MathLingua is still in development, and the documentation still needs work.
However, I wanted to get feedback earlier rather than later to help guide development and the writing of the docs.
Note that MathLingua is the language that powers https://mathlore.org.
btw "The Language of Mathematics" sounds a bit over the top ;) how about "A Language for Mathematics"?
The goal is to allow the building of a system so someone could ask, for example, "what are the known theorems that describe when a function uniformly continuous?" or "under what conditions is a group solvable?".
To enable this, it is necessary to build a system that catalogs precise statements of math theorems, definitions, and other statements (https://mathlore.org is the start of this system that uses MathLingua but currently only uses keywords and some structure for searching).
This is similar to https://mathworld.wolfram.com or the math pages in Wikipedia except the statements are not text or LaTeX, that is hard for a computer to understand the meaning of. Instead the MathLingua language is used to identify math statements by meaning.
Right now the tooling can render math statements, find duplicate definitions, and can find missing definitions. Further https://mathlore.org uses MathLingua and has some basic searching.
Further, experimental work is also underway so that the MathLingua tooling can expand a math statement by expanding the definition of items used in that statement.
Yeah, I thought about using something like "A Language for Mathematics" but I thought "The Language of Mathematics" might draw more attention :) But it is only the title of my post on hacker news, not the title of the MathLingua website.
I am interested in learning more abt your rationale for creating MathLingua? What drove u? What existing faults do you see in existing system that made you do it.
TLDR: Having a computer understand what a statement written in LaTeX means is a very difficult Machine Learning (Natural Language Processing) problem (if not impossible since math statements often have lots of hidden context).
For more details:
In LaTeX, what does "if $f$ is continuous on $[a, b]$, $f \geq 0$ there, and $\int_a^b f = 0$ then $f = 0$?
It looks clear at first, but what type of thing is $f$. From context of reading analysis books, it could be clear that $f$ must be a function because "continuous" describes functions (but can also describe other things).
Next, what does $\int_a^b f = 0$ mean? It is the integral of $f$ from $a$ to $b$, but is it the Riemann integral, the Lebesgue integral, the Darboux integral? Does the difference matter?
What exactly does $f = 0$ mean. Does it mean $f$ as a function is the same as $0$ as a function, or does it mean $f(x) = 0$ for all $x$? Are those two the same or different? Does the distinction matter?
All of these questions, show up after a computer has correctly parsed the sentence and identified the structure. However, just that step itself is a very hard natural language processing problem.
In fact, if someone comfortable in reading math was given a random theorem from a random paper and was told to explain what it means, they would likely have problems and would need to look at the context of the paper (or reference other papers) to understand the syntax. This just gives a feel for the difficultly to build a machine learning model.
That is why I decided to build a custom language. In MathLingua, for example, the theorem could be described as:
Theorem:
. for: f(x), a, b
where:
. 'a, b is \real'
. 'a \leq b'
. 'f is \continuous.function:on{\closed.interval{a, b}}'
. 'f is \nonnegative.function:on{\closed.interval{a, b}}'
. '\riemann.integral_a^b[x]{f(x)} = 0'
then:
. 'f is \identically.zero.function:on{\closed.interval{a, b}}'
For example, one could have the definition:
[\identically.zero.function:on{A}]
Defines: f(x)
assuming: 'A is \set'
means:
. for: a
where: 'a \in A'
then: 'f(a) = 0'
Metadata:
. written: "f? = 0 \textrm{ on } A?"
Other definitions can be written too, to make everything clear (or left for later. MathLingua doesn't require you to define the entire math universe before defining a more complex concept).
My response was a little long, but that is because there are a lot of subtleties that show up when trying to get LaTeX to work for describe the meaning of math statements. I hope this helps.
The dot indentation syntax seems very strange. Why not just indent? Is the intention to reduce nested indentation a little?
The dot space is used to mark arguments since some arguments span multiple lines but are connected. For example, in
Theorem:
. for: x
where: y
then: z
. 'some statement'
Theorem:
for: x
where: y
then: z
'some statement'
His descriptions of what he wants to attain align with what I would like, a formalized way to describe what is known in mathematics.
He also describes how this can be used for machine learning, and area that I am also pursuing, but my results are still in an exploratory state.
Either way, I am really glad others are looking into this area too. I think it is something that will be very beneficial to the math community.
1. The syntax is horrible (for writing by hand).
2. I'm more interested in computation than presentation, so something closer to S-expressions along the lines of Mathematica feels more comfortable to me.
3. The progress so far on defining semantics for content MathML has failed to impress me.
Concretely, I think if you are trying to do semantic mathematics, you need an implementation of the semantics in software, not just a hand-written spec. Ideally this would be backed by a formal theorem proving system. So far I just do randomized testing using symbolic evaluation and interval arithmetic. See this writeup for more details: https://arxiv.org/abs/2003.06181
The goal of MathLingua, at this time, is to match such a level of formality that is in math books and papers.
However, the syntax was designed to allow support for transpiling to a theorem proving language to be added at a later time. It is something, in the future, that I would like to do, if possible, but I don't have a time estimate if/when it will be available.
Doesn't answer the "why" question at all.
Theorem: “Fundamental Theorem of Calculus”
That is,
Theorem: "some text"
Theorem:
. "some text"
Theorem:
. "some name"
. "statement 1"
Theorem:
. "statement1"
. "statement2"
Regarding a comment on possible use cases.. One component that would really compliment this would be a programming-language specific checker of definitions. In the documentation's example of "integer" it would be straight forward to write in python a function that checks if a given variable is an integer. The same can be for other definitions such as "invertible matrix" or "set of orthonormal vectors".. other languages could provide such checks for more symbolic stuff like "commutative ring" etc.
Once you have this in place, you can load your nice data into a (e.g.) numpy.ndarray object, and checks would be triggered for all possible properties. This would automatically tag your data as, e.g. "stochastic matrix" and then you could even get suggestions of the sort of "build a time-discrete stochastic system based on this matrix?". This is would be very helpful for "data scientists" who are a bit far from theory.. at least I know I would use this very often.
There is of course the issue of expressivity to be taken into account. Much of mathematical progress involves invention of new notation, not all of which is neatly written in "left to right English with only binary operators" (think of commutative diagrams for example). While the language can of course be expanded in this direction, there will be a very big temptation (and I would guess frequent pushes from the future community) to change to more and more general languages. In this case, please always keep in mind the very expressive but useless notation of Russell & Whiteheads's Principia. There is a big theory of language and expresivity, and there are also practical examples (think of OWL or other semantic-web languages) of how to reach a compromise that allows for enough expressivity for the use cases of interest.
Everyone is welcome to join.