This year at The Institute about a quarter of my contact hours are doing QM - quantitative methods aka remedial maths. Much of this term is getting to grips with Excel, in which I've forced myself to become a bit of a whizz. Last week one of my classes was plotting out their shhoe size vs height to reveal that humans are clearly sexually dimorphic - not as extreme as Gorilla gorilla but adult males are bigger than females and have bigger feet too. One of the task was to compute the average show-size for the class and I had a bit of a rant when everyone reported the answer as 6.236783. That's crazy I raved what does ...783 mean? how can you have 3/millionths of a shoe size? Much better, I suggested to say that average shoe-size was 6.2 - about halfway between size 6 and size 6-&-a-half. And I showed them the button for controlling the number of significant figures. The rule of thumb is that, if you cannot detect a difference with the perceptual aids available then you shouldn't report it as a mathematical certainty.
This sort of issue came up the weekend we spent in Cork because Dau.II's bloke is the only person I talk to who knows, and cares about, the difference between a perfect fifth and a fifth of bourbon. This came up in the context of Philip Ball's book The Music Instinct which I eventually threw across the room in a pet. I passed the Grade I Theory exam because at the age of 9 I could count to 8 and divide it in halves and quarters. That's the level Mr Ball starts his book, but he rapidly disappears up his own orifice in a tornado of assumptions about what will be obvious all thinking people after the most cursory explanation . . . not! Anyway in Cork The W!ld Corkonial Boy pointed out that in a typical octave (12 notes on a standard piano) the normal human ear can only detect about 200 gradations of pitch, so it's absolute bloody nonsense to assert that middle C resonates at 261.625565 Hertz. The currebt convention - since about 1939 - is that the A below middle C is tuned to 440 Hz and the A one octave up is at 220 Hz. The fact that an octave results from halving the length of a vibrating string has been agreed since Pythagoras worried about beans. The best you can say about the human perception of middle C is that it is 261.65 because you can only detect any finer difference with an oscilloscope and some fancy electronics.
Same with colour. We only have 3 different photosensitive pigments in our retinas and it is their relative stimulation by different wavelengths of light that allows us to recognise cerise, magenta, viridian, and cobalt. There are actually two limits to perception with the eyes. One is about two colours, which an instrument can differentiate as to the reflected wavelengths of light, but which normal people cannot distinguish. Many people in Ireland paint their interior walls in 'magnolia' which is a super-pale cream: less austere than the flat white of, say, white lead 2PbCO3·Pb(OH)2. But Fleetwood magnolia is indistinguishably different from Dulux magnolia. The other issue is whether we are capable of resolving two dots / pixels at a given distance from the eye. Up to the age of about 40 my eyes gave stalwart service: reading very small print and discerning small objects on the horizon. As senile degeneration of the retina set in and the shape of my eye-ball slumped, I was only able to read books if held at extreme arm length. When I needed an arm-extension prosthesis to get the text in focus, the letters were too distant to see. That's when I twigged that I needed glasses. Television and colour printing rely on our inability to differentiate pixels if they are sufficiently similar in hue and sufficiently close together. We can 'see' something as grey even if it is made up of pixels covering the entire visible spectrum