
Ask HN: What OR Are there some multiple perspective books in mathematics? - davehcker
There are so many mathematical ideas that can be interpreted in multiple ways&#x2F; can be used to explain many things with the same theory. For instance, addition in Galois Field of order 2 is same as binary XOR, or that irreducible polynomials exhibit prime number like properties, etc.<p>I&#x27;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?<p>I am mostly interested in Algebra, but I&#x27;ll appreciate pretty much any suggestion.<p>&gt;&gt;&gt; From his quote &quot;You don&#x27;t understand anything until you learn it more than one way&quot;, I am guessing that Minsky might have experienced a similar thing.
======
saeranv
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.

------
suddensleep
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](https://mitpress.mit.edu/books/topology)

[2] [https://www.math3ma.com/](https://www.math3ma.com/)

~~~
eindiran
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_Theory_for_Computer_Scientists)

~~~
davehcker
Right! I have still not finished it, but it's truly enchanting and exactly as
you mentioned.

------
GaussBonnet
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.

~~~
davehcker
Just took a glimpse of the chapter on Isomorphism pg. 91
([http://www2.math.umd.edu/~jcohen/402/Pinter%20Algebra.pdf](http://www2.math.umd.edu/~jcohen/402/Pinter%20Algebra.pdf)
)and now I'm indebted to you.

------
asdf_snar
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)

------
davehcker
_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.

------
odomojuli
Maybe not books per se, but I often find more exhaustive treatments and
variations on a proof in survey papers.

