In short, any student who emerges from university with good critical thinking skills has achieved that by accident, not by the university's design. I would also argue that the courses that come closest to achieving the ideal are history, English, and philosophy.
How about abstract mathematics? I would argue that these days those might be the only courses where you learn those skills.
I suspect the author would argue that training in abstract mathematics is comparable to the other sciences in the sense of 'data analysis within a framework', subject to the problem of under-training in the ability to recognize the limitations of the framework. I suspect s/he chose History, English and Philosophy because they are fuzzier than the sciences, including mathematics, and so they require understanding how to choose - or define - a set ground rules to apply in a given situation.
understanding how to choose - or define - a set ground rules to apply in a given situation.
Which, of course, perfectly describes abstract mathematics... just not the sort that non-majors ever get to.
Many of the interesting parts of mathematics (and, I should add, computer science!) are all about coming up with the mental framework and abstractions needed to look at a problem, give you a new perspective and ultimately come up with a solution. Choosing or coming up with the logic to treat a situation is exactly what you'd be doing.
It's also particularly good at illuminating the underlying structure of different domains and giving you tools to recognize similarities between them that may not be readily apparent. This goes well beyond what most non-mathematicians regard as "mathematically accessible": just look at all the various modal logics[1] both for the breadth of their applications (belief, knowledge, time...) and their internal similarities.
Moreover, I would argue that mathematics goes further than most fields in talking about and working with its own limitations. Metamathematics and logic as fields are all about this—with greater clarity than the fuzzier subjects, in the usual mathematical style. Mathematics is also great for consistently and rationally accounting for uncertainty and limitations of other systems, fundamental to probability and any sane treatment of risk.
Mathematics, as practiced by mathematicians, also teaches another really important idea: having different levels of rigor, as appropriate. Often, an argument only needs to be convincing, and many of the important details can be left out. However, in math, this is always done with the understanding that those details can be filled in, and the understanding that you should be able to fill them in given enough time.
I figure the focus on clarity, thorough introspection and different levels of rigor, abstractions and logic all make mathematics qua mathematics one of the ultimate "humanities". The problem is that classes for non-majors (and, perhaps, many undergraduate classes for majors) often don't teach mathematics for itself; instead, they teach mathematics as a tool for engineering or solving specific problems or getting a specific result. And that, certainly, could be fairly described as "data analysis within a framework".
Though I am no mathematician, I deeply respect mathematics, and appreciate your defense of it as a humanity, and the importance of its rules for choosing rules to suit the situation.
That said, as I understand it, mathematics in the deep sense you describe gives great tools to specify patterns and how they repeat, but it breaks down somewhat when specifying exceptions to those patterns. For example, consider the debates around 'natural kind' in modal logic (Is a cat with three legs still a cat? How about Lewis Carroll's Cheshire Cat?). Wrestling with what becomes part of the 'data analysis within a framework' is essential. As it turns out, philosophy has lately turned toward the tools of mathematical logic to specify these things, while its concerns remain essentially human. That's at least one reason why I think the author preferred philosophy to mathematics in the choice of majors suggested.
Without recognizing the limits of mathematics and its abilities to describe what occurs in the real world, mathematicians are prone to being caricatured as the physicist is in the XKCD cartoon the author links.
Comments
In short, any student who emerges from university with good critical thinking skills has achieved that by accident, not by the university's design. I would also argue that the courses that come closest to achieving the ideal are history, English, and philosophy.
How about abstract mathematics? I would argue that these days those might be the only courses where you learn those skills.
I suspect the author would argue that training in abstract mathematics is comparable to the other sciences in the sense of 'data analysis within a framework', subject to the problem of under-training in the ability to recognize the limitations of the framework. I suspect s/he chose History, English and Philosophy because they are fuzzier than the sciences, including mathematics, and so they require understanding how to choose - or define - a set ground rules to apply in a given situation.
Which, of course, perfectly describes abstract mathematics... just not the sort that non-majors ever get to.
Many of the interesting parts of mathematics (and, I should add, computer science!) are all about coming up with the mental framework and abstractions needed to look at a problem, give you a new perspective and ultimately come up with a solution. Choosing or coming up with the logic to treat a situation is exactly what you'd be doing.
It's also particularly good at illuminating the underlying structure of different domains and giving you tools to recognize similarities between them that may not be readily apparent. This goes well beyond what most non-mathematicians regard as "mathematically accessible": just look at all the various modal logics[1] both for the breadth of their applications (belief, knowledge, time...) and their internal similarities.
[1]: http://en.wikipedia.org/wiki/Modal_logic
Moreover, I would argue that mathematics goes further than most fields in talking about and working with its own limitations. Metamathematics and logic as fields are all about this—with greater clarity than the fuzzier subjects, in the usual mathematical style. Mathematics is also great for consistently and rationally accounting for uncertainty and limitations of other systems, fundamental to probability and any sane treatment of risk.
Mathematics, as practiced by mathematicians, also teaches another really important idea: having different levels of rigor, as appropriate. Often, an argument only needs to be convincing, and many of the important details can be left out. However, in math, this is always done with the understanding that those details can be filled in, and the understanding that you should be able to fill them in given enough time.
I figure the focus on clarity, thorough introspection and different levels of rigor, abstractions and logic all make mathematics qua mathematics one of the ultimate "humanities". The problem is that classes for non-majors (and, perhaps, many undergraduate classes for majors) often don't teach mathematics for itself; instead, they teach mathematics as a tool for engineering or solving specific problems or getting a specific result. And that, certainly, could be fairly described as "data analysis within a framework".
But mathematics in general? Much deeper!
Though I am no mathematician, I deeply respect mathematics, and appreciate your defense of it as a humanity, and the importance of its rules for choosing rules to suit the situation.
That said, as I understand it, mathematics in the deep sense you describe gives great tools to specify patterns and how they repeat, but it breaks down somewhat when specifying exceptions to those patterns. For example, consider the debates around 'natural kind' in modal logic (Is a cat with three legs still a cat? How about Lewis Carroll's Cheshire Cat?). Wrestling with what becomes part of the 'data analysis within a framework' is essential. As it turns out, philosophy has lately turned toward the tools of mathematical logic to specify these things, while its concerns remain essentially human. That's at least one reason why I think the author preferred philosophy to mathematics in the choice of majors suggested.
Without recognizing the limits of mathematics and its abilities to describe what occurs in the real world, mathematicians are prone to being caricatured as the physicist is in the XKCD cartoon the author links.