One frustration I find with mathematics is that it is rarely presented like other ideas. For example, few books seem to explain why something is being explained prior to the explanation. They don’t start with a problem, outline its solution provide the solution and then summarise this process at the end. They present one ‘interesting’ proof after another requiring a lot of faith and patience from the reader. Likewise they rarely include grounded examples within the proofs so that the underlying meaning of the terms can be maintained. It is as if the field is constructed so that it is in the form of puzzles rather than providing a sincere attempt to communicate idea as clearly as possible. Another analogy would be programming without the comments.
A book like Numerical Recipies, or possibly Jaynes book on probability, is the closest I’ve found so far. Has anyone encountered similar books?
I agree with your remarks here and share your frustration. While books of the type that you’re looking for are relatively uncommon; over the years I’ve amassed a list of ones that I’ve found very good. What subject(s) are you interested in learning? (N.B. There are large parts of math that I’m ignorant of—in particular I don’t know almost anything about applied math and so may not be able to say anything useful—I just thought I’d ask in case I can help.)
Thank you, my main goal at the moment is to get a handle on statistical learning approaches and probability. I hope to read Jaynes’s book and the nature of statistical learning theory once I have some time to devote to them. however I would love to find an overview of mathematics. Particularly one which focuses on practical applications or problems. One of the other posts mentioned the Princeton companion to Mathematics and that sounds like a good start. I think what I would like is to read something that could explain why different fields of mathematics were important, and how I would concretely benefit from understanding them.
At the moment I have a general unease about my partial mathematical blindness, I understand the main mathematical ideas underlying the work in my own field (computer vision) and I’m pretty happy with the subjects in numerical recipes and some optimisation theory. I’m fairly sure that I don’t need to know more, but it bothers me that I don’t. At the same time I don’t want to spend a lot of time wading through proofs that are unlikely to ever be relevant to me. I have also yet to find a concrete example in AI where an engineering approach with some relatively simple applied maths has been substantially weaker than an approach that requires advanced mathematical techniques, making me suspect that mathematics is as it is because it appeals to those who like puzzles, rather than necessarily providing profound insight into a problem. Although I’d love to be proved wrong on that point.
I would second the recommendation of the Princeton Companion to Mathematics but would also warn it does not go into enough depth for one to get an accurate understanding of what many of the subjects discussed therein are about. This is understandable given space constraints.
The edifice of pure mathematics is vast and the number of people alive who could give a good overview of existing mathematics as a whole is tiny and possibly zero.
As a matter of practice, much of the information about how mathematicians learn and think about a given subject is never recorded. See this comment by SarahC and Bill Thurston’s MathOverflow question Thinking and Explaining.
On average I’ve found reading math books that adopt a historical approach to the material therein to be considerably more useful than reading math books that adopt an axiomatic approach to the material therein.
Based on my (limited) impression of applied math, it’s not uncommon for people to use advanced mathematical techniques to solve a practical problem because doing so makes for a good marketable story rather than because the advanced mathematical techniques are genuinely useful to analyzing the practical problem at hand.
There is an issue of a high noise-to-signal ratio in mathematics textbooks corresponding to the fact that many authors of textbooks don’t have the depth of understanding of the creators of the theories that they’re writing about and correspondingly do not emphasize the key points.
Concerning your suspicion that “mathematics is as it is because it appeals to those who like puzzles, rather than necessarily providing profound insight into a problem”—there’s great variability among mathematicians here. Two essays which discuss dichotomies which are not identical to the one that you draw but which I think you’ll find peripherally relevant are Timothy Gowers’ The Two Cultures of Mathematics and Freeman Dyson’s Birds and Frogs.
Those mathematicians who seek profound insight into problems often seek profound insight into problems within pure math rather than problems that arise in engineering.
Looking at your website, you might find it useful to check out the Brown University Pattern Theory Group. I don’t have any subject matter knowledge of what they do, but the group includes David Mumford who is of extremely high caliber, having earned a Fields Medal in the 1970′s for his work on algebraic geometry.
While I don’t know enough to point you in the right direction to help you with your research, if you’re interested in learning about pure math out of general intellectual curiosity then there are many books that I can recommend.
The edifice of pure mathematics is vast and the number of people alive who could give a good overview of existing mathematics as a whole is tiny and possibly zero.
In the 3,000 categories of mathematical writing, new mathematics is being created at a constantly increasing rate. The ocean is expanding, both in depth and in breadth.
By multiplying the number of papers per issue and the average number of theorems per paper, their estimate came to nearly two hundred thousand theorems a year. If the number of theorems is larger than one can possibly survey, who can be trusted to judge what is ‘important’? One cannot have survival of the fittest if there is no interaction. It is actually impossible to keep abreast of even the more outstanding and exciting results. How can one reconcile this with the view that mathematics will survive as a single science? In mathematics one becomes married to one’s own little field. [...] The variety of objects worked on by young scientists is growing exponentially. [...] Only within the narrow perspective of a particular speciality can one see a coherent pattern of development.
