detecting crackpots

Finding derivations which relate famous equations can lead to paths worn by crackpots. Well known equations like F=m*a and E=m*c^2 serve as good targets for non-experts. With sufficient math, laypersons have a difficult time distinguishing crackpots from legitimate experts.

Joseph A. Rybczyk

I was looking for the relation between E=m*c^2 and F=m*a

I found this page

http://www.mrelativity.net/relationshipef/the_relationship_between_e_and_f_p1.htm

Clue number 1: a private domain name -- mrelativity.net

Clue number 2: other papers attacking relativity

Clue number 3: lack of Latex

Clue number 4: unfamiliar jargon -- "Internal Energy of a Particle" 

visualize success and then deliver the minimum viable product

Today I applied two principles and made tremendous progress. First, I thought about what success meant for the Physics Derivation Graph. The central claim is that mathematical physics can be represented as a single graph. My insight was that I should simply focus on that. The second principle was determining the minimum viable product to achieve that vision of success.



The outcome is a hyperlinked SVG of concepts and topics:

http://allofphysicsgraph.github.io/proofofconcept/site/sandbox/topic_and_concept_graph.svg

This is a map of topics (ie Quantum Mechanics) and concepts (ie Schrodinger's Equation) in Physics.

I've drawn edges between concepts where there exists a paper detailing the mathematical connection

In the SVG, edges which are bold can be clicked to see the referenced URL


There's an equivalent PNG, but the hyperlinks aren't active:
http://allofphysicsgraph.github.io/proofofconcept/site/sandbox/topic_and_concept_graph.png


The point of this graph is that

1. the major topics in Physics are present
2. the major topics are linked to associated concepts
3. there are mathematical links between the concepts

The remaining work is to
4) fill in with additional concepts so that there is a path from any blue box (topic) to any other blue box through only red ellipses (concepts). Then the claim will be validated in theory. 
5) After that, go back and fill in the actual math using expressions and inference rules. 

6) Verify correctness of steps using a CAS

• break Latex expressions into constituent symbols and operators (ref 1, ref 2, ref 3)

Then I'm done!


The PDG isn't "complete" in accounting for all of Physics, but I didn't expect to go that far. Showing complete coverage of topics is sufficient.

What I want to do with the Physics Derivation Graph

For the past few years, completing the Physics Derivation Graph seemed infeasible in one lifetime. This meant I would get a partial result in one lifetime, or I'd have to find motivated collaborators willing to spend part of their lifetime. Working alone towards a partial result wasn't attractive, and I haven't had much success finding collaborators. Faced with these two options, I did almost no work on the project.

A few months ago I realized an exit strategy would be to get the Physics Derivation Graph to the point that I'd be comfortable handing it off to other folks. That way the project wouldn't be bottlenecked by my productivity. This insight resulted in focusing on bugs and inconsistencies which would need to be addressed prior to handing the project off to someone else.

Today I envisioned what success would look like in light of the central claim that "mathematical physics can be represented by a single graph."

• By "graph" I actually mean three levels of granularity: the symbols composing expressions as abstract syntax trees, expressions and inference rules composing derivations, and the relation between topics composed from derivations.
• The scope of Mathematical Physics is broad, but there are major topics which need to be included in order to claim completeness: quantum mechanics, classical mechanics, electromagnetics, thermodynamics, relativity, particle physics, cosmology, and astronomy. 

For each of the major topics, there are corresponding concepts that can be represented mathematically:

• quantum mechanics: particle in a box, Schrodinger's equation, uncertainty
• classical mechanics: work, F=m*a, momentum
• electromagnetics: Maxwell's equations
• thermodynamics: Planck's Law
• relativity: Dirac equation
• particle physics: Dirac equation
• cosmology
• astronomy

Finally, the claim is that each of these topics is related via mathematics. Example: harmonic oscillator occurs in quantum and classical

addressing the rendering versus verifiability challenge

For the Physics Derivation Graph, the primary output is visual. Thus rendering expressions should be beautiful. Latex is a natural choice and the entry is intuitive and straightforward.

In addition to rendering, the other task is to verify that the content is correct. This means using a computer algebra system (ie Mathematica, Octave). Latex is not amenable to CAS input because Latex can be mathematically ambiguous -- resolution depends on context.

One way to resolve this would be to stick with Latex, then convert to a CAS format for verifying correctness.

Sage

As an example, suppose I want to check that the expression "multbothsidesby" was correctly entered for input T/f=1, output T=f, with feed f. The Sage syntax looks like

T,f=var('T,f')
input_expr = T/f==1
expected_output_expr= T==f
expected_output_expr == input_expr*f

The above Sage returns true, building confidence that the step is valid. More simply,

T,f=var('T,f')
(T==f) == ((T/f==1)*f)

If Latex is to be used as the input, then we need to convert it to Sage syntax.

• declare each variable in Sage
• replace "=" in Latex with "=="
• convert the inference rule to something that can be checked

In addition to using a Sage notebook (https://cloud.sagemath.com), there's a one-time eval option -- http://sagecell.sagemath.org/
Calling a local installation of Sage is possible from Python, see http://ask.sagemath.org/question/8215/using-sage-in-a-python-cgi-script/