Based on what I’ve read, and I hope I got it right, continental philosophy is a loose term for the different schools of thought that resist the influence of the analytic tradition. Critical theory, for example, which is also known as anti-positivism in the social sciences, argues for a more qualitative and pragmatic research methodology.
I wonder if I would have a better chance of avoiding mathematics if I would concentrate on a less mainstream school of thought.
You need to know absolutely no logic, philosophy of language or mathematics to understand Heidegger, Derrida, Foucault, or most of the other "famous" continental philosophers. It would be hard to read the early Husserl without some logic, however.
But why shut yourself off from reading non-continental philosophy? The logic required to read most papers contemporary analytic isn't hard, although it requires some effort and discipline to learn. (Some specialist areas in metaphysics, philosophy of language and formal epistemology will require more advanced logical knowledge.)
If you have the wit and intelligence to absorb Derrida or Heidegger then you certainly have the wit and intelligence to absorb mathematics or formal logic. One doesn't have to absorb all of the technical apparatus or machinery to become cognisant of its value or to appreciate how it works. Certainly one thing that militates against it is poor exposition; but this charge can be equally put towards learning what Heidegger or Derrida is about - or even Kant.
One does not have to understand how to solve a PDE, a differential equation or the technicalities of Celestial Mechanics to understand the ontology of Physics, or the niceties of Model Theory to understand Gödel's Theorem or is importance in the notion of truth in its analytic guise.
One might note here that even very good mathematicians are aware of the huge investments of time and effort that it takes to learn mathematics properly (for example Vladimir Voevodsky - a Field Medallist said exactly this in an interview) and what it means for the continuation of mathematics as a tradition; but then they are in the business of creating and advancing mathematical understanding and that is a very different proposition from philosophy.
Simone Weil, who I would place in the continental tradition, for example had a good understanding of mathematics (her brother was a famous geometer) but in her work there is no mathematics in the mathematical sense; in the sense of rigorous thinking, yes.
High school/college mathematics give very little indication of what mathematics or physics is; it downplays the role of insight and emphasises solving - this is the order of an algorithm or a recipe: One does something in a certain order and one obtains a certain result.
Here is a passage in Liberty & Oppression by Simone Weil that demonstrates her understanding of the sciences in making her argument:
Nature, which is a mirror of the divine truths, offers us everywhere an image of this paradox. Catalysts, bacteria are examples of it. Compared with a solid body, a point is something infinitely small. Yet, in each body, there is one point which predominates over the entire mass, for if the point is supported the body does not fall; that point is the centre of gravity.
But a point thus supported only prevents a mass from falling if the mass is disposed symmetrically around it, or if the asymmetry in it has certain proportions. Yeast only makes the dough rise if it is mixed with it. The catalyst only acts when in contact with the reactive elements. In the same way there exist certain material conditions for the supernatural operation of the divine that is present on earth in the form of something infinitely small.
The wretchedness of our condition subjects human nature to a moral form of gravity that is constantly pulling it downwards, towards evil, towards a total submission to force. “And God saw . . . that every imagination of the thoughts of his [man’s] heart was only evil continually.”
It is this gravity which forces man, on the one hand, to lose half his soul, according to an ancient proverb, the day he becomes a slave, and, on the other hand, to command always, according to the words quoted by Thucydides, wherever he has the power to do so.
In the same way as ordinary gravity, it has its laws. When studying them, one cannot be too cold-blooded, lucid, cynical. In this sense, to this extent, one must be a materialist. However, an architect not only studies falling bodies, but also the conditions for equilibrium. The true knowledge of social mechanics implies that of the conditions under which the supernatural operation of an infinitely small quantity of pure good, placed at the right point, can neutralize gravity.
Those who deny the reality of the supernatural truly resemble blind men. Light, too, exerts no pressure, has no weight; but by its means the plants and trees reach towards the sky in spite of gravity. We do not eat it; but the seeds and fruits that we eat would not ripen without it.
Here, she mixes what would now be called Newtonian Mechanics & Biology, but for the purposes of this passage would be better called what Newton called it - that is Natural Philosophy - the Philosophy of Nature in its concrete manifestations; it suggests given the division between continental & analytic philosophy that it is this tradition that needs reinvigorating.
There's a lot of things that go by the name "continental philosophy." Critical Theory is probably one of the ones were knowledge of basic formal logic would be useful. For some of the other more famous figures in contemporary continental philosophy, you don't need to know any formal logic or math.
To be honest, depending on what you mean by "math", you don't need to know much of it to do most philosophy -- continental, analytic, personalist, or any other contemporary school.
Regarding your characterization of critical theory as seeking a more "pragmatic research methodology," I am not so sure that's an accurate or fair characterization. I might say it seeks a more ideologically-driven research methodology than the status quo. And pragmatism admits at a minimum two diverging definitions -- (a) closer to Pragmatism (the American philosophical school) and (b) practical.
But seriously, it's unwise to restrict your readings to just continental philosophy -- or even just analytic philosophy. Read good philosophy that makes solid arguments. Much of the best analytic philosophy is informed by the history of philosophy. And much of the best continental philosophy is in dialogue (and I don't mean the way Derrida was) with contemporary analytic philosophers and the history of philosophy.
As an addendum, I write this not as someone unfamiliar with continental philosophy but as someone who earned an undergraduate degree in philosophy from a US institution that included continental philosophy in what was taught. I then earned an MA from a continental/historical program in the US and finally a PhD from a program strong in continental writing under an advisor who is moderately well-known in continental circles.
While I don't make contemporary continental the primary focus of my research at this point, I do have multiple published articles that compare 20th century continental thought with 19th century continental thought...
I strongly advise against neglecting other forms of philosophy and against not developing your skills in broad areas of philosophy.
or to word this in a completely different way -- please look up what the big continental figures actually studied when they were students. E.g., Martin Heidegger studied John Duns Scotus. Foucault and Sartre both studied Hegel.
If you are going to a university in the English speaking world, enrolling in Philosophy proper usually does mean a concentration in analytic philosophy; this does not mean it is impossible to get a schooling in continental philosophy that way, but doing so will make you a bit of an outcast and will not get you out of fulfilling the basic requirements of the program which will include formal logic.
At the undergraduate level, offerings in things such as Critical Theory are more likely to be relevant in Humanities and Sociology. Most of the foremost thinkers in 20th century continental philosophy came out of places such as the École Normale Supérieure or the Frankurt School; a similar line of study here would be considered interdisciplinary or (again) Humanities, and less easily satisfied by the Philosophy Dept.
I'll restate what others have said, however, about being wary of avoiding maths for avoiding maths sake. You do not have to go toe-to-toe discussing calculus with physicists, but if the fundamental reason is you lack confidence in your ability to solve quadratic equations etc., and/or you you think such is not interesting enough to merit your time, you are giving up somewhat on your ability to think about and learn fairly fundamental things.
