Category: language

Is Hidden Figures Science Fiction?

I’ve actually written a longer piece on this film, which is still unfinished and may be unrescuable because of far too many tangents (less obvious ones including Aleksandr Solzhenitsyn, Alfred Hitchcock’s use of a toilet, and the nasty rightwing Christian ‘Focus on the Family’ group). In the meantime, this is an attempt to address the question I intended to address in the other piece but never actually reached. Somehow Ludwig Wittengstein* ended up in this one. Sorry, he gets everywhere.

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Farewell Raymond Smullyan

Martin Gardner was a kind of gateway drug. When I was a kid I went to the same secondary school that my dad taught at. That wasn’t a particular problem for me but it meant I had to hand around school until he was heading home.

This meant sitting in the school library by myself reading, which, being a bookish sort, was not any great hardship. I ploughed through the sci-fi books but on other occasions, I’d just look at random books.  It was in this way I stumbled across old Penguin (or Pelican?) editions of Gardner’s Mathematical Puzzle’s and Diversions books. Now, I was not much of one for mathematics at the time but I liked puzzles and I liked the odd, arcane nature of these books.

As an older teen Gardner led me to other writers: Douglas Hofstadter (notably I read Metamagical Themas first then Godel, Escher, Bach) but also Raymond Smullyan.

Like Gardner, Smullyan combined a love of puzzles and magic but also the absurd. That logic and absurdity are natural companions is something that people find paradoxical. People think it odd that Lewis Carroll was both an eminent logician and author of Alice in Wonderland despite the absurdism of the Alice books often relying on wordplay and uncooperative literalism.

Smullyan, who died last Monday (Feb 6th 2017) tied absurdity more closely to logic in his complex puzzle books. The connection is overt – using weird settings and strange kinds of people (knights, vampires among others) with proscriptive approaches to communication. The New York times has a substantial sample here

So why the close connection between the absurd and logic?

Two elements are at play. Firstly the necessary insistence on literalism. Exactly WHAT is being said? It is an insistence that when applied to normal conversation is a breach of normal social conventions. Secondly, the use of absurd or nonsensical propositions and conclusions helps highlight FORMAL aspects of an argument from informal and empirical aspects. To see how a syllogism functions (for example) it can be misleading to use a string of commonplace truths.

For example, Wikipedia uses this example for Felapton:

No flowers are animals.  All flowers are plants. ∴ Some plants are not animals. (SoP)

But to see the formal connection writers like Carroll might use more odd juxtapositions

No elephants are professors.  All elephants are stamp collectors.  ∴ Some stamp collectors are not professors.

Formal truths need to hold even in absurd worlds.

Smullyan wrote a whole bunch of wonderfully weird books, that used puzzles and odd juxtapositions to exemplify logic and reasoning. I think his most substantial achievement was Forever Undecided: A Puzzle Guide to Godel that used his signature style to lead the reader to Godel’s incompleteness theorems in a charmingly accessible manner.

Logician, Taoist, Magician – a 97 year life of tricks, puzzles and deep thoughts.


Reason Hell: You can’t prove a negative

Reason Hell will be an occasional series on reasoning short cuts, fallacies and internet-sayings that pertain to modes of argument.

Part of Junho Peter Whang's proof of Euclid's theorem from Wikipedia

Part of Junho Peter Whang’s proof of Euclid’s theorem from Wikipedia

You can’t prove a negative. A maxim of logic or more of a challenge? First let’s see if we can prove a negative.

  • A prime number is a whole number with exactly two factors
  • 4 is a whole number with exactly three factors: 1, 2 and 4
  • Therefore 4 is not a prime number

Well this looks like it is going to be a short post! A negative neatly proved by virtue of 2 not equaling 3.

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Reason Hell: Absence of evidence is not evidence of absence

Reason Hell will be an occasional series on reasoning short cuts, fallacies and internet-sayings that pertain to modes of argument.

“Absence of evidence is not evidence of absence” it is a neat saying and a quick comeback but like many neat sayings it is too tidy. As it stands the aphorism is either a legitimate comment or a logical fallacy. This is a term that should be used with great care or not at all.

The first thing to note about the aphorism is that it pertains to evidence. Evidence is not the same as proof. There is no contradiction in saying that there is evidence for A and also evidence for not-A. Even a short treatment of the nature of evidence would be too long for this post and I will save that for a later post. However, it is fairly safe to say that evidence are facts that lend weight to regarding a proposition to be true or to be false. The strongest kind of evidence is a factual counter-example to a more general statement. A counter example is a kind of evidence that can amount to a formal proof that a proposition is false. For statements that are more hedged (most X are Y, many X are Y, frequently X’s are Y’s etc) a counter-example does not amount to a formal proof that the statement is false but clearly adds to a weight of evidence against it.

