Could you say more about why you think real analysis specifically is good for this kind of general skill? I have pretty serious doubts that analysis is the right way to go, and I’d (wildly) guess that there would be significant benefits from teaching/learning discrete mathematics in place of calculus. Combinatorics, probability, algorithms; even logic, topology, and algebra.
To my mind all of these things are better suited for learning the power of proof and the mathematical way of analyzing problems. I’m not totally sure why, but I think a big part of it is that analysis has a pretty complicated technical foundation that already implicitly uses topology and/or logic (to define limits and stuff), even though you can sort of squint and usually kind of get away with using your intuitive notion of the continuum. With, say, combinatorics or algorithms, everything is very close to intuitive concepts like finite collections of physical objects; I think this makes it all the more educational when a surprising result is proven, because there is less room for a beginner to wonder whether the result is an artifact of the funny formalish stuff.
Personally I think real analysis is an awkward way to learn mathematical proofs, and I agree discrete mathematics or elementary number theory is much better. I recommend picking up an Olympiad book for younger kids, like “Mathematical Circles, A Russian Experience.”
This is also somewhat in reply to your elaboration in this comment. Just some data points:
In regards to this topic of proof, and more generally to the topic of formal science, I have found logic a very useful subject. For one, you can leverage your verbal reasoning ability, and begin by conceiving of it as a symbolization of natural language, which I find for myself and many others is far more convenient than, say, a formal science that requires more spatial reasoning or abstract pattern recognition. Later, the point that formal languages are languages in their own right is driven home, and you can do away with this conceptual bridge.
Logic also has helped me to conceive of formal problems as a continuum of difficulty of proof, rather than proofs and non-proofs. That is, when you read a math textbook, sometimes you are instructed to Solve, sometimes to Evaluate, sometimes to Graph; and then there is the dreaded Show That X or Prove That X! In a logic textbook, almost all exercises require a proof of validity, and you move up over time, deriving new inference rules from old, and moving onto metalogical theorems. Later returning to books about mathematical proof, I found things much less intimidating. I found that proof is not a realm forbidden to those lacking an innate ability to prove; you must work your way upwards as in all things.
Furthermore, in regards to this:
Even the math with simple foundations has surprising results with complicated proofs that require precise understanding.
In my opinion, very significant and complex results in logic are arrived at quite early in comparison to the significance of, and effort invested in, results in other fields of formal science.
And in regards to this:
I think this makes it all the more educational when a surprising result is proven, because there is less room for a beginner to wonder whether the result is an artifact of the funny formalish stuff.
I have found that in continuous mathematics I have walked away from proofs with a feeling best expressed as, “If you say so,” as opposed to discrete mathematics and logic, where it’s more like, “Why, of course!”
To my mind all of these things are better suited for learning the power of proof and the mathematical way of analyzing problems.
I think the main thrust of the article was less about the power of mathematics and more about the the habits of close reading and careful attention to detail required to do rigorous mathematics.
I’m not totally sure why, but I think a big part of it is that analysis has a pretty complicated technical foundation that already implicitly uses topology and/or logic (to define limits and stuff), even though you can sort of squint and usually kind of get away with using your intuitive notion of the continuum.
Seems like it’s precisely because of the complicated technical foundation that real analysis was recommended. Theorems have to be read carefully, as even simple ones often have lots of hypotheses. Proofs have to be worked through carefully to make sure that no implicit assumptions are being introduced. Even great mathematicians ran into trouble playing fast and loose with the real numbers. It took them about two hundred years to finally lay rigorous foundations for calculus.
Seems like it’s precisely because of the complicated technical foundation that real analysis was recommended.
What I’m saying is, that’s not a good reason. Even the math with simple foundations has surprising results with complicated proofs that require precise understanding. It’s hard enough as it is, and I am claiming that analysis is too much of a filter. It would be better to start with the most conceptually minimal mathematics.
Even great mathematicians ran into trouble playing fast and loose with the real numbers. It took them about two hundred years to finally lay rigorous foundations for calculus.
...implying that it is actually pretty confusing. There are good reasons for wanting to learn analysis because it is applied so widely. But from the specific perspective of trying to learn lessons about math and rigorous argument in general, it seems like you want a subject that is legitimate math but otherwise as simple as possible. To some extent, trying to do real analysis as a first real math class is like trying to teach physics class in a foreign language. On the one hand, you just want to learn the physics, but at the same time you always have to translate into your native tongue, worrying that you made a subtle mistake in translation. If you want to learn how to prove stuff in general, you don’t also want the objects that you’re proving stuff about to be overcomplicated to the point that it’s a whole chore just to understand what you’re talking about. That is an important but distinct skill from understanding and inventing proofs.
Oh, sure, in expressing agreement with Epictetus I was just saying that I don’t think that you get the full benefits that I was describing from basic discrete math. I agree that some students will find discrete math a better introduction to mathematical proof.
Ok that makes sense. I’m still curious about any specific benefits that you think studying analysis has, relative to other similarly deep areas of math, or whether you meant hard math in general.