Thank you very much for your great reply. I’ll look into all of the links. Your comments have really inspired me in my exploration of mathematics. They remind me of the aspect of academia I find most surprising. How it can so often be ideological, defensive and secretive whilst also supporting those who sincerely, openly and fearlessly pursue the truth.
One frustration I find with mathematics is that it is rarely presented like other ideas. For example, few books seem to explain why something is being explained prior to the explanation. They don’t start with a problem, outline its solution provide the solution and then summarise this process at the end. They present one ‘interesting’ proof after another requiring a lot of faith and patience from the reader. Likewise they rarely include grounded examples within the proofs so that the underlying meaning of the terms can be maintained. It is as if the field is constructed so that it is in the form of puzzles rather than providing a sincere attempt to communicate idea as clearly as possible. Another analogy would be programming without the comments.
A book like Numerical Recipies, or possibly Jaynes book on probability, is the closest I’ve found so far. Has anyone encountered similar books?
I agree with your remarks here and share your frustration. While books of the type that you’re looking for are relatively uncommon; over the years I’ve amassed a list of ones that I’ve found very good. What subject(s) are you interested in learning? (N.B. There are large parts of math that I’m ignorant of—in particular I don’t know almost anything about applied math and so may not be able to say anything useful—I just thought I’d ask in case I can help.)
Thank you, my main goal at the moment is to get a handle on statistical learning approaches and probability. I hope to read Jaynes’s book and the nature of statistical learning theory once I have some time to devote to them. however I would love to find an overview of mathematics. Particularly one which focuses on practical applications or problems. One of the other posts mentioned the Princeton companion to Mathematics and that sounds like a good start. I think what I would like is to read something that could explain why different fields of mathematics were important, and how I would concretely benefit from understanding them.
At the moment I have a general unease about my partial mathematical blindness, I understand the main mathematical ideas underlying the work in my own field (computer vision) and I’m pretty happy with the subjects in numerical recipes and some optimisation theory. I’m fairly sure that I don’t need to know more, but it bothers me that I don’t. At the same time I don’t want to spend a lot of time wading through proofs that are unlikely to ever be relevant to me. I have also yet to find a concrete example in AI where an engineering approach with some relatively simple applied maths has been substantially weaker than an approach that requires advanced mathematical techniques, making me suspect that mathematics is as it is because it appeals to those who like puzzles, rather than necessarily providing profound insight into a problem. Although I’d love to be proved wrong on that point.
Upvoted for a thoughtful comment.
I don’t know anything about statistical learning theory.
I don’t know what kinds of probability you’re interested in learning, but would recommend Concrete Mathematics: A Foundation for Computer Science by Graham, Knuth and Patashnik and William Feller’s two volume set An Introduction to Probability Theory and Its Applications.
I would second the recommendation of the Princeton Companion to Mathematics but would also warn it does not go into enough depth for one to get an accurate understanding of what many of the subjects discussed therein are about. This is understandable given space constraints.
The edifice of pure mathematics is vast and the number of people alive who could give a good overview of existing mathematics as a whole is tiny and possibly zero.
As a matter of practice, much of the information about how mathematicians learn and think about a given subject is never recorded. See this comment by SarahC and Bill Thurston’s MathOverflow question Thinking and Explaining.
On average I’ve found reading math books that adopt a historical approach to the material therein to be considerably more useful than reading math books that adopt an axiomatic approach to the material therein.
Based on my (limited) impression of applied math, it’s not uncommon for people to use advanced mathematical techniques to solve a practical problem because doing so makes for a good marketable story rather than because the advanced mathematical techniques are genuinely useful to analyzing the practical problem at hand.
There is an issue of a high noise-to-signal ratio in mathematics textbooks corresponding to the fact that many authors of textbooks don’t have the depth of understanding of the creators of the theories that they’re writing about and correspondingly do not emphasize the key points.
Concerning your suspicion that “mathematics is as it is because it appeals to those who like puzzles, rather than necessarily providing profound insight into a problem”—there’s great variability among mathematicians here. Two essays which discuss dichotomies which are not identical to the one that you draw but which I think you’ll find peripherally relevant are Timothy Gowers’ The Two Cultures of Mathematics and Freeman Dyson’s Birds and Frogs.
Those mathematicians who seek profound insight into problems often seek profound insight into problems within pure math rather than problems that arise in engineering.
Looking at your website, you might find it useful to check out the Brown University Pattern Theory Group. I don’t have any subject matter knowledge of what they do, but the group includes David Mumford who is of extremely high caliber, having earned a Fields Medal in the 1970′s for his work on algebraic geometry.
While I don’t know enough to point you in the right direction to help you with your research, if you’re interested in learning about pure math out of general intellectual curiosity then there are many books that I can recommend.
The mathematical experience:
Thank you very much for your great reply. I’ll look into all of the links. Your comments have really inspired me in my exploration of mathematics. They remind me of the aspect of academia I find most surprising. How it can so often be ideological, defensive and secretive whilst also supporting those who sincerely, openly and fearlessly pursue the truth.