Although numbers and mathematics seem to exist as features of the natural world, they are concepts that humans have created to model and understand the world and to do useful things by manipulating our natural environment.
How do our brains process numbers?
I remember learning a long time ago that pre-literate cultures and languages had no words for numbers. They were limited to counting one, two, lots. In fact, this is not just a feature of pre-literate cultures and languages. All living languages have the same limitation and this is a reflection of how our brains have evolved.
Languages, all over the world, have a gramatical concept for singular and for some type of plural expression. For example, in English we usually indicate plurals with -s endings (there are several examples in this sentence). Other languages do the same thing in different ways, either by altering the word, or by adding prefixes or suffixes. For example, in German the article changes from das to die and the form of the noun changes. A small number of languages also have a concept for three things, with special endings or forms of words.
Many languages also have a special form for two things. In English, we have words like both or pair (borrowed from French). We also have words for many things which may be a lot (many) or not (few).
The complete range of number concepts in all known languages is:
- One (singular)
- Two (dual)
- Three (trial)
- Few (paucal)
- Many (plural)
This concept seems to be something hard-coded in our brains. We can instantly recognise whether there are 1, 2 or many objects without any effort or counting, but almost no one can instantly recognise groups of more than 5 objects unless they are arranged into familiar patterns that we have learned.
Tallying
The first evidence we have of people trying to track a large number of objects is tally systems. These are seen on rock paintings and tally sticks dating back at least 25,000 years. Only 25,000 years! Before then, fully evolved humans, using language, fire, tools and clothing, don’t seem to have needed to keep track of objects.
There is nothing to tell us how the tally sticks were used. We can make up stories about early trading or credit systems, or other uses, but all we can assume with confidence is that one notch on the tally stick represented 1 thing in the real world. This is the first level of abstraction – using symbols to represent objects in the real world.
Although we can’t know for sure, people were probably also using tally systems based on their hands and fingers. These systems have survived into modern times and can become quite sophisticated, representing numbers much larger than 5 or 10.
Counting
At some point, people started to represent different numbers of tally marks with their own concepts. i.e. they started to use numbers like “one”, “five”, “eight”, and to put them in order.
In his Introduction to Mathematical Philosophy, Bertrand Russel spends chapters discussing the meaning of the number 1 and how it should be followed logically be the number 2, and so on. For a concept that appears obvious to us today, it doesn’t seem to be at all obvious why numbers should exist at all.
If you think about it, counting is a dramatic step in abstracting numbers from reality. It implies that 5 is always 5, regardless of what you are counting. 5 sheep is “the same” as 5 fish. If I have 5 sheep and another one comes along, then I have 6 sheep. And exactly the same things happens with those fish!
- 5 sheep and 1 sheep makes 6 sheep
- 5 fish and 1 fish makes 6 fish
- 5 + 1 = 6. Always! Regardless of what you are counting.
It’s not obvious because 5 sheep and 1 fish doesn’t make anything interesting. The abstraction only makes sense if you consider numbers as a different entity from the things they might represent.
Once the abstract concept of numbers had been accepted, the rest of mathematics followed relatively quickly, although it still took centuries to absorb concepts such as zero, negative numbers, irrational numbers and so on.
So, where do numbers come from?
Numbers are not obvious in nature and not wired into our brains, Numbers and mathematics are human inventions that we use to model the world and do useful things such as measuring and calculating. Even though mathematics has been extremely (unreasonably?) powerful as a tool to model the real world, there are no numbers out there in the world, independent of the concepts we have created in our brains.