Thoughts about Philosophy (in the Maths Classroom)

I'm currently writing an article about using philosophy techniques in the Maths classroom. It's certainly not something that springs to mind when one thinks of a typical maths lesson...but when you break down what it means to behave philosophically it becomes clear that some very important mathematical behaviours employed.

What do I mean by this?

Well to start with we have discussion: how many times has an idea become clearer because you talked it through with someone? It appears to me that this trait is inherently mathematical.

We also have conjecturing; a behaviour vital to mathematics. In order for us to make progress we have to trial ideas and then test them in some general space. A maths classroom seems to fit with that well.

Supporting arguments with examples comes through in such a context. If one makes a conjecture that is disagreed with it is important to back ideas up and this is done through giving specific examples.

I feel as though the list could go on. How do you feel about such a lesson? Students I have used it with have responded well but is this enough? Do you think that the behaviours demonstrated offer enough justification?

The Proof is in the P(udding)ierce's Lemma

Hey there I just unearthed an old proof that I liked from my degree. I figured some folk may appreciate it so here it is.

If you don't appreciate it then please enjoy the cartoon above that I borrowed from here.

Pierce's Lemma

Show that: ¦-- (((a->b)->a)->a) (=: c)

Pf: So rewrite the lemma as ¦-- c
By one of the "famous" theorems of prop. calc. we have

¦-- ((a->c)->((¬a->c)->c))

So Suff. To Prove (i)¦--(a->c) & (ii)¦--(¬a->c)

(i) by applying DT twice it is STP that

{a, ((a->b)->a)} ¦-- a

a one line pf so done

(ii) by applying DT twice it is STP that

{¬a, ((a->b)->a)} ¦-- a

One of the theorems says: ¦-- (¬a->(a->b))
So we have that {¬a, ((a->b)->a)} ¦-- ¬a
{¬a, ((a->b)->a)} ¦-- (¬a->(a->b))
so by MP: {¬a, ((a->b)->a)} ¦-- (a->b)
{¬a, ((a->b)->a)} ¦-- ((a->b)->a)
so by MP: {¬a, ((a->b)->a)} ¦-- a

So we've proved (i) and (ii) and so by the tautology we have Peirce's Lemma.

What do you get if you...

So here goes making a start with things and all. I've been wanting to do this for a while. Writing... about maths that is. You see I'm often struck by something or other to do with maths. So here is my small meaningless foray into the blogosphere *shudder*.

So where now? I guess we should get things a-going! a-rolling!

I was shown a problem today...

You have to place the numbers 1 to 7 in each of the 7 sections such that each circle's numbers sum to the same total... i.e. if you put 1 to 4 in the first 4 sections...

the first circle would sum to 1 + 2 = 3
the second circle would sum to 2 + 3 + 4 = 9
So this can't be right...

The real problem comes in when you try to solve this without resulting to trial and improvement...hmm...I found a solution I'm just not convinced that it was a very mathematical way of going about things...I'll leave it to the comments I guess...