Category Archives: Mathematics

Known Unknowns

This month’s post may make a valid point.  Or it may not.  Or it may be impossible to tell, the concept of which itself may or may not make sense by the end of the piece!

How do we handle things we don’t know?  More precisely, how do we cope with things we know we don’t know?  All right then: how do we handle things we know we can’t know?

As is the nature of this blog, the examples we’re going to discuss are (at first, at least) taken from the fields of computer science and mathematics; but there are plenty of analogies in the other sciences.  This certainly isn’t a purely theoretical discussion.

On the whole, we like things (statements or propositions) in mathematics (say) to be right or wrong: true or false.  Some simple examples are:

  • The statement “2 > 3” is false
  • The statement “There is a value of x such that x < 4” is true
  • The proposition “There are integer values of x, y and z satisfying the equation x3 + y3 = z3” is false

OK, that’s pretty straightforward but how about this one?

  • “Every even number (greater than 2) is the sum of two prime numbers”

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Santa in the Continuum

This year’s festive offering considers countable and non-countable infinities and follows (very) loosely from last year’s discussion of deterministic and non-determininstic optimisation

[Specially for Alex Irvine, who is either having trouble getting his head around infinite sets or still believes in Santa Claus: we’re not allowed to say which]

Father Christmas has a problem. The Intergalactic Department of Work and Pensions (IDWP) has threatened to cut his tax credits because he apparently only works one day a year. He’s tried to point out that he’s the head of a vast multinational organisation of elves and reindeer, who themselves work all year round, but that doesn’t wash with the IDWP boss, Ian ‘Dunkin’ Smiff, because his dad didn’t go to public school.

Planets

So, he’s been given extra work to do: a lot of extra work to do …

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Clear as Maths!

This post follows on (loosely) from a previous discussion on maths and computing and asks what it really means to ‘prove’ something in each discipline.

An apocryphal story has an Oxbridge maths don lecturing to a group of undergraduates … After some time completely filling a huge blackboard with heavy calculus – with accompanying commentary, he turns to the class and casually notes, “So then, it’s clear that …” (the exact claim isn’t important). As he turns to resume his chalk-work, a particularly bold student enquires, “Excuse me, Professor; but is that really ‘clear’?” The don steps back and surveys his work; studying the entire board from top-left to bottom-right, with numerous head and eye movements to-and-fro – even some pointing – to cross-check various parts with each other. After a full five minutes of silent contemplation, he turns back to the students, smiles, announces, “Yes!”, and carries on as before.

So who’s defining ‘clear’ here?

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The Travelling Santa Problem

Why is the annual Father Christmas world tour ‘NP-Complete’?  What is ‘NP-Complete’ anyway?  In fact, what’s ‘NP’ even?  And why can only Santa do it?  And what’s a ‘Decision Elf’?  And just how awesome is Rudolph’s nose?  What’s ‘Inspirational Magic’?  This and so much more you didn’t know about the annual Christmas Eve ritual …

We all know the problem well enough …  Father Christmas has to travel around the world, visiting all the good girls and boys, in a single night, and return to where he started at the North Pole. Sounds like a tall order!  Can it be done?

Santa

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