Math Notations, Computer Languages, and the “Form” in Formalism
perm url with updates: http://xahlee.org/cmaci/notation/index.html
Math Notations, Computer Languages, and the “Form” in Formalism
This page is a collection of essays and expositions on the subjects of nomenclature and notations in math and computer languages, in the context of facilitating human communication and theorem proving systems.
Most of these essays here are originally from email, blogs, or rants. They are not of publication quality, and they are not a coherent exposition the subject. Here's a very brief summary of of these essays's central thesis:
• Traditional math notations are very inconsistent. Edsger Dijkstra is a leader in a movement of what's called Calculational Proofs. That is, using a notation that is consistent and facilitates the calculation aspects when doing math by humans.
• Today, especially since 1990s, tremendous advances are made in computer algebra systems and theorem proving systems. In these languages, a coherent syntax, grammar, are needed for math expressions.
• In computer algebra or theorem proving systems, they are intimately tied to the math philosophies of formalism and logicism. In a sense, formalism and logicism today are tied together as a single subject, and using computer languages as foundation.
• Math expressions/syntax in computer languages are intimately tied to math notations for human reading. (e.g. Mathematical, MathML are technologies that combine the two.)
• The syntax and grammar of today's computer languages, such as Java, C, Python, SQL, Lisp, are ad-hoc and their communities have little understanding of the knowledge gained in math related fields such as computer algebra or theorem proving languages. (This applies to functional langs such as Haskell as well, but to a lesser degree.) On the other hand, mathematicians in general are illiterate about programing or using computer languages.
All of the above considered together, computer language designers and mathematicians, should be made aware of these issues, so that when they design or use computer languages, may it be math oriented or not, the language's syntax and grammar can move towards a consistent syntax system with solid foundation (as opposed to ad-hoc), and such language should have build-in markup or simple mapping to 2-dimensional notations for human reading (such as done with Mathematica or Semantic MathML), and this computer language should be in fact as a basis of theorem prover or computer algebra system (as in OCaml, Haskell, Mathematica). The languages of computer algebra and theorem prover would in fact merge together into one single subject if it is not already slowly happening today.
Progress in the above issues are made in different fields but there are little unification going on.
For example, there's Edsger Dijkstra's Calculational Proofs movement. It improves math notations towards consistency and formalism. However, people in Calculational Proofs movement are mostly math pedagogy community i think. They are not programers interested in computer languages, nor logicians interested in math formalism, or industrial and commercial organizations interested in math notation representation systems.
There's the computer algebra community, such as Mathematica, Maple, Matlab, which requires a syntax and grammar for mathematical concepts. There's the theorem proving community, such as OCaml, Coq, HOL, which not only requires a syntax for math concepts, but also made major understanding about math as a system of forms, i.e. formal systems. Both computer algebra and theorem proving systems require math notations and computer language syntax that are consistent and can represent math concepts. However, the 2 camps are largely separate communities. For example, there is as far as i know no tool that is both a practical computer algebra system as well as a theorem proving language.
Common computer languages, such as C, Java, Python, requires a good syntax, parsers, and compilers, but their community, including computer scientists and programers, are usually illiterate in typical topics of of mathematics proper. Functional languages, such as Scheme Lisp, APL, OCaml, Haskell, are more based on logic foundations (lambda calculus) but their syntax and grammar has little to do with the math notations as a logic or formal system. (these languages do not have a formal spec in the sense of Formlism, i.e. transformation of forms. In fact, almost no languages has a formal spec, formalism or not.)
There's math notation representation needs, such as TeX, MathCAD, MathML, Mathematica. These are typically commercial organizations in the computing industry. They can render math notations. In the case of MathML and Mathematica, the language also represent the semantic content of math notations. These two made major understanding about the relation of math notations and computer languages, but they in general have little to do with formalism or theorem proving. (with some exception of Mathematica)
Calculational proof notational system, Computer algebra systems, theorem prover languages, formalism and logicism as foundation of mathematics, functional languages, and computer languages in general, mathematics and its notations, all are in fact can be considered as a single subject with a unified goal. All the technologies and movements exist, but today they have mostly not come together. For example, Microsoft Equaton Editor, TeX, and various other tools does well with math notation rendering. MathML has both representational and semantic aspects (OpenMath is a new group that focus on semantic aspects), for the purpose of rendering as well semantic representation. Mathematica is a computer algebra system for solving arbitrary math equations, that is also able to represent math notation as a computer language, so that computation can be done with math notation directly. However, the system lacks a foundation as a theorem prover. Theorem provers such as OCaml (HOL, Coq), Mizar does math formalism as a foundation, also function as a generic computer language, but lacks abilities as a computer algebra system or math notation representation.
- The Codification of Mathematics
- State Of Theorem Proving Systems 2008
- The Problems of Traditional Math Notation
- Fundamental Problems of Lisp (see section 1, on the importance for regularity of syntax)
- The Concepts and Confusions of Prefix, Infix, Postfix and Fully Functional Notations
- The Moronicities of Typography
- Math Terminology and Naming of Things
- Math Jargons Explained
- Politics and the English Language, 1946, by George Orwell.
Harm Of Bad Terminologies In Computing Languages
- The Importance of Terminology's Quality In A Computer Language
- What are OOP's Jargons and Complexities?
- Java's Abuse of the Jargon “Interface” and API
- Jargons of Info Tech Industry
- The Term Currying In Computer Science
- Function Application is not Currying
- Why You should Not Use The Jargon Lisp1 and Lisp2
- The Jargon “Tail Recursion”
- What Is Closure In A Programing Language
- Jargons And High Level Languages (unpolished essay)
- Why You Should Avoid The Jargon “Tail Recursion”
- I Can Not Find A Word Better Than “CAR”
Harm of Mixing Concept of Syntax and Formatting
- The TeX Pestilence
- The Harm of hard-wrapping Lines (harm of confusing syntax with formatting)
- Tabs versus Spaces in Source Code (harm of treating syntax as formatting instead of syntax)
Applications of Regular Syntax
- A Text Editor Feature: Syntax Tree Walk (application of syntax)
- A Simple Lisp Code Formatter (application of syntax regularity)
- “Mathematica Notation: Past and Future” (2000-10-20), by Stephen Wolfram, at http://www.stephenwolfram.com/publications/recent/mathml/index.html.
- Functional Mathematics, by Raymond Boute, 2006. http://www.funmath.be/
- Formalized Mathematics, by John Harrison, 1996-08-13. http://www.rbjones.com/rbjpub/logic/jrh0100.htm
- “How Computing Science created a new mathematical style”. (1990) By Edsger W Dijkstra. EWD1073
- “Under the spell of Leibniz's dream” (2000) By Edsger W Dijkstra. EWD1298
- Abuse of notation
- Formal system