What Is Three?
You can point to three apples. You can point to the numeral 3 written on a page. But can you point to three itself? Not the apples, not the ink: the number. Where is it? This lesson ends the course with a question that has no agreed answer.
Three is not the apples
Three apples, three birds, three claps of the hands. The apples are red and round, the birds fly away, the claps are over in a second. They have almost nothing in common. Yet they share something: there are three of each.
In the Why We Count lesson, the shepherd matched pebbles to sheep, one to one. You can do the same here: match each apple to a bird, and each bird to a clap. None is left over. That is what the three groups share, whatever they are made of.
About 140 years ago, the German mathematician Gottlob Frege turned this into an answer: a number is what all the groups that can be matched one to one have in common. Three is what every group of three shares.
Three is not the numeral
We have written three in many ways in this course: tally marks, a Roman III, the numeral 3, and 11 in base two. The Persian numeral is ۳, and in Chinese it is 三. The word is "three" in English and se in Persian.
These are all names for three. Rubbing out the numeral on the page doesn't destroy the number, as burning a map doesn't destroy the city. So the number must be something other than its names.
So where is it?
Here people disagree, and they have for a long time.
- Some, like the Greek philosopher Plato about 2,400 years ago, think numbers are real things, but not things in space. Three was true before any person counted, and would be true if nobody ever had.
- Others think numbers are ideas in our minds, made by people who noticed what groups of things share.
- Others again think numbers are a kind of language: a tool we invented for counting and measuring, which works because the world has things in it to count.
Whichever is right, something strange is true. Numbers were first used for counting sheep. Now they measure stars, run computers and describe how a virus spreads. No one fully knows why an idea that began with pebbles in a bag fits the world so well.
Try it at home
- Ask someone at home: "Where is the number three?" Listen to their answer. Which of the three views is closest to it?
- Find three different groups of three things in your home. What, if anything, do they share besides being three?
Words to know
- number - what a group of things has in common with every other group it can be matched to one by one; in this course, also the counting numbers, zero, negatives and fractions
Big idea
Three is not three apples and not the numeral 3: it is what every group of three shares, and 3 is only its name. What kind of thing a number is - real, an idea, or a language - is a question philosophers still argue about.