The Plus Advent Calendar Door #9: But is it all true?
With the weather cold and the nights very dark, it's the perfect time to curl up in front of the fire and do some philosophising. Here are some of our favourite articles on mathematics and the nature of truth ... or lack of it!
Searching for the missing truth
Many people like mathematics because it gives definite answers. Things are either true or false, and true things seem true in a very fundamental way. But it's not quite like that. You can actually build different versions of maths in which statements are true or false depending on your preference. So is maths just a game in which we choose the rules to suit our purpose? Or is there a "correct" set of rules to use? We find out with the mathematician Hugh Woodin. Read more...
In the 1930s the logician Kurt Gödel showed that if you set out proper rules for mathematics, you lose the ability to decide whether certain statements are true or false. This is rather shocking and you may wonder why Gödel's result hasn't wiped out mathematics once and for all. The answer is that, initially at least, the unprovable statements logicians came up with were quite contrived. But are they about to enter mainstream mathematics? Read more...
This is not a carrot: Paraconsistent mathematics
Paraconsistent mathematics is a type of mathematics in which contradictions may be true. In such a system it is perfectly possible for a statement A and its negation not A to both be true. How can this be, and be coherent? What does it all mean? Read more...
If you like mathematics because things are either true or false, then you'll be worried to hear that in some quarters this basic concept is hotly disputed. This article looks at constructivist mathematics, which holds that some things are neither true, nor false, nor anything in between. Read more...
To read more about maths and philosophy, have a look at our package Mathematics and the nature of reality.