I’ve written about this topic before with specific reference to economics; this is more of a general piece. There seems to be a fundamental confusion among intellectual types regarding the epistemological usefulness of mathematics. Epistemology just means ‘relating to knowledge’. Maths provides a very pure kind of ‘knowledge’ in the sense that if you arrive at a mathematical proof then it is basically beyond doubt. Unfortunately, we experience reality inductively, not deductively, and we want knowledge of reality not an abstract mathematical space. That means that you have to step away from mathematics the instant you want to get knowledge about reality, and things get messy. I get a little shitty with people who bang on about the purity of mathematics, because pure maths produces, in a sense, useless knowledge.
Deduction basically involves following logic chains to arrive at statements that must follow from prior statements. For example, if all four-legged animals are cats and Betsy has four legs then she must be a cat. Seems pretty straightforward. However, a problem arises where we think about where we get our original statements from. In the example above, how do I know that all four-legged animals are cats? Following logic chains backwards you inevitably end up in one of two places. You might arrive at an axiom, which is a fact that you take on assumption. The classic example is that a line is a connection of points in a space that have no space between them. This is actually an impossibility, because mathematical space is infinite, but we can agree on it so that we can produce Euclidean geometry from there. The second place you might arrive at is an empirical statement. For example, I have only seen white swans, therefore all swans are white. Then of course you visit Australia and discover that there are black swans.
So either you make assumptions or you make empirical claims. Assumptions in practice tend to involve a simplification, like ‘people have transitive preferences’, which takes you away from reality, but might then allow you to develop a model of reality that allows for a very precise and clear articulation of a theory that you can then precisely test empirically, like demand curve theory. Much of mathematical economics has gone over into the realm of making completely unrealistic assumptions that also don’t produce models that approximate reality, like the overlapping generations model, which to my mind proves literally nothing and clearly doesn't fit reality, or the Euler equations, which most macroeconomists seem to want to get rid of. But somehow these mathematicians still feel superior because what they’re doing is ‘pure’. It’s also of very, very limited value.
Empirical claims are more interesting. The issue here is the problem of induction, which is brilliantly captured in the black swan anecdote (fully explored by Taleb in book of the same name). If you drop an egg on the floor a thousand times and it breaks every time, that doesn’t mean that on the 1001st time it won’t float up to the ceiling. Maybe you just haven’t seen enough of the probability distribution to witness a tail event. When you have a fact, like all swans are white, you can never be certain that this fact is true. The best you can do is follow scientific praxis, which is to rigorously test your hypotheses and reject them if they are refuted by empirical observation. Rigor involves having a lot of observations, hence the importance of replication in science. If a hypothesis passes the tests then you can treat it as a fact until it is refuted down the road, the way Newtonian determinism was.
Once you have some induced facts you can use deduction to develop the next round of hypotheses. For example, armed with an understanding of market dynamics, I can hypothesise that putting a price on carbon will reduce carbon emissions. Lo and behold, the carbon tax worked! Maths, mostly in the form of linguistic logic, is critical in these formulations. But note that I don’t actually get knowledge until I do an inductive empirical test of my deduced carbon tax theory. And note that this knowledge is in the form of facts, not truths.
The other way maths enters the inductive process is through probability, which is a mathematical concept. Remember the egg-dropping example? Whether or not you have seen the entire probability distribution of a phenomenon and the shape of that distribution is extremely important for claims about whether a hypothesis has passed rigorous testing. I won’t go into more detail here. If you’re interested, read Taleb or do a basic econometrics course.
So maths is extremely important for knowledge, but this superiority complex that pure mathematicians have viz scientists is bullshit because maths and science occupy themselves with two different ways of knowing (deduction vs. induction). Comparing pure maths to biology is meaningless, unlike comparing modern empirical psychology to pre-empirical claims about psychology, like those of the Pagan philosophers. Pure maths doesn’t get you knowledge in any meaningful sense because you either need to make axiomatic assumptions or rely on empirical facts that can’t be arrived at using pure maths.