Understanding Free Monoids and Universal Constructions

  Bartosz Milewski's Programming Cafe

There is an engineer and a mathematician in each of us. My blog posts, for instance, alternate between discussing write ordering in the x86 processor, and explaining abstract ideas from category theory. I might be a bit extreme in this respect, but even in everyday programming we often switch between the focused, low-level, practical thinking of an engineer and the holistic, global, abstract thinking of a mathematician. Sometimes we write a for loop; sometimes we holistically apply std::transform to a vector. Sometimes we write imperative step-by-step programs; and sometimes we write declarative ones, letting the compiler figure out the steps.

What I find fascinating is that these two approaches also manifest themselves in mathematics. There are constructive proofs that resemble the work of an engineer; and there are proofs of existence, that resemble the work of a philosopher. An entity may be defined by showing how to construct it, or…

View original post 2,174 more words

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s