A thread on the 2+2 thing. It’s a bit of a slow burn:

In high school physics, you learn that acceleration due to gravity is -9.8 m/s^2. You can use that fact to model projectile motion.

Is that true? Well, the answer is “you’re asking the wrong question”
1/2

There are situations where this is an appropriate model to use, but there are other situations where it isn’t. Maybe you need to take air resistance into account. Or (in baseball) the rotation of the ball matters. Or (in rocketry or planetary motion) you need to use
2/n

the more general Newton law for gravity. Or maybe you need to use relativity.

It doesn’t make sense to say that any of these models are “right” or “wrong” in an absolute sense. Each model is appropriate in some situations and inappropriate in others.
3/n

Math provides abstract frameworks that can be components of a model. *Whenever you apply math to a real-world problem, you are using math as part of a model.*

4/n

I’ll repeat myself because it’s important:
It doesn’t make sense to say that a model is right or wrong in an absolute sense. It only makes sense to say that a model is appropriate in some situations and inappropriate in others.

5/n

Basic arithmetic (where 2+2=4) is a model that’s useful in a lot of situations! That’s why we teach it to the children!
6/n

But even little children encounter situations where more subtlety is required. My 5 year old would count 2 pennies and 2 quarters and conclude that he has “4 moneys”. If you tell him to wait for 5 minutes he’ll count to 5 and say “are you ready now?”
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If I tell him that 22 is “2 and then 2” he will say “I thought 2 and 2 was 4”.

There are some situations where 2+2=4 is an appropriate model and there are other situations where it isn’t.
8/n

I would argue that it’s less important to know that 2+2=4 than it is to know when it is the appropriate model.

The reason? It’s in your hands right now. Phones/computers/calculators can do addition for you. They still aren’t that good at telling you when you should add.
9/n

“Hey Google! What’s 2+2?”
“The answer is 4.”
“Hey Google! What’s 2 pennies and 2 quarters?”
“Sorry, I don’t have any information on that.”

This was a real conversation that just happened.
10/n

These sorts of issues are things that kids can and should learn about!
11/n

Math is, roughly, the subject that studies abstract models and their inner logic. Math contains multitudes.

Basic arithmetic is and always will be considered one of the most important models. But mathematicians also study operations where 2+2=22, or
12/n

or where 2 pennies + 2 quarters = 52 cents, etc. All of these models have their place.

So is 2+2=4? You’re asking the wrong question.
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Well this is my first tweet with more than 6 or 7 likes. I think I’m supposed to plug something? Here’s a recent paper that I wrote with an amazing undergrad student: https://arxiv.org/abs/1907.00702 

Also I’m a big fan of the Invisibilia podcast. No personal connection, just something I like and that helps me think about the world in different ways.