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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/
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/
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.
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.
