Ernst Mayr in his last book titled "What Makes Biology Unique?" argues that many of the theories in biology do not need any mathematical support. He says that much of biology is only conceptual and cannot be describe by mathematical formulations. In the meantime, he also argues that these properties of biology does not prevent this field of knowledge to be considered as a science.
Here are a couple of relevant quotes:
Mathematics remained the earmark of true science. Kant certified this opinion by saying “there is only that much genuine [richtig] science in any science, as it contains mathematics.” And this greatly exaggerated evaluation of physics and mathematics has dominated science until the present day. What would be the scientific status of Darwin’s Origin of Species (1859), which contains not a single mathematical formula and only a single phylogenetic diagram (not a geometric figure) if Kant had been right? And this was also the philosophy of science of the leading philosophers (e.g., Whewell, Herschel) that affected Darwin’s thought (Ruse 1979). Yet several recent philosophers of science have published science and sciences a Philosophy of Biology strictly based on the conceptual framework of the classical physical sciences (e.g., Kitcher 1984, Ruse 1973, Rosenberg 1985) while ignoring the autonomous aspects of biology (chapter 2).
The philosophers of logical positivism, and indeed all philosophers with a background in physics and mathematics, base their theories on natural laws and such theories are therefore usually strictly deterministic. In biology there are also regularities, but various authors (Smart 1963, Beatty 1995) severely question whether these are the same as the natural laws of the physical sciences. There is no consensus yet in the answer to this controversy. Laws certainly play a rather small role in theory construction in biology. The major reason for the lesser importance of laws in biological theory formation is perhaps the greater role played in biological systems by chance and randomness. Other reasons for the small role of laws are the uniqueness of a high percentage of phenomena in living systems as well as the historical nature of events.
Owing to the probabilistic nature of most generalizations in evolutionary biology, it is impossible to apply Popper’s method of falsification for theory testing because a particular case of a seeming refutation of a certain law may not be anything but an exception, as are common in biology. Most theories in biology are based not on laws but on concepts. Examples of such concepts are, for instance, selection, speciation, phylogeny, competition, population, imprinting, adaptedness, biodiversity, development, ecosystem, and function.
(sections from pages 14 and 28)
Is it possible that a scientific theory cannot be supported by any mathematical formulations? Or stated differently: Can a field of knowledge be a field of science if it does not contain any mathematical formulations?