Friday, December 15, 2017

Mangan's Moment of Truth

There is a moment, recognisable to an experienced operative, when, after weeks of preparation and cultivation, months of effort, years even, an operation flickers into life, when the hard little bead of possibility breaks open to reveal the actual. A journalist knows the moment, too, that second when the shape of the story starts to show itself, its scope, its layers of meaning. The moment may come with the acquisition of a specific piece of information, or the making of a connection, or an event. Or it may be something far smaller, an inconsistency, a gesture, a look on someone's face that tells of a lie.

Adam Brookes, The Spy's Daughter, page 162

Saturday, November 18, 2017

Do We Systematically Misunderstand Our Own Language?

Suppose that God had created Adam and also, in an identical space-time, an exact copy of Adam. Such a God could swap Adam with that other person, then swap them back, and do that over and over, and it would make no difference, empirically: they have identical memories and everything. One difference it would make, though, is that any direct reference to Adam in this space-time would keep failing. And similarly, what if fundamental particles (or other substances) are constantly being replaced by identical particles (substances)? It would make no difference empirically, which is why we cannot rule it out, logically. We can of course assume that it does not occur. Consequently we can assume that reference is direct; but, surely language would work normally even if such replacements were occurring.

Friday, November 03, 2017

The Trouble with Transporters

In Star Trek people are beamed from place to place using transporters (their molecules are converted into energy which is then beamed to another place where it is turned back into the original people), even though logic dictates that if so then people should not eat food, whereas in Star Trek they do eat food. The logic is as follows: When people eat they replace some of their molecules with ones from food, so that we can easily imagine that all their molecules are replaced, with the old ones being crapped out. If that crap was collected and its atoms rearranged into an exact copy of those people, then that would be just that: copies. So, if that is how things are, then transporters would kill people and create copies of them.

Thursday, October 19, 2017

Isserley's Feeling For Snow

The sensation of fresh snow crunching underfoot was deeply satisfying to Isserley. Just the idea of all that water vapour solidifying by the cloudful and fluttering to earth was miraculous. She couldn't quite believe it, even after all these years. It was a phenomenon of stupendous and unjustifiably useless extravagance. Yet here it lay, soft and powdery, edibly pure. Isserley scooped a handful off the ground and ate some. It was delicious.

Michel Faber, Under the Skin, page 56

Monday, October 09, 2017

The Bible Explained

222111 is the number of The Royal Mint,
and it equals 1 + 2 + 3 + ... + 666, where
666 = 1 + 2 + 3 + ... + 36 is The Number
Of The Beast, which is Spooky, because
money is the root of all evil. 36 = 1 + 2 +
+ 3 + ... + 8, and 8 = 2 x 2 x 2 x 1 x 1 x 1

Tuesday, February 14, 2017

Easy as 1, 2, 3 (in base 5 + 5)

12 = 3 x 4
56 = 7 x 8
90 = 360/4

      0 + 12 = 3 x 4
      5 + 67 = 8 x 9 = 360/5

(there are 60 = 360/6 seconds in a minute)

4 x 5 x (6 + 6 + 6) = 360 = (5 + 5) x 6 x 6

                    1 + 2 + 3 + 4 = 5 + 5
1 + 2 + 3 + ... + 8 = 9 x 4 = 6 x 6

    1 + 2 + 3 + ... + (5 + 5) = 55
    1 + 2 + 3 + ... + 6 x 6 = 666

Paradox of Expectation

When you expect a lot of some promising event, thing or person, it tends to be more disappointing than it would have been had you not, so in order to maximize your pleasure you need to avoid expecting so much: If you expect to be disappointed, then you won't be disappointed!

Saturday, December 17, 2016

Is 'No' the Answer to this Question?

More precisely, is “No, it is not” a correct answer to this question?

If it is a correct answer, then by the meaning of “No, it is not” that answer is not a correct answer. And of course, it cannot be correct and not correct; the meaning of “not” rules that out. But, if it is not a correct answer, then “No, it is not” would be a correct answer to our question. Could it be the case that there is no correct answer? But then “No” would still be a correct answer to our question. We therefore need a correct answer. Now, if “No, it is not” was as correct as not, then it would be as incorrect as not, which would solve our problem; and since we have ruled out all the other sorts of answers, hence that does solve our problem. A good answer to our question is therefore: It is as correct as not.

