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Author of book here. Nice comment, and very interesting question. I am not worried about using "informality" to get more people studying mathematics. The book

Author of book here. Nice comment, and very interesting question. 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.

dustfinger 7y ago
To be fair, I have not read the article. I am to busy with work today, but I did bookmark it for later. Also, thank you for not being insulted by my question. It wasn't meant as an insult, I was just pondering what the net effect might be if in formalism became dominant: > 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?
rrss 7y ago
My pages apparently don't align with yours (I see page 46 has only a single exercise), but I don't see where Rudin says anything is "good enough." He states the definition of convergence, meaning that if a sequence satisfies the property then we choose to call it convergent. There is no question of 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."
- gregpetrics 7y ago
  Whoops. Page 47. I paraphrase: Sequences "converge" if there exists an N such that the sequence stays within epsilon of p (the mark) for all indexes larger than N. 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.
  - lonelappde 7y ago
    Most US schools teach formality in Geometry with 2-column mechanical proofs, following Euclid. Formal and informal belong side by side throughout the curriculum.

dustfinger 7y ago

To be fair, I have not read the article. I am to busy with work today, but I did bookmark it for later. Also, thank you for not being insulted by my question. It wasn't meant as an insult, I was just pondering what the net effect might be if in formalism became dominant: > 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?

rrss 7y ago

My pages apparently don't align with yours (I see page 46 has only a single exercise), but I don't see where Rudin says anything is "good enough." He states the definition of convergence, meaning that if a sequence satisfies the property then we choose to call it convergent. There is no question of 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."

gregpetrics 7y ago
Whoops. Page 47. I paraphrase: Sequences "converge" if there exists an N such that the sequence stays within epsilon of p (the mark) for all indexes larger than N. 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.
- lonelappde 7y ago
  Most US schools teach formality in Geometry with 2-column mechanical proofs, following Euclid. Formal and informal belong side by side throughout the curriculum.