20 November 869 AD was the day King Edmund of East Anglia was killed by Danish invaders. He became the patron saint of England, but was replaced by St George in the fourteenth century because Edward III wanted to be more involved with Europe. Two centuries earlier, Richard I had been one of many Europeans uniting around St George, as they defended Europe during the Crusades. It is therefore a tad ironic that English nationalists are so fond of the flag of St George. But this is the Irony Age.
Friday, November 20, 2020
Thursday, November 19, 2020
(a) the maths
Since adding zero to any amount does not change it, we can keep adding zeroes forever, and it will make no difference.
Such additions always amount to adding zero. We might write that as 0 = 0 + 0 + 0 + 0 + 0 + …, or, when spread out:
0 = 0 + 0 + . . .
Each 0 on the right-hand side can be replaced by 1 – 1, to give:
0 = (1 – 1) + (1 – 1) + . . .
In the next equation, the brackets have been removed.
0 = 1 – 1 + 1 – 1 + . . .
In the next equation, brackets have been put back in, in different places.
0 = 1 + (–1 + 1) + (–1 + . . .
We now replace each (–1 + 1) with 0.
0 = 1 + 0 + 0 . . .
All those zeroes on the right-hand side add up to zero, of course. But that means that:
0 = 1
Clearly 0 = 1 is false, and so the equation above that was also false. As was the one above that, 0 = 1 + (–1 + 1) + (–1 + 1) + ..., each of those “(–1 + 1)” being zero.
Going the other way from 0 = 0 + 0 + ..., which was clearly true, the next equation, 0 = (1 – 1) + (1 – 1) + ..., is similarly true, because each of those “(1 – 1)” is zero.
In between those equations we have the infinite sum 1 – 1 + 1 – 1 + …, which was originally described by the Italian theologian and mathematician Guido Grandi (1671–1742). Grandi was interested in the calculus, as described by Leibniz. Newton had earlier devised his own version of the calculus, and used it to defend the heliocentric view of the solar system that many theologians opposed. Puzzles like Grandi’s could have been used to argue for limiting its use. Nevertheless, the calculus actually explains this puzzle.
In the calculus, an infinite sum is equal to the limit of the initial finite sums as their length tends to infinity. Grandi’s infinite sum 1 – 1 + 1 – 1 + ... has initial sums that alternate between 1 and 0 = 1 – 1 endlessly (the next are 1 = 1 – 1 + 1 and 0 = 1 – 1 + 1 – 1). Since the initial sums tend to no limit, there is Grandi’s infinite sum is not given any value by the calculus. By removing the brackets, we moved from an infinite sum of zeroes, which is equal to zero, to Grandi’s infinite sum, which has no value. Adding brackets differently then took us from Grandi’s infinite sum to a sum that is one plus an infinite number of zeroes, which is equal to one.
(b) the physics
You may be familiar with the idea of a particle/antiparticle pair appearing out of the vacuum. Such pairs give rise to Hawking radiation from a black hole, but all we need to know here is that such pairs can, in theory, appear from the background fields of the vacuum. Once formed, the particle and antiparticle are moving away from their point of origin, so we might picture them moving downwards, like this: /\ (near a black hole, one of them might be swallowed by the black hole, while the other flies away from the black hole, giving rise to Hawking radiation).
Space does not seem to be infinite, but an infinite space is a physical possibility. And in such a space, an endless line of such particles/antiparticle pairs is a possibility, for all that it is highly unlikely. We might picture them like this: /\/\/\/\/\... (the zig-zag continues to spatial infinity).
The top of that zig-zag pictures a line of particle/antiparticle pairs appearing, which might be modelled mathematically by modelling each particle as +1 and each antiparticle as –1. We then get this equation:
0 = (1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) + ...
Each (1 – 1) represents a particle/antiparticle pair appearing, and then moving downwards.
They move downwards in such a way that each antiparticle collides with the particle from the pair to the right, so that they are both annihilated. The particle at the extreme left is not annihilated. The bottom of the zig-zag pictures events that would therefore be well-modelled by this equation:
1 = 1 + (–1 + 1) + (–1 + 1) + (–1 + 1) + (–1 + 1) + (–1 + ...
Each (–1 + 1) corresponds to an antiparticle and a particle annihilating each other.
In between those two equations, there is no sum total, neither 0 nor 1. That corresponds to infinitely many particles and antiparticles just being there, in between their creation and their almost total annihilation.
The highly improbable, but physically possible, appearance of this particle from an infinite vacuum is therefore so well-modelled by Grandi’s sum that it is clearly an instance of 0 = 1, much as Jack and Jill being a couple is an instance of 1 + 1 = 2.
(c) the questions
Could this be a true contradiction?
In order to think about that question logically, should we use paraconsistent logic?
(d) my answers
Yes and no. Contradictions are never true, by definition, so: no. Although it does seem that a contradiction can be used as a description that is such a good description, it should count as a true description, and so: yes, in that sense. This is similar to the two obvious ways in which 1 + 1 = 2 is true: by definition of 2, and as a description of Jack and Jill.
In order to answer that question correctly, I needed to think logically. So although mathematical models of reasoning that are not very good models of logical reasoning may have their uses (in computing, perhaps), this could hardly have been one of them.