This poll seems to have amused some people and annoyed others: my guess is that the former mainly voted "No" or "It's complicated" and the latter mainly voted "Yes". Interestingly, "Yes" was in a clear minority for the first few thousand votes, but ended at just over 50%. 1/ https://twitter.com/wtgowers/status/1295992753831149568

I think that's because at first most voters came from my followers, who are largely mathy and techy, but then @ConceptualJames sent his troops over to save the day.

What follows is a brief (if I can manage it) statement of my philosophical position about mathematics. 2/

Incidentally, for those worried that my views will dangerously infect my teaching, I can provide some assurance: it's basically impossible to deduce from the way a university-level mathematician teaches what his/her philosophical views about mathematics are. 3/

Also, for those who'd like me to cut to the chase, let me start with the question of whether 2+2=4. I'll go this far straight away.

1. If it's clear that we're talking about addition of positive integers as normally understood, then yes of course.

4/

2. If the context is not specified, then it's very likely that addition of positive integers is what is being referred to.

3. I know of no mathematical contexts in which people routinely declare that 2+2=5.

So does that mean that I agree after all that 2+2=4? 5/

Yes it does, as long as you'll allow me to qualify it very slightly by adding the words "within a certain standard number system". What I don't like is the idea that the statement "2+2=4" is describing a fact about the abstract realm -- that is, I am not a Platonist. 6/

Many mathematicians and philosophers *are* Platonists, and I respect their views, but disagree with them. Many others are more like me.

I'm more drawn to formalism -- the view that mathematics is a kind of game where we play around with symbols according to certain rules. 7/

I adopted that view after learning that the continuum hypothesis (the statement that there is no infinite set that is bigger than the natural numbers but smaller than the real numbers) can neither be proved nor disproved. 8/

The idea that it must nevertheless either be true or false seems to me ridiculous. But it also seems ridiculous to me that more basic mathematical statements have objective truth values and that somehow only when the statements get a bit exotic does that all break down. 9/

It's far simpler just to talk about whether a statement can or can't be proved given a certain system of axioms and certain rules of deduction.

To be clear, I don't object to the notion of truth in mathematics, but for me it's always truth within a system. 10/

Where I slightly depart from full-on formalism is that I'm happy to acknowledge that some parts of mathematics have a more direct connection to physical reality than others. For instance, if my wife and I invite another couple round to dinner, I'll put out four plates. 11/

Why? Because (i) 2+2=4 in standard arithmetic and (ii) standard arithmetic, at least when the numbers involved are small, is very useful for situations like this. (If you thought I would put out five plates, then you've not been following the discussion properly.) 12/

But sometimes the relationship between 2+2=4 and reality is a bit more complicated. For example, if I put two pints of water in a jug and then another two pints, will I have four pints? 13/

In a sense, yes, but if we dig down a little and ask what "two pints" means, it turns out that the notion of "exactly two pints" is a complete fantasy. So what does the fact that 2+2=4 in the real number system tell us about pints of water? 14/

It makes us confident that if we put approximately two pints of water into a jug and then approximately another two pints into the jug, then we'll have approximately four pints. So the real numbers are a great model for this physical set-up, but not a perfect one. 15/

Here's another example. Suppose that there are three trains on a straight track and that the first is going at 2mph relative to the second and the second is going at 2mph relative to the third. Does that mean that the first is going at 4mph relative to the third? 16/

Since Einstein's special theory of relativity, we've known that the answer is NO. The first is going at very slightly under 4mph relative to the third. That's very counterintuitive, but I'm afraid it's the truth. 17/

So that's a situation where one might have expected 2+2=4 to be a very direct model for reality, but in fact it isn't. (Does that threaten the status of 2+2=4? Not if we do as I advocate and talk about the truth *within a system*.) 18/

I'll end by justifying my answer of "It's complicated" for the poll. The basic point of the question of whether 2+2=4.00 is that the numbers on the left-hand side look like integers and the number on the right-hand side looks like a real number. 19/

And while in elementary mathematics it's fairly harmless to think of the real number 2 as being "the same" as the integer 2, in higher mathematics that's very inconvenient for all sorts of reasons. And in fact, even non-mathematicians treat them rather differently: 20/

typically 2 the integer is used for counting, whereas 2 the real number is used for measuring.

But in higher mathematics there's a technical sense in which integers *aren't* real numbers -- we say instead that they can be *identified* with real numbers. 21/

This matters a lot in computer programming too. So while there's certainly a sense in which 2+2=4.00, there's also a useful and sensible sense in which 2+2 and 4.00 are simply different kinds of objects (or "types", to use a more technical term) so can't be compared. 22/

In other words, it's complicated.

PS I don't think @kareem_carr would disagree with any of this, except perhaps minor details.

Actually what I meant by that PS was something stronger -- I see what I've written above as an elaboration of part of what @kareem_carr has already said.