an early age by excessive formality.
I've Seen it time and again: Great mathematicians who are awful mathematics students get turned away. They end up making great full stack engineers.
I am not worried about using "informality" to get more people studying mathematics.
The book is informal, but by the end of the book, the integral that gets presented is the correct definition of the integral. I've just collapsed as much of the technical language at possible and focused on the core idea. My thinking is: if someone is hooked, sure they'll run up against walls if they try to use my book and only my book, but that would be the time to turn to Stewart (famous Calc text) or comparable. My thinking it that at that point the student is ready for "rigor" and "formality", and they won't even think twice about. They might even appreciate it. I've seen it happen over a decade of calculus teaching. It happens more than you think.
But to take this a little further, I believe the "formality" you mention actually hides a fundamental and insidious truth about mathematics: Mathematics fundamentally is informal. Burrow down deep enough into the epsilon/delta of limit definitions, and you'll see at the bottom is what amounts to an informal "this is good enough I guess".
For instance, at the bottom of epsilon/delta definition of what it means to converge in Baby Rudin (pg. 46), he essentially says "if you can get sequence within epsilon of the target anywhere past N" that's good enough. But why?! There is no more unpacking or additional fundamentalism at that point. How can we be sure we can make a claim about an infinite set of inequalities? Do if/then statements work this way? How can we be sure we can use the natural numbers this way? That fundamental informality then persists throughout the text. It's fine of course, and this is the agreed upon way to do mathematical calculus, but it's also a fundamental informality.
From my point of view (and this is part of what got me writing this book in the first place): why bother going all the way "down there" just to say "good enough"? Why not say "good enough" a lot higher up the ladder closer to where the problem originated.
I'm hardly the final arbiter on this matter. But that's my opinion.
> why bother going all the way "down there" just to say "good enough"? Why not say "good enough" a lot higher up the ladder closer to where the problem originated.
That is exactly what motivated me to pose the question. Will would-be amateur mathematicians, that might have been great, find themselves lacking the necessary motivation to pursue a deeper understanding of the material? Will they stop climbing the ladder and just say -- good enough?
I don't see a claim about an "infinite set of inequalities" - I see an infinite set of I equalities that must be satisfied.
> Do if/then statements work this way? How can we be sure we can use the natural numbers this way?
Could you be more specific?
I feel like what you call "fundamental informality" I might call "assumption of mathematical maturity."
I know it seems formal because it adheres to a certain structure, but even this is informal at a fundamental level.
Specifically, how can we be sure we can perform a countably infinite number of distance measurements in the metric space to be sure the sequence stays close to p? (this is the infinite stack of inequalities I alluded to)
He doesn't say. Implicitly, Rudin is saying here that this definition of "converges" is good enough. And he's not wrong. It is a very good definition. To me at least this is Rudin, the towering statue of formality, being informal.
He could/should have actually gone down to a more fundamental level and whipped out mathematical induction as an axiom to assure us that we can do such things, but then that would have taken him off his narrative goal, and also probably lost even more readers. Furthermore, even if he did so, an axiom is an assertion that "you just have to trust me on this one."
Now look, I'm not bashing formality. I'm a huge fan of it, and teach upper level math classes formally. But it has it's place and it is NOT in Calculus 1. Furthermore, I think folks need to realize that even the most formal of treatises have informalities buried in them at the very least in the form of stated axioms.