Ask HN: What OR Are there some multiple perspective books in mathematics?
9 comments
Plug for a friend's book that is forthcoming later this summer, "Topology: A Categorical Approach". [1]
I can't speak to its contents per se, because there isn't a preview yet, but I can speak to the quality of exposition in the lead author's math blog. [2]
I haven't ever dug too much into category theory for its own sake (usually just one-off chapters or appendices that get included in books on other topics), but my understanding is that it unites a lot of mathematical topics. As such, this book might be of more interest to you than, say, a classical point-set topology text, given your desire to uncover connections. That being said, there may be other category-theory-flavored books on other more strictly algebraic topics that would suit your fancy more.
[1] https://mitpress.mit.edu/books/topology
[2] https://www.math3ma.com/
I can't speak to its contents per se, because there isn't a preview yet, but I can speak to the quality of exposition in the lead author's math blog. [2]
I haven't ever dug too much into category theory for its own sake (usually just one-off chapters or appendices that get included in books on other topics), but my understanding is that it unites a lot of mathematical topics. As such, this book might be of more interest to you than, say, a classical point-set topology text, given your desire to uncover connections. That being said, there may be other category-theory-flavored books on other more strictly algebraic topics that would suit your fancy more.
[1] https://mitpress.mit.edu/books/topology
[2] https://www.math3ma.com/
If you have a CS background and are looking to dig in to category theory a bit more, I would recommend checking out Benjamin Pierce's Basic Category Theory for Computer Scientists. It's a nice, concise introduction to the topic that examines a few very interesting applications to CS.
https://www.goodreads.com/book/show/1810837.Basic_Category_T...
https://www.goodreads.com/book/show/1810837.Basic_Category_T...
Right! I have still not finished it, but it's truly enchanting and exactly as you mentioned.
This one is for statistics.
I have found the explanation of statistical concepts through the lens of linear algebra immensely intuitive. A simple, short and clear illustration of this is in 'The Geometry of Multivariate Statistics' by Thomas D. Wickens, which I purchased solely based on it's title. It goes through the geometric interpretation of univariate, and multivariate linear regression, then goes into the geometric interpretation of correlation, collinearity impact on prediction, PCAs, and statistical tests. Warning: This book assumes you have some very basic statistical background.
Funnily enough, recently I've been going through Strang's 'Introduction to Linear Algebra' textbook, and he also goes through derivation of mulitvariate statistics in the same fashion. I like the way he builds up the geometric interpretation of regression by building up from a exploration of column/row spaces, orthogonality, projection matrices, and from there, seamlessly introduces solving the LLS as a problem that can be solved with a projection matrix. That being said, I find Wicken does a better job of illustrating his concepts, which is most intuitive modality to interpret this.
I have found the explanation of statistical concepts through the lens of linear algebra immensely intuitive. A simple, short and clear illustration of this is in 'The Geometry of Multivariate Statistics' by Thomas D. Wickens, which I purchased solely based on it's title. It goes through the geometric interpretation of univariate, and multivariate linear regression, then goes into the geometric interpretation of correlation, collinearity impact on prediction, PCAs, and statistical tests. Warning: This book assumes you have some very basic statistical background.
Funnily enough, recently I've been going through Strang's 'Introduction to Linear Algebra' textbook, and he also goes through derivation of mulitvariate statistics in the same fashion. I like the way he builds up the geometric interpretation of regression by building up from a exploration of column/row spaces, orthogonality, projection matrices, and from there, seamlessly introduces solving the LLS as a problem that can be solved with a projection matrix. That being said, I find Wicken does a better job of illustrating his concepts, which is most intuitive modality to interpret this.
Perhaps one of the most well-known such gems is Milnor's "Topology from the Differentiable Viewpoint".
https://math.uchicago.edu/~may/REU2017/MilnorDiff.pdf
https://math.uchicago.edu/~may/REU2017/MilnorDiff.pdf
Maybe not books per se, but I often find more exhaustive treatments and variations on a proof in survey papers.
_Introduction to Information Theory: Symbols, Signals and Noise (Dover Books on Mathematics)_ by Pierce is another such book.
It is of course only an introduction to the field, but it had an immense impact on how I saw information (hence, the universe) after I read. Primarily because it showed to me the many faces of Information Theory- music, psychology, geometry, language, cybernetics, etc.
It is of course only an introduction to the field, but it had an immense impact on how I saw information (hence, the universe) after I read. Primarily because it showed to me the many faces of Information Theory- music, psychology, geometry, language, cybernetics, etc.
The phenomenon you are talking about is essentially that of a homomorphism, or homomorphic structures. That is, structures that appear superficially different but share an underlying common structure.
The concept of a 'functor' was invented to describe a higher order 'homomorphism of homomorphisms'. An example most people miss is the total derivative in multivariable calculus: the chain rule implies that the total derivative is a functor that maps the composition of differentiable functions on a manifold, to matrix multiplication (of matrices acting on the tangent space).
You might also be interested in various 'dual' concepts, like that between tangent spaces and cotangent spaces in differential geometry.
For algebra, I'd recommend Pinter's Book of Abstract Algebra.
The concept of a 'functor' was invented to describe a higher order 'homomorphism of homomorphisms'. An example most people miss is the total derivative in multivariable calculus: the chain rule implies that the total derivative is a functor that maps the composition of differentiable functions on a manifold, to matrix multiplication (of matrices acting on the tangent space).
You might also be interested in various 'dual' concepts, like that between tangent spaces and cotangent spaces in differential geometry.
For algebra, I'd recommend Pinter's Book of Abstract Algebra.
Just took a glimpse of the chapter on Isomorphism pg. 91 (http://www2.math.umd.edu/~jcohen/402/Pinter%20Algebra.pdf )and now I'm indebted to you.
I'm sure that there is a general process of consolidation and generalization to come up with better i.e. a more inclusive theory. I am curious if there are books of this kind in mathematics that try to explain to the reader the same concept from different angles?
I am mostly interested in Algebra, but I'll appreciate pretty much any suggestion.
>>> From his quote "You don't understand anything until you learn it more than one way", I am guessing that Minsky might have experienced a similar thing.