Is Geometric Algebra inconsistent/circular?

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Discussion Overview

The discussion revolves around the consistency and definitions within Geometric Algebra, particularly focusing on the geometric product of vectors as presented in the textbook by Doran and Lasenby. Participants explore the axioms defining the geometric product, the implications of these definitions, and the challenges in computing the product of vectors and multivectors.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant questions the circularity in defining the geometric product based on the inner and outer products, suggesting that the definitions do not uniquely determine the geometric product.
  • Another participant proposes that the geometric product should be defined axiomatically, mentioning that the full list of axioms includes conditions beyond those presented in the textbook.
  • Concerns are raised about the implications of the geometric product yielding a scalar and a bivector, complicating the computation of products involving multivectors.
  • Several participants discuss the need for a rigorous definition of the geometric product that accommodates all forms, not just scalars.
  • There is a suggestion that the book by Dorst may provide a more intuitive understanding of the geometric product, despite initial concerns about its focus on detailed definitions.
  • One participant expresses a preference for physics applications over rigorous mathematical treatments, questioning the necessity of detailed definitions for practical understanding.
  • Another participant highlights confusion regarding the "usual" rules for multiplication in the context of the geometric product, specifically addressing the non-commutativity of the geometric product.

Areas of Agreement / Disagreement

Participants express differing views on the definitions and implications of the geometric product, with no consensus reached on whether the definitions provided in the textbook are sufficient or if they lead to circular reasoning. The discussion remains unresolved regarding the best approach to define and compute the geometric product.

Contextual Notes

Participants note limitations in the textbook's explanations, particularly regarding the computation of the geometric product for multivectors and the need for clarity on the axioms involved. There is also uncertainty about the applicability of certain definitions in practical physics contexts.

Who May Find This Useful

This discussion may be of interest to students and practitioners of Geometric Algebra, particularly those exploring its foundational definitions and applications in physics and mathematics.

  • #31
micromass said:
I'm actually planning to do an insight series about it too!

Wonderful idea! I am really looking forward to see it published soon.
 
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  • #32
Although I've postponed learning geometric algebra (GA) [putting my efforts into differential forms instead],
I have found myself having to learn to it in order to understand/decode some calculations.

In writing an Insight,
it might be useful to include translations between statements and calculations in GA
and statements and calculations in (say) more-standard tensor-calculus and/or vector-calculus,
accompanied by some diagrams.
To me, the notation found in various sources in quite dense, rather abstract for a beginner, and not quite standardized,
forcing the reader to juggle the various objects and operations.

My $0.03.
 
  • #33
I guess the original idea for the insight was not to focus only on the details of GA, but more generally on "Non-mainstream but useful math to know". As micromass said:
micromass said:
There are many other things in mathematics that I feel should be basic knowledge for undergrads, but that is somehow something that is only known to a select few.
 

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