Epistemic status: Half-certain, half-skeptical. This mental model feels a bit too compressed to me to be maximally useful.
I’ve spent a surprising (=non-zero) number of hours studying mathematics during my summer break, and noticed an interesting pattern in undergraduate textbooks.
With a certain amount of remarks, exercises, and concrete examples thrown in and seasoned to taste, most traditional mathematics textbooks will start out with some rigorous definitions, then state some theorems that use those definitions, and finally prove those theorems. Then the process starts all over again—the “Definition → Theorem → Proof” pipeline, as it’s often called by pure mathematicians.*
I think this lens is very obvious when pointed out, but I’ve also found it useful to structure my thinking around my newfound hobby. Let’s see if it pays out.
First, how do we most effectively deal with each step of the core pipeline? Here are some things I’ve found useful, although YMMV:
Definitions are usually the easiest step, but it’s easy to get windshield-wiper eyes when reading a technical work (=reading, understanding it in the moment, but not retaining it once the book closes), so I recommend constructing simple examples of whatever is being defined. Of special value is trying to construct objects that break one and only one interesting parameter of the definition if it has multiple different conditions.
What counts as an interesting parameter is a matter of your own taste. For example, the set-theoretic definition of a topological space (X,T) had 4 important parameters for me to tweak before I felt like I had a handle on it.
Ditto for theorems. Usually, a good theorem shows me something at least a little surprising, so I’m more motivated to find an example (and to try to find a counter example) than definitions. More useful than that, however, is to take a minute and try to strategize about how you might prove it for a minute or two. Actually writing a proof out for a new theorem is likely more trouble than it’s worth, not least because your plan is often not going to pan out perfectly, but doing this gets your brain into the right gear for the next part of the pipeline.
Proofs are where you spend the bulk of your mental energy when doing higher mathematics. Proofs are hardcore applications of definitions and theorems in service of a goal, and the arguments can easily get involved and subtle, so expect them to take much more time to grok.
When the going gets tough in a proof, it is very easy to get windshield-wiper eyes and just gloss over the rest of the proof. One piece of advice I have found routinely helpful in combating this is to “[read] in multiple overlapping passes”, because with each new pass you can clear a new mental roadblock towards full understanding of the proof.
One big advantage I see with the DTP pipeline is that, since mathematics is remarkably conservative with its definitions and theorems, someone who’s read one book on subject X can often blaze* through significant portions of another book on subject X. The time cost on the duplicating part of the reading is lower than one would expect.
If you’ve already taken a course on Real Analysis, but you used the easier Understanding Analysis by Abbott instead of Principles of Mathematical Analysis by Rudin, for example, you are still probably going to find Rudin’s works much easier to go through because of that background. This, combined with a generally increased ability to read mathematics works thanks to practice, makes the experience more enjoyable, and makes you more likely to stick with it.
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*Axioms are kind of halfway between theorems and definitions. They also show up very infrequently, since they’re supposed to be foundational to everything, so I don’t include them explicitly.
*Relatively speaking. Blazing through a math textbook is still going to feel like a snail’s pace compared to blazing through the HPMOR archives. :)
Math: Textbooks and the DTP pipeline
Epistemic status: Half-certain, half-skeptical. This mental model feels a bit too compressed to me to be maximally useful.
I’ve spent a surprising (=non-zero) number of hours studying mathematics during my summer break, and noticed an interesting pattern in undergraduate textbooks.
With a certain amount of remarks, exercises, and concrete examples thrown in and seasoned to taste, most traditional mathematics textbooks will start out with some rigorous definitions, then state some theorems that use those definitions, and finally prove those theorems. Then the process starts all over again—the “Definition → Theorem → Proof” pipeline, as it’s often called by pure mathematicians.*
I think this lens is very obvious when pointed out, but I’ve also found it useful to structure my thinking around my newfound hobby. Let’s see if it pays out.
First, how do we most effectively deal with each step of the core pipeline? Here are some things I’ve found useful, although YMMV:
Definitions are usually the easiest step, but it’s easy to get windshield-wiper eyes when reading a technical work (=reading, understanding it in the moment, but not retaining it once the book closes), so I recommend constructing simple examples of whatever is being defined. Of special value is trying to construct objects that break one and only one interesting parameter of the definition if it has multiple different conditions.
What counts as an interesting parameter is a matter of your own taste. For example, the set-theoretic definition of a topological space (X,T) had 4 important parameters for me to tweak before I felt like I had a handle on it.
Ditto for theorems. Usually, a good theorem shows me something at least a little surprising, so I’m more motivated to find an example (and to try to find a counter example) than definitions. More useful than that, however, is to take a minute and try to strategize about how you might prove it for a minute or two. Actually writing a proof out for a new theorem is likely more trouble than it’s worth, not least because your plan is often not going to pan out perfectly, but doing this gets your brain into the right gear for the next part of the pipeline.
Proofs are where you spend the bulk of your mental energy when doing higher mathematics. Proofs are hardcore applications of definitions and theorems in service of a goal, and the arguments can easily get involved and subtle, so expect them to take much more time to grok.
When the going gets tough in a proof, it is very easy to get windshield-wiper eyes and just gloss over the rest of the proof. One piece of advice I have found routinely helpful in combating this is to “[read] in multiple overlapping passes”, because with each new pass you can clear a new mental roadblock towards full understanding of the proof.
One big advantage I see with the DTP pipeline is that, since mathematics is remarkably conservative with its definitions and theorems, someone who’s read one book on subject X can often blaze* through significant portions of another book on subject X. The time cost on the duplicating part of the reading is lower than one would expect.
If you’ve already taken a course on Real Analysis, but you used the easier Understanding Analysis by Abbott instead of Principles of Mathematical Analysis by Rudin, for example, you are still probably going to find Rudin’s works much easier to go through because of that background. This, combined with a generally increased ability to read mathematics works thanks to practice, makes the experience more enjoyable, and makes you more likely to stick with it.
----
*Axioms are kind of halfway between theorems and definitions. They also show up very infrequently, since they’re supposed to be foundational to everything, so I don’t include them explicitly.
*Relatively speaking. Blazing through a math textbook is still going to feel like a snail’s pace compared to blazing through the HPMOR archives. :)