What came first Addition or Multiplication?

Maths is fascinating. These days we see it as value-free beyond social concepts. I could write a number like 46587612165684612, but even if that number has never been written before in history people would think it odd for me to claim I invented it. In western thought 46587612165684612 has always existed, even if no-one has observed it, between 46587612165684611 and 46587612165684613. It can’t be destroyed, broken or damaged. It’s existed from the start of time and will continue to exist to the end of the universe as it is today. Perfect and uncorrupted. Which is a very Platonic idea, and that’s a problem if you dealing with cultures which didn’t have a Plato, like the Greeks before the fourth century BC.

A lot of thought on numbers requires assumptions which we don’t even acknowledge existing. A problem I’ve been thinking about for a while is the origin of mathematical operations. Which came first addition or multiplication? It would seem to be a no brainer, but it’s not. Clive is not entirely happy with this and it still needs a lot of work but it’s a problem worth thinking about. Why do we use mathematical operations? This first came to me while reading either Walter Burkert on Pythagoreanism or Carl Huffman on Philolaus.

Before Plato numbers had gender. Two and the even numbers were female. Three and the odd numbers were male. This lead to some interesting properties and conclusions. For instance you could mathematically prove males were more creative than females. Adding female numbers together could only produce female numbers. But add a couple of male numbers together and you had a female number. Add three male numbers together and you had a male number.

Numbers also had form. Female numbers were perceived as rectangular. Male numbers were phallic ‘gnomon’ shapes. These gave numbers other properties. For instance four was the number of justice, because it was mutually reinforcing being two high and two wide. But what number represented marriage, the union of a male and female? Five or Six? Aristotle said five. Theophrastus said six.

The forms of Greek numbers.

The author of the book, who I now think was probably Burkert, said five was likely to be the older belief. Partly because Aristotle was a more ancient source than Theophrastus, but also partly because addition was simpler than multiplication and so being less abstract was likely to be earlier. I’m not so sure. Look at the way the numbers are drawn.

Five is definitely a male number in this system, but six can be cut a couple of ways. It’s either two threes or three twos. When you see six you automatically perform a multiplication seeing two lots of three. Why perform addition?

Addition is the summing of two numbers. You can get the same answer by counting the total of the two numbers. You don’t need to know two plus three equals five, you can recount the sum, one, two, three, four, five. You only need the operation of addition when the numbers are so large that a recount isn’t practical. I don’t know how big a number that would be.

Linear B tablet showing the number Six (coloured)

Unfortunately the evidence one way or the other is slight. There’s this Linear B tablet in the Ashmolean which shows a six as two times three. However I’ve a feeling that five in the same system is shown two scratches above three, so I’m not sure if that really is evidence in favour of either system. In addition there’s a thousand years between this tablet and later Pythagorean thought and no guarantee of a staright connection between the two. Pythagorean philosphy was also influenced by Orphic thought. There’s even a possibility of Italian cosmologies having an effect as the big centre of Pythagoreanism was southern Italy.

Definitely something that needs more work before it’s convincing.