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Ask HN: What OR Are there some multiple perspective books in mathematics?

13 点作者 davehcker将近 5 年前
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.

6 条评论

saeranv将近 5 年前
This one is for statistics.<p>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 &#x27;The Geometry of Multivariate Statistics&#x27; by Thomas D. Wickens, which I purchased solely based on it&#x27;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.<p>Funnily enough, recently I&#x27;ve been going through Strang&#x27;s &#x27;Introduction to Linear Algebra&#x27; 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&#x2F;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将近 5 年前
Plug for a friend&#x27;s book that is forthcoming later this summer, &quot;Topology: A Categorical Approach&quot;. [1]<p>I can&#x27;t speak to its contents per se, because there isn&#x27;t a preview yet, but I can speak to the quality of exposition in the lead author&#x27;s math blog. [2]<p>I haven&#x27;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.<p>[1] <a href="https:&#x2F;&#x2F;mitpress.mit.edu&#x2F;books&#x2F;topology" rel="nofollow">https:&#x2F;&#x2F;mitpress.mit.edu&#x2F;books&#x2F;topology</a><p>[2] <a href="https:&#x2F;&#x2F;www.math3ma.com&#x2F;" rel="nofollow">https:&#x2F;&#x2F;www.math3ma.com&#x2F;</a>
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GaussBonnet将近 5 年前
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.<p>The concept of a &#x27;functor&#x27; was invented to describe a higher order &#x27;homomorphism of homomorphisms&#x27;. 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).<p>You might also be interested in various &#x27;dual&#x27; concepts, like that between tangent spaces and cotangent spaces in differential geometry.<p>For algebra, I&#x27;d recommend Pinter&#x27;s Book of Abstract Algebra.
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asdf_snar将近 5 年前
Perhaps one of the most well-known such gems is Milnor&#x27;s &quot;Topology from the Differentiable Viewpoint&quot;.<p><a href="https:&#x2F;&#x2F;math.uchicago.edu&#x2F;~may&#x2F;REU2017&#x2F;MilnorDiff.pdf" rel="nofollow">https:&#x2F;&#x2F;math.uchicago.edu&#x2F;~may&#x2F;REU2017&#x2F;MilnorDiff.pdf</a>
davehcker将近 5 年前
_Introduction to Information Theory: Symbols, Signals and Noise (Dover Books on Mathematics)_ by Pierce is another such book.<p>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.
_5659将近 5 年前
Maybe not books per se, but I often find more exhaustive treatments and variations on a proof in survey papers.