As the author admits in his update, this document is not very interesting. When I saw the title I was mainly interested in perhaps a formalism of mathematical exposition and proof notation.
The problem with the author's proposals is this: he proposes replacing well-known, well-distinguished symbols (alphabets and digits) with symbols that reflect a certain geometric interpretation.
First, the geometric symbols are difficult to read, and a small error (either in printing or in reading) can mean a huge change in meaning.
Second, the obvious criticism that they are too radical a departure from the current convention for them to be widely adopted.
Most salient, however, is that they offer no benefit to the development of mathematics. The mathematical enterprise is based on abstraction, generalization, and symbolic manipulation. The symbols proposed reflect only the most banal representation of the mathematical objects they discuss.
For example, he discusses representing the natural numbers with a number line. But this is an elementary schooler's notion of natural numbers. Mathematicians would like to discuss other representations and constructions of the natural numbers, say, their Peano axiomatization, as Church numerals, as an algebraic structure, as indices, as a countable set, and many more. Hence it is honestly more convenient, and less intrusive to the discussion of novel mathematical ideas and perspectives, to simply use a flavorless symbol of the alphabet, rather than one that is associated with a particular geometric interpretation.
Finally, I'd like to point out that while many things have changed in mathematics since ancient times, we still follow Aristotle in using letters to denote mathematical objects, and Euclid's rigorous exposition and proof, and actually use them today with far more frequency than they ever did.
Hi, I'm the author. Please keep in mind that I wrote this when I was 16.
Thanks for your comments -- I agree with many of them. The only part of this that I still think is at all interesting is the section on quantifiers, and to a lesser extent the octal numbers. And I don't think they're that interesting. In general, while I do think there is interesting potential for improving notation -- Feynman diagrams come to mind as a somewhat modern example of notation impacting research -- I don't think this is what it looks like.
> he proposes replacing well-known, well-distinguished symbols (alphabets and digits) with symbols that reflect a certain geometric interpretation.
That's what most of it is, although the quantifiers are a more "grammatical" change. I think the best defense I can level of the symbol proposals is that the people most effected by low-level notational choices like symbols are probably young students learning them for the first time. It seems plausible there might be significant pedagogical gains in other symbols.
Math major here. I too came out of your paper thinking that the quantifiers are the only interesting part of you paper; interesting enough, in fact, that I may start using them myself. One criticism is that there is no way to use them for variables that occur in more than one place. That is, you could not use them to right "for all x in R, x<x+1". I propose a simple addition of an optional subscript for the variable name in the quantifier.
My other criticism is that it seems less natural to read that notation. With the standard notation, you can simply replace the symbol with English words and read it. While the transliteration in your notation is likely possible, it seems more complicated. In this sense, I would probably view this notation more as a shorthand. For pedagogical purposes, I feel like separating the qualifier from the usage of the variable is a good idea.
It is not the only problem, unfortunately. For example, there is no way to specify the order of quantifiers, which is very important (if you commute existiential and universal quantifiers, in general you change the meaning of a formula). So, for example, the definitions of pointwise and uniform continuity would we written identically, and that is not tolerable.
Probably you could solve also this by adding another index to quantified quantities, but at that point are you really solving the problem of complexity? Complexity is not only measured in terms of verbosity (i.e., length of the expressions you produce); ability to quickly locate information and ease of manipulation are also important, and it seems to me that this formalism fails completely. In the "traditional" formalism you separate the "final fact" you want to assert from its "conditions", given by the quantifiers; usually, you want to process those pieces of information at different times. Also, you have rules for quantifier introduction and elimination, which seems to be a nightmare with the proposed formalism.
That said, I do encourage new proposals and experimentation with notations, as with anything else in mathematics, even from young students. There have been cases in maths in which some good idea for a new notation made an entire field much easier (I am thinking for example to Einstein notation for tensor calculus).
Hi, thanks for replying! I'm sorry if I come off as overly critical. I did indeed think that this was the work of a very curious teenage mind who nevertheless still had insufficient experience with academic-level mathematics.
The intention is good, e.g. 1, 2 and 3 can be written in as many strokes as they represent, an S already looks like a cosine-curve (as it's depicted on e.g. synthesizer interfaces). But after that it becomes tedious because there is only so much space for entropy in a small area even with printed symbols. Bigger symbols, ie. diagrams are embraced already in many areas. Geometric representations are embeddings of mathematical objects in the real world.
Of course improved symbolism had the potential to improve learning. Another way to achieve that is reduction of formalism, structure is much more important to avoid repetition. Mathematical didactic is very much concerned with that, but I see no didactic arguments in the paper, its only stylistic.
The problem with the author's proposals is this: he proposes replacing well-known, well-distinguished symbols (alphabets and digits) with symbols that reflect a certain geometric interpretation.
First, the geometric symbols are difficult to read, and a small error (either in printing or in reading) can mean a huge change in meaning.
Second, the obvious criticism that they are too radical a departure from the current convention for them to be widely adopted.
Most salient, however, is that they offer no benefit to the development of mathematics. The mathematical enterprise is based on abstraction, generalization, and symbolic manipulation. The symbols proposed reflect only the most banal representation of the mathematical objects they discuss.
For example, he discusses representing the natural numbers with a number line. But this is an elementary schooler's notion of natural numbers. Mathematicians would like to discuss other representations and constructions of the natural numbers, say, their Peano axiomatization, as Church numerals, as an algebraic structure, as indices, as a countable set, and many more. Hence it is honestly more convenient, and less intrusive to the discussion of novel mathematical ideas and perspectives, to simply use a flavorless symbol of the alphabet, rather than one that is associated with a particular geometric interpretation.
Finally, I'd like to point out that while many things have changed in mathematics since ancient times, we still follow Aristotle in using letters to denote mathematical objects, and Euclid's rigorous exposition and proof, and actually use them today with far more frequency than they ever did.