“Absence of evidence is not proof of absence” is a stronger statement. That there is no evidence of a deity does not prove there is no deity (the deity might be intentionally hiding and being omnipotent can ensure there is no evidence). This though is not aphorism. Applied to the case of a deity an absence of evidence for one would amount to evidence that there isn’t one because we would otherwise expect there to be evidence for a deity if one existed.

Put another way the existence of an X implies that there will be evidence of an X. Framed that way the aphorism looks like a logical fallacy in most cases – the exception being an X that has no material or observable impact on our world. If P implies Q and we can show not-Q then we can conclude not-P. With less the less tight logical implications of evidence, evidence for not-Q is evidence for not-P. Interestingly the reverse is less logically valid (i.e. evidence for Q implying P) but that also is a discussion for another time when I get to empiricism more generally.

Why then, is this aphorism so popular in discussion? Applied correctly it is not fallacious but it all very much depends on why there is an absence of evidence.

I am currently typing this on a train. Is there a man called James Smith on this train? As things stand I do have evidence that there are men on the train (I can see them) and my general knowledge about the country I am travelling in tells me that ‘James’ and ‘Smith’ would not be unusual names for a man on the train. However, apart from that I have no specific evidence on whether there is a James Smith on this train. Is this absence of evidence for James Smith on my train evidence that James Smith is absent from my train? Well, no or rather it isn’t very good evidence for one key reason: I haven’t actually looked.

What I should do if I really wish to establish whether there is a James Smith on the train is to seek him out. I could enter each carriage and call out “James Smith! I’m looking for James Smith!”. If a James Smith answers me then I have strong evidence that there was a James Smith on the train. If no James Smith answer then yes, this absence of evidence is evidence of absence. Not proof of course – James Smith may be hiding his true identity from the crazy person shouting on the train.

More generally absence of evidence is not evidence of absence when we either haven’t looked or we haven’t looked exhaustively or our experimental design or instrumentation may not have been sensitive enough.

Objective facts about intersubjective things

N-grams and metaphor maps, marking rubrics and lists of award winners all have a common theme: extracting objective facts from data about intersubjective entities. We can add to that list voting systems as well.

The net effect is, I believe, a way in which the nature and boundaries of science are changing in a world in which data collection and data structures become more common place. We can discover facts about purely invented things. Arguably this has always been the case with mathematics but now we have more definite cases.

Metaphor map

The University of Glasgow have produced an extraordinary thing – a giant map of metaphors.

A team who have been working on a historical thesaurus that tracks the interrelation of word meanings in English overtime, have used that work to plot network connections of word meaning.

The image below is one I generated from their map by looking at connections from the category of ‘Supernatural’ to the category of ‘Animals’.


A more complex graph below the fold of time and space!

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This is one of those posts on a somewhat abstract topic that I’m putting here for later so that I can point at the post and say “I mean that”.

This blog is interested in both logic and fiction. That is a problematic combination because you would assume logic is about objective truth and the qualities of fiction are subjective.

Objective truths pertain to things whose truth (in theory) is independent of any one person. There are issues with that but for the sake of argument let’s assume that we all know what we mean. If I say that a given novel was written in English, I am making a claim about an apparently objective truth. It isn’t enough to say that it is my opinion.

Subjective truths pertain to things whose truth rest with one person’s feelings and thoughts. I can know when I’m bored by a book but it is a subjective truth that the book is boring.

The gulf isn’t unsurmountable. Feelings and mental states can be observed in various ways and with the right experimental design I can establish truths that are objective but rest on subjective experiences. For example a claim that 90% of a given set of people find a book boring is a claim about an objective truth that can be investigated empirically.

Intersubjectivity is something else. Intersubjective truths rest on agreements between people. They behave in some ways like objective truths but at the same time they seem to rest on quite arbitrary grounds. The most obvious example is the meaning of words.

Words do not gain their meaning from some objective physical properties. The etymology of words might suggest meanings but there is no guarantee that the meaning of the word can be found from looking at it parts. If you find that hard to understand then just consider the word ‘understand’.

So while it seems like you could just pick any arbitrary meaning for a word and change how you would like, in reality you can’t because a word is only useful if other people know what you mean by it.

Intersubjective truths can feel a little baffling – they aren’t necessarily the same as a generally held opinion but they don’t seem to have anything thing else very much underneath them. And many important things like morality, social conduct, word meanings and grammar may rest on this notion of intersubjectivity.