7 ms·

One possible consequence of a trend towards less formal ways of presenting mathematics is that the ease of entry to the informal might not motivate beginners to

One possible consequence of a trend towards less formal ways of presenting mathematics is that the ease of entry to the informal might not motivate beginners to learn the formal. Consider that amateur mathematicians from the past, particularly those whom made great contributions to science, might not have been motivated to teach themselves the rigorous formal notation if everything they read was explained in layman's terms. Similarly, those that would not be motivated to teach themselves the formal notation may read plenty of laymen explanations about mathematical theory, but also never be motivated to learn the formal notation. Consequently, a trend towards "presenting material in a more down-to-earth way" may might lead to a global average decline of amateur mathematicians with knowledge of formal notation. 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?

oneepic 7y ago
This comment keeps saying "[rigorous] formal notation" -- are you implying that new mathematicians might lack rigorous mathematical technique, or just that they won't be motivated to communicate their results in a formal way that other mathematicians will generally understand?
- dustfinger 7y ago
  My intention was to pose the following questions for arguments sake: 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.
  - ksdale 7y ago
    It seems like there could hardly be fewer amateur mathematicians making great contributions to science. 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.
    - gregpetrics 7y ago
      This. 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.
deleted 7y ago
[deleted]
gregpetrics 7y ago
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?

oneepic 7y ago

This comment keeps saying "[rigorous] formal notation" -- are you implying that new mathematicians might lack rigorous mathematical technique, or just that they won't be motivated to communicate their results in a formal way that other mathematicians will generally understand?

dustfinger 7y ago
My intention was to pose the following questions for arguments sake: 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.
- ksdale 7y ago
  It seems like there could hardly be fewer amateur mathematicians making great contributions to science. 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.
  - gregpetrics 7y ago
    This. 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.

deleted 7y ago

[deleted]

gregpetrics 7y ago

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?