Kahn’s Studying Mathematics
P. Kahn Studying Mathematics and its Applications Palgrave 2001.
This is of interest in terms of what a user of mathematics, as against a specialist, might reasonably be expected to understand.
[I]f you are studying one of the sciences, engineering, business studies, economics, statistics, computer science or any other subject involving mathematics, then this is the book for you.
1 Setting the Scene
What is mathematics?
Much mathematics concerns mathematical structures as such, but two more significant views of mathematics are given:
- Mathematics is a complex framework of ideas which enables you to make sense of the world.
- Mathematics is concerned with solving problems about numbers and using the results to help you understand the world.
This blog is about ‘so what’ questions, and hence such views.
Facts or connections?
The guide contrasts views of mathematics as about facts, versus views that it makes connections. In my view, mathematics can be about the facts of mathematics, but in the real world there are no ‘given’ facts. Even if we are given licence to treat something as if they were facts, the facticity of any mathematical deductions is still contingent on the ‘given’ facts. Applications are only ever connections: ‘if you think this then you ought to think this’. Mathematics can produce no facts outside of mathematics, except by connection to supposedly ‘given’ facts, which cannot all be entirely mathematical: there must be some mathematically ground-less assumptions somewhere. (And, I believe, identifying them can be useful.)
A second approach to studying mathematics and its applications is to concentrate on making sure that you understand what is going on. [How] to solve an equation is less important than understanding why the solution is valid and how the solution relates to other mathematical ideas.
Effective study of mathematics is characterised by looking for connections between ideas rather than just memorizing facts and procedures.
(This seems to be particularly true of applications.)
2 Using Examples
Why use examples?
[A] mathematical idea can be conveyed by considering a varied collection of examples of the idea.
(Sometimes it is important that the examples should be not only be varied, but that that variety should be representative of the whole.)
It is important to note that mathematical ideas model our concept of the real world, rather than the real world itself.
[P]roviding a genuinely varied set of examples of an idea is not always as easy at it sounds.
3 Thinking Visually
Make sure that you link your images both to mathematical ideas and with the real world.
4 Coping with Symbols
Finding meaning in symbols
Learning to pay attention to definitions is a key to success in mathematics.
Using symbols in calculations
The crucially important skill … is the ability to reintegrate the symbols with their meanings at will, in order to interpret or check the details or results … .
(This seems particularly true of applications: we can never be sure that our symbology is adequate, and it will only have been checked in certain circumstances. We should check that our results are still ‘on safe ground’, and especially cautious if we don’t know what they mean.)
5 Taking Ideas Apart
Include all of the basic ideas
- Some of the basic ideas are easy to overlook …
- These are usually the ideas that you most need to revise!
(Here ‘revise’ is possibly intended in the context of learning an academic subject. But in practical applications it is also tends to be true that if one is ‘stuck’, one often needs to revise=change one’s deepest assumptions.)
6 Thinking Logically
Why think logically?
[T]ruth is established on the basis of agreed starting points, called axioms. New results then follow … according to agreed rules.
(Note that something may be ‘agreed’ and yet not appropriate for the intended application.)
Inductive approaches can … lead to error.
The formal and deductive [not inductive] is central to mathematics. Even if you intend to apply mathematics in science and engineering, you still need to understand some basic logic.
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.