This is not a critic of all geometric algebra and especially not of its more basic parts. Therefore it is not an excuse for not grasping e.g. what Hestenes has written about GA, or what Eric Lengyel himself has written, e.g. in his new book that is advertised in this article.
It is a critic of many books about geometric algebra, which have made attempts to expand and further develop some parts of its theory, but those attempts have not been thought carefully and they have produced various inconsistent or useless definitions.
It is also a critic of attempts of presenting geometric algebra as preferable for applications where in fact it is not optimal, by showing misleading "benchmarks". Unfortunately this tactic is not at all specific to geometric algebra, but it is frequently encountered for almost any kind of algorithm known to mankind when it accumulates for one reason or another some kind of fan base.
Calculus had poor foundation and was thus logically incoherent from Newton/Leibniz' discovery to roughly the middle of 19th Century. None-the-less it was a powerful tool and most of the key theorems were discovered then.
The basic situation, I think, is a set of tools can be consistent in the way mathematicians use them but in the way the mathematicians explain them. And the tools can be very useful despite this.
So my guess is saying "it has poor foundations" isn't saying "I'm against it, it's worthless"
Yeah I’m pretty well-versed in free constructions of various algebraic objects and how this would interact with things like a quadratic form, etc, but couldn’t sort out GA (I think the authors of “GA4CS” had a very different sort of computer scientist in mind). When I saw the geometric algebra constructions clash with constructions I was already familiar with, I generally got suspicious and lost interest.
I’m actually quite interested in checking out Lengyel’s book. It looks rock solid.
