(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 + …, which can be spread out like this:
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. So, where did we go wrong? Well, since the last equation was false, the equation above it must also have been false (the only difference between those two equations is the first equation, which was clearly true). And the next one, going upwards, 0 = 1 + (–1
+ 1) + (–1 + 1) + ..., must have been false too, as each of those “(–1 + 1)” does equal zero.
Going the other way, from the first equation, 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 two equations, one false and one true, 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). And 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, 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.
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 of the zig-zag is not annihilated. The bottom of the
zig-zag therefore pictures events that are modelled rather well 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 mathematical sum, 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 0 = 1 – 1 + 1 – 1 + ... = 1, that it is essentially an instance of it. It is in a very similar way that Jack and Jill being a couple is an instance of 1 + 1 = 2.
Such equations as 1 + 1 = 2 only exist because they are such good descriptions of any collection of two things. It is the physical instantiation that ultimately justifies the mathematical equation. And of course, to say of what is, that it is, is to say something that is true. Which raises the following question.
(c) the questions
Could 0 = 1 – 1 + 1 – 1 + ... = 1 be a true contradiction?
In order to think about that question logically, should we use paraconsistent logic?
(d) my answers
Although a contradiction can be used as a description that is such a good description, it should count as a true description, that does not mean that the contradiction is true. Consider how there are two ways in which 1 + 1 = 2 is true. It is true as a description of Jack and Jill, and it is, in a different way, true by definition (of 2). Contradictions are false (as a rule). And it is not at all contradictory for there to be no particle and then, at a later time, one particle.
In order to answer that question correctly, I needed to think logically. Why would anyone think that a mathematical model of reasoning that is not a very good model of logical reasoning would help?
