Almost every discipline in humanities and science requires some level of math. It seems to me that math majors can specialize in any field they like, while people who are good with words are stuck with editorial or journalistic jobs. I’m not trying to make an unwarranted dichotomy between linguistic intelligence and mathematical intelligence. I’m quite sure that mathematicians are good with words too; otherwise they won’t be able to explain their ideas in a persuasive and elegant manner. What I’m trying to say is that many writers are just not interested in math, to say the least. But I don’t think this should hinder them from studying our world the way scientists do. Physics, Chemistry, and Biology, which are by far the most mathematical disciplines, may not be for someone who loves words but hates math. But there must be some other objective disciplines (maybe in the social sciences or law) that a writer can specialize in.
The problem is that if you want to be objective, you usually have to present hard numbers. Which means that at a bare minimum you need to know some very basic statistical analysis to determine if your numbers are actually interesting. (To be nitpicky: this is calculations, not math. Everyone can learn to do calculations, there is no special "mathematical intelligence" needed. Not everyone can be Paul Erdős. Anyone can learn to carry out a t-test with the help of a computer.)
As a mathematician in academia, I can't afford to say that writing is boring or teaching is boring. It's a part of my job, and if I don't enjoy it, I shouldn't be in academia. You seem to want to work in a "hard" discipline, but without dealing with the numbers that make it "hard". The world doesn't work like that.
Mathematicians in general might not be good at writing, but the ones that are also successful at communicating their ideas and hence successful academics most definitely are good at writing. They didn't write annoyed posts online about how it's unfair that people don't just understand what the meant: they spent some time and learned it. A mathematician who wants to be an academic has to learn to communicate efficiently. A writer who wants to study our world like a scientist has to learn to use the main tool scientists use: math. If the write is not interested in math, maybe they should reconsider going in to a scientific field. It's like a scientist saying that they want to be a writer, but they don't want to deal with pens or keyboards. Science without math is no longer science.
I'm only starting out as a grad student, but one of the things I've learned about academia, and one of the reasons I wanted to be in academia in the first place is that it's an extremely demanding, varied job. A good academic is excellent at doing research in their area, and then very good at other things. These other things may include writing, teaching, mentoring students, coding, and analysing data, depending on their field. (Interestingly enough, I think the only field where you can get away without doing any data analysis ever is math. I did a poll in my office, and only 4 out of 40 grad students in math could carry out a t-test off the top of their head). You can't just pick a single thing to be good at: you need the whole package. That includes some math skills if you want to work in an objective discipline. Suck it up and learn some math. Math is great.
Your question answers itself in the last sentence: law and the social sciences could be a good fit. Of course you should probably stay away from tax law if you want to avoid mathematics entirely, but most legal studies don't involve mathematics. Quantitative work in the social sciences involves statistics, but there's plenty of qualitative work (e.g., based on ethnographic methods). And I don't believe your assertion that mathematics is widespread in the humanities. What about history, for example? Mathematical techniques in the humanities are the exception, not the rule.
There's no way to make a comprehensive list of topics that don't involve mathematics. If you're considering your future academic career and worried about the prevalence of mathematics, you could address this fear by looking over the books or papers of potential advisors. Someone who uses lots of statistics or formal mathematical models wouldn't be a good choice if you don't like that approach. If you can't find anybody who takes a qualitative approach to the topics you are interested in, then you're looking at the wrong department or wrong field for you. If you can, then you have a potential path forwards.
But let me put in a plug for learning a little statistics. If you avoid statistics because you don't understand it, then that will limit the scope of your work. If you avoid it because you don't like it, but you can handle it when necessary, then you'll have more flexibility and freedom. (And in practice I've talked with many more people who regret not having studied statistics than people who regret having wasted time studying it.)
This probably isn't the answer you're hoping for, but hear me out. As a scientist with a healthy interest in the humanities, I think it's a bad idea to limit yourself from vocations that remotely involve math. You would be surprised how little mathematics (read: work) is needed to have a relatively good understanding of the physical world.
As such, highly recommend for you to check out the book: A Mind for Numbers by Barbara Oakley. It is written by a linguist who hated mathematics and science until she decided to give it a fresh shot in adulthood. Don't fear, it doesn't aim to make you a theoretical physicist or a mathematician.
The world is only going to continue becoming more scientific minded, and in my opinion, it'd be good for a writer to plan for that direction :)
Good luck !
As a social scientist my academic research is generally mostly qualitative and in social sciences such as socio-cultural anthropology and qualitative sociology, you can get away with doing 0% math.
However, I also complete pro-bono/paid contracts for applied social research, in which some basic quantitative skills (math) are required. I have no experience in working with large data sets or 'big data' but I know enough to conduct small-scale surveys and build on these with qualitative analysis. Currently, one of the projects I recently completed is being turned into a journal article, where my co-author will be helping out with the math portion (survey results).
Really, the only profession other than writing novels/literature that I can see a 100% absence of math would be something like visual art, and even then, there's a 'kind' of math involved in understanding how to use different materials (such as painting/mixing with other types of chemicals such as oils etc). Even music and dance are highly reliant on a form of math.
I have a dyscalculia which is a learning disorder around math and numbers, so I can 100% empathise with a hatred for the subject. I quadruple check all equations etc and have a really good handle on excel to calculate things for me and get my partner to check for me as well, whose pretty good at math.
But I'm not letting this disorder get to me, and I use math constantly for marking, my research and so on, I just don't specialise in that area. The minute a project calls for a high level of stats/data/quant, I get someone else involved.
As others have said, math is everywhere and you can't get away from math, but sometimes it can be helpful and doesn't require intense equations or the like. Just having a basic handle on it is enough in many disciplines!
While it is understandable that you would not want to work with hardcore maths if you don't like the subject, I think you need to get over the mental block that you cannot handle anything even remotely connected to mathematics. Most humanities subjects, apart from economics, geography, geology, or modern philosophy would not require a high level of mathematical application. Subjects within the social sciences, such as sociology, history, politics, etc. might work well for someone who can write well but not want to study mathematics. However, if you completely shut yourself off the moment you see even a simple calculation, your choice of subject might become very restricted.