Very intriguing article here: “The Structures of Computation and the Mathematical Structure of Nature“ by Michael S. Mahoney: rutherfordjournal.org/article030107.… 
Paradoxical constructs “application” and “abstraction” brought together leading the field forward.
Algebraic structures like lattices, rings, fields, and semigroups as structures that have cross-domain application in computer science.
Difference between classical and modern variants of algebras from Birkhoff and Bartee’s book is quoted here:
Good questions probed up here: Is software, a science? Is science mathematics? Or can mathematics be considered a science? Also how does one distinguish between human made artifacts and nature? cc @paulportesi
Neat piece of wisdom here by Francis Bacon:
How computation gave metaphorical windows to see the world through. DNA as code, L systems for plant growth and bio molecules as automata:
The uncanny bidirectionality between the Platonic realm/mathematical models and physical realm. Exploring mathematical models paved way to certain patterns before they were located and named in physical reality:
Dynamic simulation enabled by computers is said to be a new paradigm that enables us to understand nature better:
Did some typography changes and it looks a bit better to read.
Here’s Mahoney talking about the considerable shift in the meaning of applied mathematics from putting numbers on theory to modeling as abstraction of structure that happened in mid-20th century.
Here’s Mahoney talking about the considerable shift in the meaning of applied mathematics from putting numbers on theory to modeling as abstraction of structure that happened in mid-20th century.
A quick illustration that was sparked by the paragraph on homomorphisms vs. isomorphisms.
Importance of fix-point theorems in economics equilibrium theory figured out by Von Neumann. This guy is a polymath in the truest sense of the word.
Physical world as a semantic model of an uninterpreted Platonic abstract realm as conceived in the Von Neumann style word view.
Bifurcation in research of Von Neumann is stressed here. One towards combinatorial (all or one) concepts and the continuous concept of the real/complex numbers in analysis:
This was quoted in @pressron’s articles: pron.github.io/posts/computat… and is an important divide that has changed the course of CS history:
Von Neumann was looking for a middle ground between all-or-one automata and continuous analytical concepts and it is said to have emerged over the next 15 years:
These two papers by Mahoney are said to delve deeper into the history of this evolution:
1/ The Search for a Mathematical Theory: web.archive.org/web/2017042504…
2/ What Was the Question? The Origins of the Theory of Computation: web.archive.org/web/2020032813…
1/ The Search for a Mathematical Theory: web.archive.org/web/2017042504…
2/ What Was the Question? The Origins of the Theory of Computation: web.archive.org/web/2020032813…
Mahoney gives an outline of how the different research agendas influenced each other. Particularly interesting here is the emergence of parallels between structures, automata, and their corresponding languages:
It’s very interesting to see that Mahoney made the previous diagram by fusing together two of his earlier sketches. I love the visual schemata of the first image here.
This preliminary draft of the paper might also be worth a read to understand how it differs from the original: princeton.edu/~hos/Mahoney/a…
Michael Rabin and Dana Scott unifying the works of Stephen Kleene, Von Neumann, and E. F. Moore on Finite Automata.
All of these had common root in the work of McCullough and Pitts via Von Neumann:
All of these had common root in the work of McCullough and Pitts via Von Neumann:
