Tag Archives: mathematics

How maths is different from science

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“It doesn’t matter how beautiful your theory is, it doesn’t matter how smart you are. If it doesn’t agree with experiment, it’s wrong.”

— Richard Feynman

In science we first observe a phenomena (e.g. the Moon orbiting Earth) – and then come up with a hypothesis (e.g. objects with mass attract each other) to explain it. That hypothesis is then tested by experiment; if the evidence from the experiment contradicts the hypothesis then it is disproved and must be rewritten. If the evidence does not contradict the hypothesis then it is supported, not proved.

All the evidence in the world cannot prove a hypothesis, whereas only one piece of evidence is needed to disprove it. It would only take one instance of something “falling” upwards to disprove the theory of gravity.

In maths the situation is very different: once something has been proved, it is proved forever. No further research is necessary, no tests need to be performed. Pythagorus’s theorum about the lengths of the sides of right-angled triangles has been proved in many different ways (Cut the Knot has 84 different proofs), but will never be disproved.

The dangers of extrapolation

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Physics isn’t for everyone; every year there’s a pretty good chance that at least one person will drop out of a group (sometimes you get drop ins too). This year one group in particular has lost more pupils than average so I decided to start collecting data.

I plotted the data and added a line of best fit.

The R2 value is 0.934 so there’s fairly good correlation in the dataset. Now that I have an equation linking the date (x) and the number of pupils in the class (y) I can work out when the next pupil will leave the group by setting y to equal seven.

y = −0.0165x + 671.63
7 = −0.0165x + 671.63
x = 40280.61

Excel handles dates in a strange way, so the figure of 40280.61 needs to be converted into a regular date: now I know that the next pupil will leave 6PH3 on the 12th April 2010 (at half past two).

I can even find out when the class will be empty by solving for y=0.

0 = −0.0165x + 671.63
x = 40704.84

The class will finally empty out at 10th June 2011 at twenty past eight.

I thought that perhaps the linear fit wasn’t the best approach so I tried a polynomial fit, which turned out to have an R2 value of exactly 1.

It turns out that this line only has one real solution, and according to that, I won’t run out of pupils until 27th June 2103.

On calculators

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The Casio fx-992s is, without a doubt, my favourite calculator.

calculators-front calculators-case

I’ve had my fx-992s since I did my GCSEs and it’s served me very well. I own three – the original (shown on the left), the backup (shown on the right) and the backup-backup (not shown). I try to avoid having all three in the same place in case of catastrophic calculator loss.

I like the fx-992s because it doesn’t have any of the fancy, difficult to use “functions” that I see in my pupils’ calculators: it doesn’t do surds, it doesn’t give every answer as a fraction or require you to use Natural Textbook Display and it doesn’t have a weird cursor-controller joysticky thing.

Most of the pupils I teach can’t use their calculator properly – the calculator companies’ efforts to make calculators easy to use have done exactly the opposite.