You may be wondering: Can answers be as correct as not? Another reason why they must be able to be comes from the following question: Is “It is an apple” a correct answer to the question “What is A?” where A is originally an apple but has its molecules replaced one by one with molecules of beetroot? Originally it is a correct answer, but eventually it is not, and so if it must be either correct or else not, then an apple could be turned into beetroot by replacing just one of its molecules with a molecule of beetroot. But of course, that is not what “apple” means.

Saturday, October 15, 2016

Telling the Truth

The following sentence occurs in the preface of a book: “All the sentences in this book are true.” That sentence is saying that all the other sentences in that book are true, whence it is too. Were all the other sentences true, it would be silly to say that that one was false just because it was logically possible for it to be false (if it is false then not all the sentences are true), and so if the other sentences were all true, then that one would be too.
The Truth-teller is the self-referential bit of that sentence: “This sentence is true.”
The Truth-teller is saying only that what it is saying is true (which all sentences implicitly assert anyway), so it is not saying much, and so there is not much for it to be true or false about. It would be consistent for it to be true, and consistent for it to be false, but what could determine which it is? Maybe there is no fact of the matter, what it is. So, is that sentence neither true nor false? But then “This sentence is true” would clearly be false (and the fact that it would refutes the idea that it says nothing at all). Still, it is not saying much, and so it is not very true and not very false. It is therefore fairly false that it is true, and so what little it is saying – that it is true – is fairly false.
The Truth-teller is not saying much, but what it says is more false than true. It may therefore be the case that the Truth-teller is not very true and only about as false as not. To see why that might be possible, consider a man going bald: As he goes bald he will not, by the loss of just one or two hairs, become bald and so there might be an intermediate or overlap stage at which he is about as bald as not, when it would make sense for it to be about as true as not that he was bald, and about as false as not. There is a lot to be said for, and against, such a possibility; here it would be most apposite to look at the Truth-teller’s paradoxical companions.
The following sentence occurs in the preface of a work of fiction: “None of the sentences in this book are true.” If that sentence, say Sent, was true, then none of the sentences in that book would be true, and so Sent in particular would not be true; and that contradiction means, by reductio ad absurdum, that it is not the case that Sent is true. So either Sent is false – in which case at least one of the other sentences would have to be true – or else it is neither true nor false; and it could of course be the case that none of the other sentences are true, so Sent is neither true nor false. But that cannot be because Sent is meaningless, because we have just been reasoning logically with its meaning.
Sent does have two obvious meanings, though. As well as the one we have been working with (its literal meaning), it was clearly supposed to be saying that none of the other sentences in that book were true. And since it did express that latter meaning (which we might call its literary meaning) clearly enough for us to notice, hence it did also have that meaning. So, if none of the other sentences were true, then it would be true that none of them were true, and so in a sense Sent would be true (with the literary meaning) and not false; and it would false that there was not even that truth, and so in a sense Sent would not be true (applying the literal meaning) but would be false. It follows that Sent is false: If it is not false because one of the other sentences is true, then it is still false because it is in another sense true.
Working with only the literal meaning was paradoxical, though; it left us with something like a reductio of the logical presumption that sentences are either true or else not true. Logic, and language itself, suit veracity and falsity being black and white, but there is also vagueness; and we do want to think something veridical about paradoxical sentences. So, the actual reductio above should be replaced by: Insofar as Sent is true, Sent is not true. It follows that Sent, taken literally, is about as true as not. Still, it is more realistic to read it as being true if none of the other sentences are. So for simplicity, let us consider the paradoxical bit by itself: “This sentence is not true.”
That sentence (the Liar) is saying that what it is saying is not true. Unlike the Truth-teller, it is not just saying something that all sentences implicitly assert, so it does not seem to be saying nothing; and note that if it was saying nothing, then it would not be saying anything true, whence it would clearly be true. So, Liar is paradoxical; insofar as Liar is true, it is not true, and insofar as it is false it is true. It follows that Liar is about as true as not, and about as false as not: It is to some extent true, that it is not true, because it is only to some extent true; and it is not exactly false that it is not true, because it is to some extent false.
For a final complication: “This sentence is not true, and not even about as true as not.” If that sentence, say Even, is about as true as not, then Even is saying something false. Nevertheless, Even, by saying that Even is far from true, is thereby saying that it is far from true that Even is far from true. So although Even is saying something that is false, it is something that is also true, and about as true as not. Consequently Even can be about as true as not, and about as false as not.

