1 + 1 = MORE(or LESS) Story

It was an ordinary day.  Reading a story titled 1+1=More (or LESS).  But...why does one plus one equal More?  One possible answer is 'it just does!"  That is really a variation on "Because I said so!" - and answer that has been frustrating children for generations.  It is quite right to feel frustrated by that idea.  Math can seem like a world of rules you just have to follow, which makes it seem rigid and boring.  Whereas my love of math is somewhat driven by my love of breaking rules, or at least pushing against them.  both of those impulses play an important role in advancing human understanding, and in particular mathematical understanding.  So other than think about why one plus one is two, let's go a little further and question whether it's even true all the time.  Sometimes, one plus one can equal more than two. If you have enough cash on you to buy one cup of coffee, and your friend has enough to buy one, then together you still might have enough to buy three.  Because if you have 1.5 or even 1.9 times the money needed for a cup of coffee, that still only gets you one cup on your own.  One plus one can also equal more than two because of reproduction: Say you put one rabbit and another rabbit together.  You might well end up with a lot of rabbits.  Or sometimes it's because the things you're adding together are more complicated : If one pair of tennis players gets together with another pair for an afternoon of tennis, there ends up being more than pairs of tennis players because they could play each other in all sorts of different combinations.  If the first pair are called A and B, and the second pair are called C and D, then we have the following pairs in total: AB, AC, AD, BC, BD, CD.  So one pair of tennis payers plus another pair makes six pairs.  Sometimes, one plus one is just one, like if you put a pile of sand on top of another pile of sand, then you get one pile of sand.  Or, as an art student of mine pointed out, if you mix one color with one color, you get one color.  Or, as I saw in an amusing meme, if you put a lasagna on top of another lasagna, it's still just one lasagna (a taller one).  And, in some situations, one plus one is actually zero.  If I say "I'm not not hungry," that means "I'm hungry,"  Some children find it very funny to say "I'm not not not not not not not not not not not hungry!" and dissolve into hysterics because they know nobody has been able to keep track of how many times they said "not."  The point here is that one "not" plus one "not" is the same as zero "nots."  Now, you might thing that these aren't really situations in which one plus one equals something else, because they aren't really addition, or because those aren't really numbers.  You're welcome to think that, but that's not what math does.  Math instead says: Let's work out the context in which one plus one really does equal two, and contexts in which it doesn't.  And in doing so, we'll understand something about the world more deeply than we did before.  Math isn't really about getting the right answer; it's about building good justifications.  This gets implemented in schools as children needing to learn different "strategies" for doing the same thing, and I often hear parents complaining how pointless this is because if they can do something one way, why do they need to know all these other ways?  But, having different ways to think about something constitutes a deeper understanding of that thing, and it gives you more ways to check that what you're doing is secure.  Imagine we were designing a jungle gym for children.  We'd want to test it in every possible way to make sure it's safe.  We wouldn't test it by just playing on it in sensible ways: We'd want to jump on it, swing from it, bash into it, fall from it, and try to pull it out of the ground, rather than simply trusting that we built it well.  The solidity of math comes from not wanting to trust things, but wanting to jump and swing and know that our framework will hold up.  One of the reasons the framework is so strong is precisely because we question it so deeply.  I hope that we will start seeing mathematics as a place to pose questions and explore answers, rather than a place where the answers are fixed and were supposed to know them.  And I hope we will place more emphasis on those who are curious, and who follow their curiosity on a journey that may be slow and without a clear destination, a quiet walk through the countryside rather than a race to the finish.  It was another extraordinary day in the life of an ordinary guy.