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Ben edited this page Apr 1, 2020 · 4 revisions

In order to visualize the connections between equations in mathematical physics, directed graphs are created from a finite set of inference rules.

Components:

  • database of statements (equations, inequalities) expressed in latex
  • database of inference rules which act on statements to produce other statements. Example: "multiply both sides of equation by 1"
  • database of connections between adjacent statements by a specific inference rule scripts to convert the information in the three databases to a user-friendly format, such as a directed graph or Latex-generated PDF.
  • database of connections between adjacent statements can be checked by a computer algebra system (e.g. SymPy). Problem: Godel's incompleteness theorem says meta-mistakes will not be caught.

The equation indices are my Godel numbers.

Example of an inference rule: Suppose A=B and B=C and we want to claim A=C. To carry this out, a fourth statement, C=B, is needed.

declare A=B as statement 1

declare B=C as statement 2

inference rule: swap sides of statement 1. Result: B=A is statement 3

inference rule: left-hand-side of statement 2 is same as left-hand-side of statement 3. Result: C=A is statement 4

Here two inference rules have been used: "swap sides of statement" and "left-hand-side of one statement is identical to left-hand-side of another." Both of these applications of the inference rules can be checked by a computer program. Jargon:

I use "statement" to encompass both equation (=) and inequalities (<,>,<=,>=) I use "inference rule" to distinguish from standard operators (+,-,,/,^,\int). Inference rules are operators in the sense that they take at least three arguments and return statements. However, I find it confusing to use "operator" to refer to both an action acting on statements and standard operators.

Keyword descriptions:

  • grammar of physics
  • grammatical rules for physics
  • inference rules for physics
  • systemization of physics

What I am trying to do:

  • build a framework capable of describing and relating all of physics in a centralized electronic format.
  • have an intuitive GUI with Latex-based input. (Doesn't exist yet.)
  • keep everything human-readable.
  • use only open source, stable software with significant community (user+dev) base. (Python, XML, dvi2png, Latex)
  • minimize the number of external dependences (i.e., only use Python rather than Python+Perl+Mathematica) while not re-inventing the wheel (XML rather than complicated flat text file)

The intent is to show graphically how various subfields within physics are related. It will also show which operations are necessary in physics, along with which equations are most commonly used. Ideally, given enough content, patterns will become apparent.

I should also say what I am not trying to do:

  • explain all of physics (and math). is done better in textbooks and research papers.
  • build a framework capable of describing all of math. [I'm not sufficiently familiar with math notation.]
  • make money. don't think what I'm doing constitutes a viable financial plan.
  • The purpose is not to replace Principia Mathematica or thinking for oneself. The results require human verification. The above example (A=B, B=C, thus A=C) is logical, whereas most content I'm interested in describing is physical. Starting from an statement which is physically testable, how does one get to another (related) physically testable mathematical statement?
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