ALL OUR MATHEMATICS, and consequently all our science, is based on the logical concept that philosophers call the Law of the Excluded Middle. This states that every proposition is either true or not true, but not both. It sounds obviously the case, but it doesn’t always apply to the real world.
There are degrees of truth. Take the following statements:
|Sherlock Holmes lived at 221a Baker Street||Not true. He lived at 221b Baker Street. But that isn’t true either; he never lived at all. Nonetheless, the second statement is more true than the first.|
|It’s going to rain tomorrow||By tomorrow we’ll know if this is either true or false. Right now, however, the statement is neither.|
table is smooth
|Compared to a sheet of sandpaper, yes. Compared to a sheet of ice, it’s rough. There is no absolute.|
|Yellow penguins have two heads||There are no yellow penguins. So we can’t say the statement is false, even though we can be reasonably sure it isn’t true.|
In mathematics, 1 + 1 = 2. But it’s the same problem. The equation works for solid objects, such as apples. If you have one apple, and I give you another, you’ll have two apples. But the rule doesn’t apply with less discrete concepts.
Take a salt cellar and sprinkle two piles of salt on the table. Count them: two piles. Now scrape them together to make a single pile. How many piles do you get when you add them together? One. It’s just a slightly bigger pile.
While mathematics works well enough for abstract numbers, the real world isn’t made of apples. It’s made of piles of salt.