Comment by joshuamorton
5 years ago
A more interesting question.
I think there are two answers:
1. This style of notation was, I believe first really developed in Principia Mathematica in 1910ish. Some of the proofs in that book are big, and would be vastly longer if they were in plain english.
2. Part of the goal of that book was precision. They could reuse existing words, but those have pesky connotations that might mislead. So instead they'd either be left to invent new terminology "sqaggle x, y lemmy ror, x grorple y" or pick something symbologicial.
There are really good conversations about intuitiveness of notation (see, e.g. https://mathoverflow.net/questions/366070/what-are-the-benef...), and I think that comes down to things being hard. Notation is almost always created by an expert who is making something useful to them. It's not clear that the correct expressive notation would even be the same as the correct layman's/teaching notation. Those two concepts may simply be at odds.
The linked MathOverflow answer was written by Dr. Terence Tao, and it really is an excellent dissection of the many needs levied upon notation. I find the two "Suggestiveness" items to be most related to concision as practices in research papers.
This bit from the tail end should not be missed, in our context:
> ADDED LATER: One should also distinguish between the "one-time costs" of a notation (e.g., the difficulty of learning the notation and avoiding standard pitfalls with that notation, or the amount of mathematical argument needed to verify that the notation is well-defined and compatible with other existing notations), with the "recurring costs" that are incurred with each use of the notation. The desiderata listed above are primarily concerned with lowering the "recurring costs", but the "one-time costs" are also a significant consideration if one is only using the mathematics from the given field X on a casual basis rather than a full-time one. In particular, it can make sense to offer "simplified" notational systems to casual users of, say, linear algebra even if there are more "natural" notational systems (scoring more highly on the desiderata listed above) that become more desirable to switch to if one intends to use linear algebra heavily on a regular basis.
It's a great explanation, thank you. Yes, as a casual user I'm concerned with "one-time costs", ie. the barrier to entry. I'm glad that I'm not the only one who noticed that these costs are real and make "casual usage" harder than it could be otherwise.
Your previous comment, though, is much less optimistic (to me): you say that, even if they are aware of this, the experts have no incentives to use "simplified notational systems". That means that I can either become an expert myself, find an expert who will translate for me, or be unable to make use of the knowledge contained in materials for experts. I wish there was some other choice, like Google Translate from math to English...
Yes, I would like to see more efforts out there like Distill in making cutting-edge research more accessible. I don't think we can wish away the significant gap in expertise and knowledge embodied by research, though. Somebody has to put in the effort to understand it for themselves and then unpack it for others. What may be one page in a research paper could easily turn into dozens.
I myself have been studying category theory (CT) off and on for a few years now, trying to get enough of an understanding to be able to explain it to others in the software engineering (SE) field. I think there's a lot to gain from CT, but it's so strongly founded on distant mathematical fields like algebraic topology that it's very hard to get at its essence and find tractable connections from an SE perspective.
Finding good explanations is really a hard research problem of its own; it just isn't funded that way.
> That means that I can either become an expert myself
I don't think there's any way around this. If you understand a topic, you have obtained some amount of expertise thereof. An expert cannot simply translate for you; it's not a merely notational difference. There's a body of knowledge that must be transferred.
I would recommend finding an expert who's willing to correspond with you on occasion, and look for more introductory materials like textbooks and position papers in the field. If there's a particular goal you have in mind -- say there's a specific paper that achieves something you have an interest in -- be up front about that; it helps focus the explanations and recommendations.
Being an expert doesn't mean you have to have a very broad base of expertise. In fact, it could be argued that most experts are expert in a very, very small and focused niche. The smaller the niche, the less you need to digest; but the more rarefied the niche, the more stable the successive foundations below you need to be.