As a pure mathematician, whatever field you turn to, you find a vernacular
that
people use -- be it physics, or economy -- the various concepts
from temperature to assets price, have rather specific definitions,
in context of specific instrumentation and accounting standards.
(Measuring asset
price, needs equilibrium of supply and demand
of your asset in some specific market. Measuring temperature
may need equilibrium of energy exchange with your mercury
inside the thermometer).

Suppose you want to stay at abstract level, and not load
your brain with all that vernacular (like "Market price,"
"Temperature," etc.), and just want to model input-output
scenarios to apply decision theory to decide what to do and
what's most valuable, and yet communicate with the society, in a
language that they immediately understand.

For example, business valuation layer of society may rely on
vernacular, with words like EBITDA, CAGR, NOPAT, NWCTR,
FATR, FCFF, NIBD,.. etc. which are very specific terms, in
context of specific accounting systems, and they **mean** very
specific things, like acceleration to a physicist, that doesn't want
to think about each molecule, because their brain has layers of
neurons trained to take those metrics as input to make decisions.

Being that these characteristics are just derivative features of
Input-Output processes, and you have input-output data, you can
compute them, no problem. You just need a convenient reference
to what are the concepts that a target population makes
decisions with, and apply them on your data types.

A library, that takes a Wikipedia page URL, and returns a formula
corresponding to the index of that formula, with well-defined
domains and codomains, so you can conveniently use it in your
CAS system, or your programming language.

NOTE: Could be not limited to Wikipedia, but with capability to support
many other URLs with computational formulae described in a less
unstructured, less rigorous way.

// So - does your “mathematician” in this case understand the concepts
of other disciplines or are they simply manipulating symbols?

Well, the idea is pragmatic, in a sense, that mathematician does
understand some low level concepts, like data record and basic data
types, which gives her the ability to compute arbitrary sums and
differences, like cumulative sums, matrix products, differentials, etc.,
which is enough to compute the values for concepts that others care
about, if she has the definition of that concept provided in a form of a
formula or equation.

So, basically, she would compute temperature, just give the records of
particle movements, and tell what temperature definition is. So, when
you use that library, and do something like:

And you had a record of Brownian motion, and associated particles over
time, in form of a database of atom paths over time, with sufficient
granularity, you'd be able to compute it from the definition, without
having to go through and interpret other literature.

So, with this library, we're talking. You giving me the link to very specific
concept that your talk needs, and I give you the value for that concept,
with respect to a data sample.

So, if I understand this correctly, if I was a human computer and
someone asked me to solve a fluid dynamics problem with some given
data, could ask your library system for the formulas and solve the
problem using them. But I wouldn't have to know what the input or
output data means, right? I could use the formulas just the same, and
get the right answers, without even knowing what fluid dynamics is. So
this seems conceptually equivalent to a collection of downloadable
algorithms to run on an electronic computer.

// formula corresponding to the index of that formula //

Hang on; is that a typo, or does this idea depend on either (a) the
prior construction of an infinite collection of formulae referring
to each other without resolution or (b) some entirely vacuous
formulae which resolve to their own addresses?

Have you consulted Hofstadter? If not, I would urge you to do
so.