Friday, September 09, 2016

Proof's Nearest Kin

Paradoxes are akin to proofs: We have a paradox when we have a very good argument for something that is beyond belief, a proof when we have a logical argument for something not too odd. Many a paradox is therefore a proof by reductio ad absurdum of the negation of its weakest premise. Georg Cantor’s famous paradox of the 1890s was exceptional, being a logical argument for a contradiction, but it thereby proved that human reasoning is not perfectly logical. In response the twentieth century saw a proliferation of formal logics, but as we develop such calculi with mathematical precision we might easily forget that some illogicality is unavoidable in our reasoning. To remind us, then, the following is the essence of Cantor’s paradox.
Consider any 3 things, e.g. a chair, a plate and a fork. There are clearly 3 different ways of making a pair from those 3 things (e.g. the chair and the fork are, collectively, a pair), each of which derives from, and is defined by, the presence of 2 particular things in our original collection of 3 things: Given those 2 things, we have that way of making a pair. Now, making a pair is just one way of making a selection. There are 8 different ways of making selections from our original collection because there are 23 ways of assigning the labels “In” and “Out” to 3 things (e.g. the chair has “In,” the plate has “Out,” and the fork has “In”). Given our original collection, there are also those 8 combinatorial possibilities, those 8 ways of making selections, ways that are entirely grounded in our original things and which are therefore distinguished from each other whether there is a selector who can make those selections or not. Let us call them possible selections, from our original collection, and say that they are collectively the selection collection for our original collection. In general, for any natural number n, if there is a collection of n things, then there is also a selection collection of 2n possible selections from it.
We can safely assume that there are 2 things (e.g. you and I are 2 people), so there is also the selection collection of 4 possible selections from those 2 things, and the selection-collection of 16 possible selections from those 4, and so on. All those possible selections are there already, intrinsically distinguished from each other, and so there are infinitely many things, which are certainly things in the weak sense that there are numbers of them, in the fairly weak sense of cardinal number. Two collections of things have the same cardinal number of things when there are 1-to-1 mappings from each collection onto the other. Cardinality is fairly weak, e.g. there are clearly fewer prime numbers than natural numbers in some stronger sense, but it is an equivalence relation – it is reflexive, symmetric and transitive – and so it partitions collections into equivalence classes. For any collection of things, T, there are possible selections from T – each corresponding to some combination of as many “In” and “Out” labels as there are things in T – even if T is infinite, and so there is a selection collection, S(T), of all the possible selections from T. And for the following reason (which is essentially Cantor’s diagonal argument) every selection collection is cardinally bigger than its original collection.
Suppose that S(T) has the same cardinality as T. There would then be 1-to-1 mappings from T onto all of S(T). So let M be one such mapping. We might then use M to specify D as follows: For each thing in T, if the possible selection that M maps that thing to includes that thing (in other words, if that thing has the label “In” in that possible selection) then D does not include it (i.e. it has the label “Out” in D), but otherwise D does; and there is nothing else in D. Since the only things in D are things in T, D should be in S(T); but according to its specification, D would differ from every possible selection that M maps the things in T to, and so D would differ from everything in S(T). D is therefore contradictory, and so there is no such M. Consequently S(T) does not have the same cardinality as T. And since S(T) contains at least one thing for each thing in T – e.g. the possible selection whose only “In” label is assigned to that thing – and cardinality is an equivalence relation, hence S(T) is bigger than T.
Given you and I, then, there is some infinitely big collection, say N, and so by the diagonal argument above with T = N there is also a bigger collection, S(N), and by the diagonal argument with T = S(N) there is the even bigger S(S(N)) = S2(N), and there is similarly S3(N), and so on. All the things in all those collections are collectively the union of those collections, U, which is bigger than each of those Sn(N), for natural numbers n, because it contains all the things in each S(Sn(N)). Furthermore S(U) is even bigger, and so on; so there is also the union, say V, of all the Sn(U) for natural numbers n. And again, S(V) is even bigger, and so on and so forth. Now, each of the possible selections that such endlessly reiterated selection collections and infinite unions would or could ever show to be there is already there, as a combinatorial possibility that is implicitly distinguished from all the others of that kind, and so they are collectively some collection, C, of all the possible selections of that kind. But by the diagonal argument, their selection collection, S(C), would contain even more things of that kind. That contradiction shows that we went wrong somewhere, but why should even highly evolved primates know where?