Conversation

I'm all hyped up about hyperrings. In the old days people treated the square root as a 'multi-valued function':

√4 = ±2

But for over a century, mathematicians have fought against multi-valued function, arguing that a legitimate function sends each input to one and only one output. Thanks to this, we now understand the square root in a new way! We say it's a single-valued function whose inputs are not numbers, but points in a 'Riemann surface' that has 2 different points sitting over the number 4, say 4' and 4'':

√4' = 2
√4'' = -2

This attitude turns out to be immensely profitable. But we only bother to explain it to people who take an advanced course in complex analysis. And I sometimes forget how removed from ordinary reality mathematicians are: when I was in college a fellow student saw me holding Lang's book Complex Analysis and said "oh, Freudian psychology?"

Now the tide has turned. Multi-valued functions are catching on! Of course we know they aren't really functions in the official sense: they're 'relations'. But mathematicians are starting to study 'hyperrings', where multiplication is single-valued as usual, but addition is multi-valued.

For example the hyperring of signs:

S = {+,0,-}

We can multiply these in the obvious way, like

- × - = +

since multiplying two negative numbers gives a positive one. Sometimes addition is unproblematic too:

- + - = -

since adding two negative numbers gives a negative number. But what's a negative number plus a positive number? It could be positive, zero or negative! So we say

- + + = {+,0,-}

In other words, this sum takes all 3 possible values!

See how I hope to use this in applied math:

https://johncarlosbaez.wordpress.com/2024/11/05/polarities-part-3/