I voted Maths professor, for the following reason:
A bin man is certainly useful within society and a necessity, but they do not advance society. If everyone only performed the tasks necessary to run society and not advance it, then we'd still be the same society we were hundreds of years ago. Maths however progresses society, it allows us to move from one stage to some more advance stage which is incredibly beneficial. Think of the things we have to help us now compared to hundred years ago. Computers, mobiles, the internet, medicine etc etc. A lot of major technological advances stem from mathematical theorems developed by theorists which while at the time seemed irrelevant, but become incredibly useful when someone figures out how they can apply to them create some new technology.
Some examples: Non-Euclidean geometry: Developed in the 19th century. Seemingly completely useless to the real world. Developed and incorportated by Einstein into the theory of General Relativity. Now essential to GPS, a system used in many, many technological devices. Also essential to calculating space flight trajectories.
Fourier Transform: Developed in the 1700s. Allows for a mapping of some variable function into a new domain, along with the representation of any function with an infinite series of sins and cosine functions. Now used in practically every area of communication to condense information into a sendable form.
List of applications: Fourier analysis has many scientific applications – in physics, partial differential equations, number theory, combinatorics, signal processing, imaging, probability theory, statistics, option pricing, cryptography, numerical analysis, acoustics, oceanography, sonar, optics, diffraction, geometry, protein structure analysis and other areas.
So yeah, without the mathematicians, the world would be a lot less advanced than it is now.