Classic question on the lines of "someone should do something", and easily incorrectly answered by an argument of why the respondent would do something.

The question is about a "taxpayer or a private organization ". The latter is vague; it could be just because the private organisation is a foundation chartered to fund mathematics.

But, if by "taxpayer", we really mean a government, I will assume to start with that we mean government of a rich country.

Let's look at a few different departments:

• Science, let's ask Roger Bacon:

"“If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.”

This has been true throughout the history of science and remains true today; genetics is pushed forward by mathematicians who may not have originally commenced their career in that area at all.

• Industry

The modern information economy is based on technologies whose key founders such as Turing and von Neumann were mathematicians. In the first instance their research (prior to the war, see below) would surely be classed as pure rather than applied. In addition of course, countless areas have made use of applied maths.

• Arts

As covered elsewhere, mathematical truth is beauty.

• Defence/Defense/War or however referred to in the said country.

Maths had a major role in WW2 in both code breaking and in the modelling supporting the Manhattan project, as well as many other areas. In the future, code breaking is even more important as it underpins more and more of our infrastructure. In the US, how does the NSA's budget compare to federal funding of mathematics? To misquote Dr Strangelove, can any country really afford a "maths gap"?

• Finance/ Treasury/ Economics

Well maybe most Western governments could use a few more mathematicians there?