The Map of Mathematical Models
A referencible repository of mathematical models.
While number theorists had catalogued the integer sequences by building "The On-Line Encyclopedia of Integer Sequences" - oeis.org (so today, we can say "Positive Integers" is the ID: A000027, or "Non-negative integers" is the ID: A005843, and so on), the rest of mathematics, physics, chemistry and lifesciences don't seem to have something like that. Even biologists have catalogued each gene ID, but there seem to be no IDs for the important relationships like "Taylor's expansion", "Ohm's Law", "Bayes theorem", "Combined gas law", "Pythagora's theorem", "Relativistic rocket equation" or pretty much any other important mathematical relationship in sciences, there's no ID or a coordinate for this or that formula. A little more about this problem is described here, in the issue of Cataloguing Mathematical Models.
The below is one possible way that we could start creating a map for mathematical formulae (or let me know if I'm oblivious to it, and such a map already exists).
The Map of Mathematical Models
The idea would be to introduce such coordinates and such a map for mathematical functions and equations as models: an index of mathematical models with implementations in various computing languages and CAS systems, or at least expressions in common mathematical notation, that can be automatically rendered to the notation of multitude of target CAS languages. It could be like oeis.org ("The On-Line Encyclopedia of Integer Sequences®"), but unrestricted to integers.
Considerations for Realization
1. Reuse of OEIS.org
This could work as an extension of oeis.org, but with new letters besides letter
A. We could give out different letters for new classes of models, like, e.g.,
C for continuous functions,
D for differentiable functions,
E for systems of equations, etc., defined on various groups and fields. A thing to notice, is that the integer sequences already define a good deal continuous functions, - all you need to do is to look at them as defined on field of reals, complex, hypercomplex or other fields of numbers, and there's no need to repeat the same entries in such an index. For example, if we have the definition for squares
a(n)=n^2, which defines
1, 4, 9, 16,..., it already defines the quadratic function, all you need to do is replace the
n with continuous
c(x)=x^2. It is not always trivial reuse of expressions though. For example, Gamma function and factorial are closely related, but arguably, would merit separate entries.
2. List all representations
To be able to find the formulae just like we can find the sequences of numbers, we'd have to have two kinds of search: 1) by entry of symbolic formula, 2) by entry of list of points to find models by fit.
For the symbolic search, each entry would have to implement the search by a graph of relations that each formula defines. For example, while the "Ohm's Law" defines a hypergraph of atomic points [I, V, R], connected by hyperedges that summarized all alternative forms of formula:
R=V/I. In essence, such a hypergraph defines an equivalence class of parse trees.
When phenomena are described by multiple equations, like in the case of Maxwell's equations, Einstein's field equations, or other, we would treat the entire model as a single entry in the database, and the gaps between the equations as "AND" operator, collapsing them into one complex model, but each of the equations may have their own separate entries.
Perhaps though, we should not call these relations by name of physical phenomenon that they describe, because, for example, "Ohm's law" is so simple, that you'll find a lot of other things in nature behaving that way. For example, the "Combined gas law" is just like "Charles's law" (
V=kT), which is just proportionality to (a product of) two quantities. There are thousands of phenomena from physics to social sciences that follow the proportionality. In such compendium of mathematical models we would simply mention that there are these known laws that follow such proportionalities, giving each one a section with the choice of letter notations and restrictions to domains and ranges of each variable, known based on scientific knowledge.
3. Technical implementation:
One way of doing it would be to collaborate with oais.org, who may provide the necessary expertise and mathematical-social network effects. On the other hand, doing such a thing as a brand new initiative, we could reuse GitHub's wikis functionality as backend, which would enable easy participation as well as long-term sustainability, because GitHub is used very widely across software industries, and is a piece of technology that is unlikely go down because of wide use across broad range of industries.