‘2+2=4’ can be causally linked to reality. If you take 2 objects, and add 2 others, you’ve got 4, and this can be mapped back to the concept of ‘2+2=4’. Computers, and your brain, do it all the time.
This argument falls when we start talking about things which don’t seem to actually exist, like fractions when talking about indivisible particles. But numbers can be mapped to many things (that’s what abstracting things tends to do), so even though fractions don’t exist in that particular case, they do when talking about pies, so fractions can be mapped back to reality.
But this second argument seems to fall when talking about things like infinities, which can’t be mapped back to reality, as far as I know (maybe when talking about the number of points in a distance?). But in that case, we are just extrapolating rules which we have already mapped from the universe into our models. We know how the laws of physics work, so when we see the spaceship going of into the distance, where we’ll never be able to interact with it again, we know it’s still there, because we are extrapolating the laws of physics to outside the observable universe. Likewise, when confronted with infinity, mathematicians extrapolated certain known rules, and from that inferred properties about infinities, and because their rules were correct, whenever computations involving infities were resolved to more manageable numbers, they were consistent with everything else.
So our representations of numbers are a map of the territory (actually, many territories, because numbers are abstract).
‘2+2=4’ can be causally linked to reality. If you take 2 objects, and add 2 others, you’ve got 4, and this can be mapped back to the concept of ‘2+2=4’. Computers, and your brain, do it all the time.
This argument falls when we start talking about things which don’t seem to actually exist, like fractions when talking about indivisible particles. But numbers can be mapped to many things (that’s what abstracting things tends to do), so even though fractions don’t exist in that particular case, they do when talking about pies, so fractions can be mapped back to reality.
But this second argument seems to fall when talking about things like infinities, which can’t be mapped back to reality, as far as I know (maybe when talking about the number of points in a distance?). But in that case, we are just extrapolating rules which we have already mapped from the universe into our models. We know how the laws of physics work, so when we see the spaceship going of into the distance, where we’ll never be able to interact with it again, we know it’s still there, because we are extrapolating the laws of physics to outside the observable universe. Likewise, when confronted with infinity, mathematicians extrapolated certain known rules, and from that inferred properties about infinities, and because their rules were correct, whenever computations involving infities were resolved to more manageable numbers, they were consistent with everything else.
So our representations of numbers are a map of the territory (actually, many territories, because numbers are abstract).