the maths of uncertainty

3 02 2012

This is definitely peculiar. Absolutely no idea how useful it might be. I used to be a math teacher, and only recently did I realise that an equation is not read like a normal sentence. To keep things simple, consider the following expression, or sum:

Any school kid will tell you, you don’t start out from the left, like it was a sentence, with eg 2×3. You follow the procedure, BODMAS, in order to unpack the expression.

Consider what this means in time. The procedure of BODMAS sets the order by which the equation is unpacked in time. The mental sequence of processes. So, in the above expression, you complete (4+7) and (6-3) simultaneously, then multiply them respectively by 3 and 2. and then you subtract the second from the first, and then multiply the whole lot by 2, which gives 54. Or I guess you can multiply all the brackets and then simply sum them up. Whichever way you do it, the answer is the same, 54.

Now consider the following statement:


If we consider it from left to right, we get (10+4) then multiply by 3, which gives 42. Or, if we follow our usual BODMAS protocol, we do the multiplication first (4×3) and then add 10, the correct answer.

But what would a maths look like that holds both correct, at the same time?


That is, where both answers, 42 and 22, were correct?

And if this doesn’t strike you as being particularly useful, you are in popular company. However the following expression does pique a curious line of inquiry:


Which derives a possible simultaneous answer of 0 and 1. And because the digit in the middle can be a 1 or a 0, it might be more clearly written as:


And if we multiply this expression by itself:

And the answer to this is the truth table for the AND function:

AND (x)

0

1

0

0

0

1

0

1

And if we add it to itself:

We derive the boolean OR function:

OR (+)

0

1

0

0

1

1

1

1

This is not entirely unsurprising, since our original expression simply changes the order of the operations, thus producing the effect of sequenced logic gates.

What is interesting, at least notionally, is that this is contained in a single arithmetical expression. I know that variables can perform this function in maths, and set theory, but to have it in a simple sum, is… peculiar. And I can’t help but relate this to chaos equations, which after a suitable number of iterations, tend to a constant, or two constants, or a set of constants. That, and qubits.

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