More Mathematical Weirdness


What’s your favour­ite? Photo by Claudecf

This week’s New Scientist tipped me off that I’d missed an import­ant press release. Prof. Dan Everett at the University of Manchester has press release that I missed. He has a paper out in the new edi­tion of Current Anthropology that claims the Pirahã are a tribe that has no words for num­bers or even a concept of count­ing. That seems to dove­tail nicely with my own thoughts that math­em­at­ics is social as well as real. Nevertheless is it pro­foundly strange. The final sen­tence is also some­thing I agree with on translation.

The rel­ev­ant extract from the CA press release is below the fold. Prof. Everett has put a couple of brief items on the Pirahã, one in English and one in Italian, on the web and has also a short dis­cus­sion on the LanguageLog.
Continue read­ing

What came first Addition or Multiplication?


Carnivalesque ButtonMaths 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.

Numbers (Corrected)
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
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.