Merry Xmas

Merry Custard (and a Happy New Year ;-)

(yesterday was the shortest day)

Looking back to what was once new, "Custard" :

Harold Budd (1936–2020)

The Village Duck-Pond, Winking

Living inside a rainbow!

 

Happy St Edmund's Day ;-)

20 November 869 AD was the day King Edmund of East Anglia was killed by Danish invaders. He became the patron saint of England.

St Edmund was replaced, as patron saint of England, by St George in the fourteenth century, mainly 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).

Ironically, English nationalists are fond of the flag of St George.

A true contradiction?

(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?

And 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 (as when we say that something is and isn’t a certain way, meaning that it is that way in one sense but not in another, or that it is that way about as much as it is not), that does not mean that the contradiction is true. Not too dissimilarly, 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). And it is, in any case, not at all contradictory for there to be no particle and then, at a later time, one particle.

Why would anyone think that a mathematical model of reasoning that is not a very good model of logical reasoning (because if something is not the case, then it cannot also be the case: it not being the case means that it cannot) would help them to think logically?

Monochrome Monday

The same cat, who I used to meet

when out walking around the village

Sleepy Sunday

 Another photo from google+

SepiaCaturday

Have a Harrowing Halloween!!

Three years ago today,

a honeybee was contemplating the cosmos

Four years ago,

it was Trump and Mrs. Clinton, slugging it out. The general feeling over here was that Mrs. Clinton was a shoo-in. Indeed, her supporters called her "the only qualified candidate." But while she was certainly the most experienced candidate, I wondered at the time whether her experience would disqualify her, in the minds of a lot of the voters. Would they not feel that her two terms in the East Wing meant that her standing for president went against the spirit of the Constitution (amendment 22)?

Suppose that it had been Mr. Clinton in the East Wing for eight years, and that he had won against Trump (suppose that Mrs. Clinton only lost because she was a woman). If Mr. Clinton went on to win a second term, what would stop him divorcing Hilary at the end of his second term and marrying someone who became president for two terms? What would stop him continuing to remarry people who became president for two terms? The voters would: they would surely regard it as unconstitutional for him to stay indefinitely in the White House (as the traditional head of the presidential household).

As it is, of course, Trump won. "Will the same thing happen again this time?" the liberal elite wonder. I don't see how it could: neither Mr. nor Mrs. Biden have been president before. Nevertheless, if the liberal elite of America do not even know how America thinks about the unity of married couples, then how qualified could any of them be to embody America as its necessarily short-term president?

Five years ago today,

this Painted Lady butterfly was drinking from this butterfly bush:

Five years ago today,

this duck was ducking for pond-weed in the village duck-pond:

 

Six years ago this month

we had the coldest August for twenty years. It was like Autumn.

On the plus side, there were lots of clouds for me to photograph

Gusty Auguries

Unstoppable force shatters immovable object

By July bejewelled

Variant of Clark's Third Law

Richard Dawkins, in The God Delusion, page 235, observed that cargo cults were an example of Arthur C. Clark's third law, which is that: Any sufficiently advanced technology is indistinguishable from magic.

It seems that [...] the islanders were bowled over by the wondrous possessions of the white immigrants to their islands [...] The islanders noticed that the white people who enjoyed these wonders never made them themselves. When articles needed repairing they were sent away, and new ones kept arriving as 'cargo' in ships or, later, planes. No white man was ever seen to make or repair anything, nor indeed did they do anything that could be recognized as useful work of any kind (sitting behind a desk shuffling papers was obviously some kind of religious devotion). Evidently, then, the 'cargo' must be of supernatural origin.

The islanders' culture included belief in the supernatural, but did not include much technology, and so they developed a cargo cult. But our culture includes knowledge of technology, and disbelief in miracles. So, if aliens arrived from another planet, we would be more likely to explain what they were capable of doing in terms of their superior technology. Clark's third law has not been true for a while now.

And what if the aliens had also developed superior social structures to ours? Who knows what such things might actually be like. We all have our political views, which we take to be true, but science has shown how different the truth can be from our expectations. So, such alien explorers might, just possibly, be saintly missionaries, who could perform miracles.

Of course, Dawkins would hardly believe that to be possible; but he should not forget Clark's first law, which is that: When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong.

I suppose that we would try to explain such miracles in terms of the aliens' superior technology, as though it was now true that: Any sufficiently advanced spirituality is indistinguishable from technology. Some evidence for that variant of Clark's third law is von Däniken's infamous Chariots of the Gods: Was God an Astronaut?

The Way of Things

1. Introduction

    the lie of the land | low expectations

2. The Way of Things

    infinities | sets | proofs | explanations

3. Extraordinary Evidence

    many things | impossibly many things

4. A God Hypothesis

    the shape of time | authorial authority

5. Exceptional Logic

    heaps | Russell's paradoxes | truth