We might mean many things by “2 + 2 = 4”. In PA:
“PA |- SS0 + SS0 = SSSS0″, and so by soundness “PA |= SS0 + SS0 = SSSS0”
In that sense, it is a logical truism independent of people counting apples. Of course, this is clearly not what most people mean by “2+2=4″, if for no other reason than people did number theory before Peano.
When applied to apples, “2 + 2 = 4” probably is meant as:
“apples + the world |= 2 apples + 2 apples = 4 apples”.
the truth of which depends on the nature of “the world”. It seems to be a correct statement about apples. Technically I have not checked this property of apples recently, but when I consider placing 2 apples on a table, and then 2 more, I think I can remove 4 apples and have none left. It seems that if I require 4 apples, it suffices to find 2 and then 2 more. This is also true of envelopes, paperclips, M&M’s and other objects I use. So I generalise a law like behaviour of the world that “2 things + 2 things makes 4 things, for ordinary sorts of things (eg. apples)”.
At some level, this is part of why I care about things that PA entails, rather than an arbitrary symbol game; it seems that PA is a logical structure that extracts lawlike behaviour of the world. If I assumed a different system, I might get “2+2=5”, but then I don’t think the system would correspond to the behaviours of apples and M&M’s that I want to generalise.
(On the other hand, PA clearly isn’t enough; it seems to me that strengthened finite Ramsey is true, but PA doesn’t show it. But then we get into ZFC / second order arithmetic, and then systems at least as strong as PA_ordinal, and still lose because there are no infinite descending chains in the ordinals)
We might mean many things by “2 + 2 = 4”. In PA: “PA |- SS0 + SS0 = SSSS0″, and so by soundness “PA |= SS0 + SS0 = SSSS0” In that sense, it is a logical truism independent of people counting apples. Of course, this is clearly not what most people mean by “2+2=4″, if for no other reason than people did number theory before Peano.
When applied to apples, “2 + 2 = 4” probably is meant as: “apples + the world |= 2 apples + 2 apples = 4 apples”. the truth of which depends on the nature of “the world”. It seems to be a correct statement about apples. Technically I have not checked this property of apples recently, but when I consider placing 2 apples on a table, and then 2 more, I think I can remove 4 apples and have none left. It seems that if I require 4 apples, it suffices to find 2 and then 2 more. This is also true of envelopes, paperclips, M&M’s and other objects I use. So I generalise a law like behaviour of the world that “2 things + 2 things makes 4 things, for ordinary sorts of things (eg. apples)”.
At some level, this is part of why I care about things that PA entails, rather than an arbitrary symbol game; it seems that PA is a logical structure that extracts lawlike behaviour of the world. If I assumed a different system, I might get “2+2=5”, but then I don’t think the system would correspond to the behaviours of apples and M&M’s that I want to generalise.
(On the other hand, PA clearly isn’t enough; it seems to me that strengthened finite Ramsey is true, but PA doesn’t show it. But then we get into ZFC / second order arithmetic, and then systems at least as strong as PA_ordinal, and still lose because there are no infinite descending chains in the ordinals)