Patterns

Children are often given mathematical puzzles that involve working out how to continue a sequence of numbers.  Wittgenstein focuses on this kind of situation when discussing rules and mathematics.

There can, of course, be no 'right' answer to how to continue any finite sequence, since there will be an infinte number of potential rules that its generation might instantiate.  Given this, it is odd that children are - generally explicitly, and always implicitly - told that there is a 'right' answer.  Valid, but peculiar, rules will be rejected (e.g. suggesting that the sequence '2, 4, 6 ...' instantiates listing even numbers in the odd hundreds and odd ones in the even hundreds).

This is an essential part of teaching mathematics - this weeding of peculiar rules.  In fact, a sound intuition about which rules would be peculiar is a component of 'mathemtical ability'.

A child that is being given this mathematical training is getting two signals from the teacher with each exercise: one is the list of numbers, and the other is a reassurance that there isn't something 'tricky' going on - that this list really is a sort of message, a hint that some simple and obvious solution is the 'correct' one.

We handle numbers using manageable rules - which comprise a subset of the formally possible rules.  All the mathematical 'observations' made by humanity so far comprise a finite set, and so could instantiate an infinitely large number of mutually incoherent sets of rules.  We have selected from these the humanly manageable ones which appear to be coherent.

We have no way - we could have no way - of demonstrating that the set we have selected is the only possible one.  Such a demonstration would depend upon a set of rules which would have to have mathematical meanings.

If we had two sets of rules which were internally consistent (and so were consistent with all our 'observations') but inconsistent with each other, we could only choose between them on the basis of an observation we had not previously made - but the generation of mathematical 'observations' is a rule-driven business ...

Maybe we cannot state a complete set of rules, because we modify their meanings as we do mathematics - because doing mathematics is partly a matter of resolving rule ambiguities.

And that's apart from any open question issues which would arise.