As a discipline philosophy in many ways seems like a hybrid of math and science. Like math, philosophy often is conducted through “proofs”, which in philosophy are attempts to show that a premise follows from certain assumptions. Like science, philosophy has the world we experience as its subject matter, and its results are supposed to be relevant to that world. So philosophy is unlike math because it is about the world directly, while mathematical ideas must be shown to model some physical process, by experiment, in order to tell us anything interesting about the world. And unlike math there are no universally accepted rules for constructing a philosophical proof. The lack of such a formal structure is not a weakness of philosophy, but a necessary consequence of its subject matter, since what we can deduce from premises, and how those deductions can be made, is something that must be argued for. And philosophy is unlike the sciences because it deals with subjects that are not directly available to experiment. How could physics ever address what a cause, in general, is?
By thinking of philosophy as between math and physics some inadequate approaches to philosophy stand out. One such failed approach is conceptual analysis, and some versions of natural language philosophy. These approaches take philosophy as a study of our concepts or our words as we use them. Thus a philosophical theory about causation under one of these approaches would be considered successful if it succeeded at reflecting our idea of causation, or how the words associated with causation are used in our discourse. Let us assume that such an account is possible (although the differences between different people in their use of abstract words such as causation, and their conceptualizations of those words, might give such a project difficulties). The problem with it is not that it is “wrong”, given the standards it is holding itself to, but that it is too much like math and too little like science. Such an approach has abandoned the idea of having the world as its object, excepting of course our ideas and language themselves. When we wonder about what causation is we don’t want to know more about our idea of causation, we want to know more about what it really is, and thus we must be open to the idea that our current ideas about it may be mistaken.
Another problematic philosophical approach is taking everything as it seems. The phenomenologists as a whole might be accused of having such an approach, but the philosopher who springs to mind first is Heidegger. In his book Being and Time Heidegger attempts to explain being in the world as we experience it. And in his book there is little argument for his ideas, instead he presents his observations about the world as though they should be obvious when pointed out. And perhaps they do agree with our experience, even if not all the distinctions he makes seem natural. The problem with such approach is, again, that it lacks the necessary scientific character. Skepticism plays a central role in both philosophy and science, as we wonder if things are really as they seem, and thus are motivated to uncover the truth behind our experience. So the approach taken by Heidegger (and a number of other philosophers too) is problematic because it isn’t sufficiently skeptical, and thus doesn’t attempt to approach reality. And again such theories may be valid descriptions of experience, but such descriptions aren’t what we want philosophy to produce. When we wonder about causation we don’t want to know about our experience of causation, or how causation seems to us, we want to know what causation really is.
Finally there are thought experiments / intuition pumps. Not all thought experiments are bad, after all they are used in science too. In general though thought experiments should motivate our initial assumptions or be provided to make a conclusion argued for in other ways seem more acceptable. By themselves thought experiments aren’t a valid way to argue for a conclusion because they often revolve around our pre-existing intuitions about the subject of our investigations, and as pointed out above our intuitions may not always be correct. They are allowable only to justify initial assumptions because assumptions are just that, assumptions. This means that it is always possible to challenge the validity of the assumption, and thus the thought experiment. This is why the Chinese room / Chinese nation thought experiments are poor arguments for biological naturalism. If one puts those thought experiments forwards as an argument for biological naturalism then what is really being done is simply asserting biological naturalism as an assumption. But this is not what those arguing for biological naturalism take them to be, they take them as arguments that show machines cannot be conscious, and this is a mistake. So, while the previous mistakes were examples of being insufficiently scientific, this mistake is probably best thought of as being insufficiently mathematical, and not providing a sufficient argument for the conclusions.
They say describing philosophy is one of the hardest tasks of philosophy. Of course this may be because it usually isn’t a primary concern to philosophers, who pass it by on the way to more interesting problems, with the exception of those who would like to define it so that they can prove there is no such thing (for example, the early Wittgenstein). But philosophy probably could benefit from a more rigorous definition of what is and is not philosophy, so that philosophy could adopt a more rigorous method, like math and science, in order to simply eliminate some kinds of bad philosophy.