I think that analysis is actually the easiest entry point to the kind of mathematical reasoning that I have in mind for people who have learned calculus. Most of the theorems are at least somewhat familiar, so one can focus on the logical rigor without having to simultaneously having to worry about understanding what the high level facts are.
I see. That could be right. I guess I’m thinking about this (this = what to teach/learn and in what order) from the perspective of assuming I get to dictate the whole curriculum. In which case analysis doesn’t look that great, to me.
Could you say more about why you think real analysis specifically is good for this kind of general skill? I have pretty serious doubts that analysis is the right way to go, and I’d (wildly) guess that there would be significant benefits from teaching/learning discrete mathematics in place of calculus. Combinatorics, probability, algorithms; even logic, topology, and algebra.
To my mind all of these things are better suited for learning the power of proof and the mathematical way of analyzing problems. I’m not totally sure why, but I think a big part of it is that analysis has a pretty complicated technical foundation that already implicitly uses topology and/or logic (to define limits and stuff), even though you can sort of squint and usually kind of get away with using your intuitive notion of the continuum. With, say, combinatorics or algorithms, everything is very close to intuitive concepts like finite collections of physical objects; I think this makes it all the more educational when a surprising result is proven, because there is less room for a beginner to wonder whether the result is an artifact of the funny formalish stuff.
Personally I think real analysis is an awkward way to learn mathematical proofs, and I agree discrete mathematics or elementary number theory is much better. I recommend picking up an Olympiad book for younger kids, like “Mathematical Circles, A Russian Experience.”
This is also somewhat in reply to your elaboration in this comment. Just some data points:
In regards to this topic of proof, and more generally to the topic of formal science, I have found logic a very useful subject. For one, you can leverage your verbal reasoning ability, and begin by conceiving of it as a symbolization of natural language, which I find for myself and many others is far more convenient than, say, a formal science that requires more spatial reasoning or abstract pattern recognition. Later, the point that formal languages are languages in their own right is driven home, and you can do away with this conceptual bridge.
Logic also has helped me to conceive of formal problems as a continuum of difficulty of proof, rather than proofs and non-proofs. That is, when you read a math textbook, sometimes you are instructed to Solve, sometimes to Evaluate, sometimes to Graph; and then there is the dreaded Show That X or Prove That X! In a logic textbook, almost all exercises require a proof of validity, and you move up over time, deriving new inference rules from old, and moving onto metalogical theorems. Later returning to books about mathematical proof, I found things much less intimidating. I found that proof is not a realm forbidden to those lacking an innate ability to prove; you must work your way upwards as in all things.
Furthermore, in regards to this:
In my opinion, very significant and complex results in logic are arrived at quite early in comparison to the significance of, and effort invested in, results in other fields of formal science.
And in regards to this:
I have found that in continuous mathematics I have walked away from proofs with a feeling best expressed as, “If you say so,” as opposed to discrete mathematics and logic, where it’s more like, “Why, of course!”
I agree with Epictetus’ comment.
(See reply there.)
I think the main thrust of the article was less about the power of mathematics and more about the the habits of close reading and careful attention to detail required to do rigorous mathematics.
Seems like it’s precisely because of the complicated technical foundation that real analysis was recommended. Theorems have to be read carefully, as even simple ones often have lots of hypotheses. Proofs have to be worked through carefully to make sure that no implicit assumptions are being introduced. Even great mathematicians ran into trouble playing fast and loose with the real numbers. It took them about two hundred years to finally lay rigorous foundations for calculus.
What I’m saying is, that’s not a good reason. Even the math with simple foundations has surprising results with complicated proofs that require precise understanding. It’s hard enough as it is, and I am claiming that analysis is too much of a filter. It would be better to start with the most conceptually minimal mathematics.
...implying that it is actually pretty confusing. There are good reasons for wanting to learn analysis because it is applied so widely. But from the specific perspective of trying to learn lessons about math and rigorous argument in general, it seems like you want a subject that is legitimate math but otherwise as simple as possible. To some extent, trying to do real analysis as a first real math class is like trying to teach physics class in a foreign language. On the one hand, you just want to learn the physics, but at the same time you always have to translate into your native tongue, worrying that you made a subtle mistake in translation. If you want to learn how to prove stuff in general, you don’t also want the objects that you’re proving stuff about to be overcomplicated to the point that it’s a whole chore just to understand what you’re talking about. That is an important but distinct skill from understanding and inventing proofs.
Oh, sure, in expressing agreement with Epictetus I was just saying that I don’t think that you get the full benefits that I was describing from basic discrete math. I agree that some students will find discrete math a better introduction to mathematical proof.
Ok that makes sense. I’m still curious about any specific benefits that you think studying analysis has, relative to other similarly deep areas of math, or whether you meant hard math in general.
I think that analysis is actually the easiest entry point to the kind of mathematical reasoning that I have in mind for people who have learned calculus. Most of the theorems are at least somewhat familiar, so one can focus on the logical rigor without having to simultaneously having to worry about understanding what the high level facts are.
I see. That could be right. I guess I’m thinking about this (this = what to teach/learn and in what order) from the perspective of assuming I get to dictate the whole curriculum. In which case analysis doesn’t look that great, to me.