The most obvious politician's argument against funding research in pure mathematics is that it costs money and it isn't useful. I'm aware that there is a point of view that holds that the state should not fund research of any kind, but it is very alien to my milieu, so I won't presume to explain it. As I'm closer to mathematicians than to politicians, the bulk of my answer will take their point of view.
The very definition of research is that you don't know in advance what you'll find. If you take researchers and tell them “invent something useful”, what they'll only find very small increments to what already exists. The only way to make significant advances to science is to have enough researchers and tell them to search in whatever direction they feel their efforts will lead to something. Research is like art production in that way: you can't have all books be bestsellers.
Even mathematicians themselves cannot reliably know how useful their work will turn out to be. This is true not only while they're searching, but even after the fact. G. H. Hardy was famously proud that his work in number theory was particularly pure mathematics, where pure is to be understood as both noble and useless. Yet some years later number theory turned out to have important practical applications to cryptography, with considerable military and economic importance.
Additionally, science proceeds faster when scientists share their ideas. (I don't have a citation for that, but it is a commonly shared sentiment among researchers.) In order to maximize the productivity of scientists, there has to be some who do work in more theoretical fields with no obvious applications, and who are there to discuss with more applied scientists, teach them, validate or refute their ideas, etc. Diversity pays.
This drives the conclusion that in order to be efficient, research requires a large enough number of cooperating researchers, at least a fraction of which have no imposed goal. This requires a sufficiently large organization that applies policies that encourage sharing rather than competition and do not cull the least productive units. This is better suited to a state than to a private enterprise (but it can also work in a patronage system such as practiced by private universities).
On a different track (but related to the remark above about cooperation between ivory-tower scientists and economic-value scientists), having a reputable pure mathematics department attracts students and fellow researchers. The best students tend to be attracted to the best professors. Some of these students will end up being pure mathematicians themselves, but others will produce economic value in a more direct way (applied researchers, engineers, …). In terms of economic competition between countries or other environments, there is an advantage to being the place with the best mathematicians.