Most of this blog is about technical issues, but a key finding is that perception matters. My experience is that many, even numerate people, treat mathematics (and statistics and science) as if it were magic, and in this respect are ‘uneducated’. It is almost as if mathematics, statistics and science are being taught and are accepted, so they must provide all the answers.
Minor Public Library view
These are guides to hand from my local library, intended for undergraduates or other intelligent interested persons for whom mathematics is not their main interest.
P. Kahn Studying Mathematics and its Applications Palgrave 2001.
This guide takes a common-sense view of ‘the real world’ and possibilities for making objective sense of it. There are two types of mathematics:
- That which seeks to formalise concepts, which may happen to be ideas about the real-world, mistaken or not.
- Those which claim some relationship with reality.
Thus one could envisage a valid mathematics of some astrological theory, irrespective of our views about the legitimacy of astrology. But a mathematical model of astronomy implicitly makes greater claims. It is not just that to operate the model one uses some branch of mathematics (e.g., calculus). There are some constraints on the supposed relationship to ‘reality’:
- The model has been tested against the fullestt variety of examples.
- All necessary assumptions have been identified.
My own view is that mathematics (beyond computation) can be most helpful in uncovering hidden assumptions. The guide might usefully have put more emphasis on this. In practice, one can never be sure that one has covered the full variety of cases or uncovered all assumptions, and the link between the model is scientific, rather than mathematical. But one should try. In particular, one needs to be careful about induction, whose limitations might have been spelled out more.
A. Graham Understanding Statistics Teach Yourself, 2010.
- We are warned about extrapolations, such as assuming that statistics will be stable.
- Here, probability is only defined for gambles, trials and experiments.
There is clearly no warrant for deductions about, for example, economies, and particularly not for probabilistic judgements about the future.