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The Mathematics Autodidact’s Aid (2005) [pdf]

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Re: The Mathematics Autodidact’s Aid (2005) [pdf]

#21

Post by ExternalCenter »

imakecomments wrote: post #16

I'd like to see a list like this that included the field of mathematical logic. For whatever reason mathematical logic no longer seems to be a "popular" area of research, despite its deep connection to theoretical computer science. But there are distinction in study, as computer scientist tend not to go deeply into computability theory like a traditional mathematician would.

I think I understand why. Mathematical logic research has largely become focused on problems that, while important, seem very arcane from an outsiders perspective (even by the standards of other fields of math). The fundamental results of classical recursion/computability theory can be developed and presented with a fairly small amount of logic (see something like Cutland or Cooper). I'm not really super informed on the current state of the field though, so I may be misjudging the situation.

Re: The Mathematics Autodidact’s Aid (2005) [pdf]

#22

Post by plinkplonk »

imakecomments wrote: post #16

I'd like to see a list like this that included the field of mathematical logic. For whatever reason mathematical logic no longer seems to be a "popular" area of research, despite its deep connection to theoretical computer science. But there are distinction in study, as computer scientist tend not to go deeply into computability theory like a traditional mathematician would.

Perhaps not quite what you are looking for, but I've found the book reviews etc on this site very useful

http://www.logicmatters.net/tyl/

Re: The Mathematics Autodidact’s Aid (2005) [pdf]

#23

Post by rguzman »

cle wrote: post #10
lunchladydoris wrote: post #9

I enjoy a mathematics textbook as much as the next person, but what annoys me is the lack of solutions to the problems in so many of the books I've skimmed outside of classes.

I prefer solutions to some, but not all problems. Being able to independently and confidently verify your own solutions is an important skill to develop.

having the solutions does not stop you from developing that skill. however, the lack of solutions makes it difficult to make progress when you have time constraints, which is the common case.

Re: The Mathematics Autodidact’s Aid (2005) [pdf]

#24

Post by openasocket »

MichaelMoser123 wrote: post #5

maybe i am just stupid, but I find that many texts in mathematics like to skip over details, or I get stuck because there is an ambiguity in the text and there is no one to ask about it.. What do we do in this kind of situation?

Others have already given some good answers, but I'll add that a part of it is that this sort of thing is to be expected, especially when dealing with proofs. All proofs have at least some ambiguity because whether or not to leave certain parts out is subjective. Don't expect to just get a nontrivial proof on the first read. You have to convince yourself that the proof is valid. It's easy to think of this as a waste of time, because the author could have just been more clear, but to a certain extent it's a good thing. It forces you to understand the context around this proof, and ultimately is more illuminating than just reading and memorizing the proof.

Re: The Mathematics Autodidact’s Aid (2005) [pdf]

#25

Post by mathgenius »

lunchladydoris wrote: post #9

I enjoy a mathematics textbook as much as the next person, but what annoys me is the lack of solutions to the problems in so many of the books I've skimmed outside of classes.

Sometimes the problem is more important than the solution.

Re: The Mathematics Autodidact’s Aid (2005) [pdf]

#26

Post by gms7777 »

cle wrote: post #10
lunchladydoris wrote: post #9

I enjoy a mathematics textbook as much as the next person, but what annoys me is the lack of solutions to the problems in so many of the books I've skimmed outside of classes.

I prefer solutions to some, but not all problems. Being able to independently and confidently verify your own solutions is an important skill to develop.

This is absolutely a crucial skill. One of the most valuable exercises I had to do in college was during my introductory physics courses. Before solving any problem, we had to write what we expected the solution to look like, then after solving the problem, we had to write whether our solution seemed plausible. Did it fit our initial expectation? I not, could we explain the disparity? If we applied this value or equation to something else, would we get reasonable results (say, if the problem were estimating the gravitational force of the sun, what would this value give us for the length of the earth's year). Note that we'd get credit for this part even if our solution was wrong, as long as we recognized that it was in fact wrong.

It was a huuuge pain at the time and often took longer than the initial problem took to solve, but it did force me into the habit of critically evaluating my work, and its been one of the most valuable life skills I learned in college. It also helped develop my intuition, and significantly improved my teaching and presentation skills.

Re: The Mathematics Autodidact’s Aid (2005) [pdf]

#27

Post by lumberjack »

mathgenius wrote: post #25
lunchladydoris wrote: post #9

I enjoy a mathematics textbook as much as the next person, but what annoys me is the lack of solutions to the problems in so many of the books I've skimmed outside of classes.

Sometimes the problem is more important than the solution.

Yes, but if you're cramming for an exam it is so much more productive to just read through a fully worked out solution instead of trying to figure out how to do it yourself.

Re: The Mathematics Autodidact’s Aid (2005) [pdf]

#28

Post by mrcactu5 »

one day I woke up and realized the mathematics curriculum is almost completely arbitrary. there is no reason to teach algebra, geometry, trigonometry in that order

by the middle of graduate school everyone is self-teaching and you may know more than a professor from time to time about a given topic. And certainly about the basics since professors forget to do basic integrals at the board.

any good study group has to retain momentum and keep the discussion moving forward.

Re: The Mathematics Autodidact’s Aid (2005) [pdf]

#29

Post by fmap »

imakecomments wrote: post #16

I'd like to see a list like this that included the field of mathematical logic. For whatever reason mathematical logic no longer seems to be a "popular" area of research, despite its deep connection to theoretical computer science. But there are distinction in study, as computer scientist tend not to go deeply into computability theory like a traditional mathematician would.

What exactly are you trying to learn? Mathematical logic is a huge field in its own right, with plenty of topics that are of historical interest and a lot of active research areas.

If you want to learn modern mathematical logic you're in for a rough time, since you'll basically have to learn category theory in order to understand the few really excellent textbooks which exist (e.g. Sketches of An Elephant). If you are interested in type theory you should try reading the Homotopy Type Theory book, which is (mostly) self contained.

Re: The Mathematics Autodidact’s Aid (2005) [pdf]

#30

Post by xemdetia »

medymed wrote: post #8
MichaelMoser123 wrote: post #5

maybe i am just stupid, but I find that many texts in mathematics like to skip over details, or I get stuck because there is an ambiguity in the text and there is no one to ask about it.. What do we do in this kind of situation?

One approach is to look at a few different but overlapping primers on the same matter and do the exercises. If one treatment doesn't click, another might. And for well-known subjects, topics are often covered in roughly the same order. Maybe not for graduate classes, not sure, but linear algebra, set theory, group theory, analysis it's generally the case. Unless you can afford a tutor to teach it several different wa…

It is also important to look for primers from multiple decades because what sometimes can be confusing is the current state of the art, thesis papers (PhD students are struggling to learn it the first time too, but often they hold the complete supporting material with references), and the original paper for the field. A lot of those papers are struggling to explain the new concept so they draw more analogies and you can see where they are going compared to a worked example.
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