How do we know that 3×4 = 4×3 ?

Why even ask a question like that?  Everyone knows that 3\times4=4\times3, just like everyone knows that 3+4=4+3.  Mathematicians would say that multiplication or addition of two numbers is commutative, meaning that you get the same answer regardless of the order in which you do the multiplication or addition.  But what does that really mean?  When we say 3\times4, we literally mean “3, 4 times,” or 4 sets of 3 objects each, or 3+3+3+3 objects.  Likewise, when we say 4\times3, we mean “4, 3 times,” or 3 sets of 4 objects each, or 4+4+4 objects.  Most people would think it’s obvious that 3\times4=4\times3, but it’s equivalent to saying that 3+3+3+3=4+4+4, which seems quite a bit less obvious.  So how do we know it?  Is the fact that multiplication is commutative just something we assume about numbers, or is there a way to prove it?

Continue reading

How do we know the brain is where thoughts occur?

Anyone with a brain knows that our thoughts happen in our brains.  But our metaphors belie our confidence in this assertion:  “I know it in my heart,” “I have a gut feeling about it,” “I use my muscle memory to play this game.”  Each of these statements says a little bit about what it feels like to process information.  Do we know that our thoughts occur inside our heads because we can feel them happening there?  Or do we feel them happening there because we’ve been taught very early on that our brains are where our thinking happens?

Continue reading