My suspicion is that most people who don't like philosophy have lots of philosophical ideas - they are just really dogmatic about them and don't like to be challenged.

Philosophy is like maths. You can build a lot of bridges with a crummy understanding of maths. The Romans did it with the most pathological notation for numbers imaginable, and lacking all sorts of basic mathematical concepts. So you can say all that maths stuff is just nonsense and you can do it all by these seventeen-hundred ambiguous rules of thumb you really carefully follow. It's just you're not actually avoiding maths - you're just doing it in a really ad-hoc, inflexible, and inelegant manner. That's what people are doing when they say philosophy is a load of bunk but they still believe a whole load of things about the universe.

Here's a paper by Tarski, widely cited by both mathematicians and philosophers and containing both formal and informal reasoning: http://www.thatmarcusfamily.org/philosophy/Course_Websites/R... I don't know how one could "remove the philosophy" from this work without making it far less useful to mathematicians. The entire reason the T-schema is used in model theory is because of Tarski's philosophical argument that it provides a meaningful definition of truth.