The second view is the correct one, but your characterization of it is not quite right:
The second intuition is based on the self-consistence of mathematics: mathematics have its own ways how to decide between truth and falsity, and those ways never defer to the external world; therefore the first interpretation must be false.
It is not that the methods of determining mathematical truth “never defer to the external world”; it is that they do not depend on the external world on the object level. You do not verify that 2+2=4 by observing apples, and then earplugs, and then generalizing; instead, you verify it by writing down—on paper, or in your mind, or wherever and however, but yes, somewhere in the actual world—a formal proof within a formal system. You may then observe the behavior of apples to conclude that that formal system is a useful model of (some aspect of) the world—but that proposition is a distinct, meta-level proposition, and it is not the same as the object-level, within-the-formal-theory proposition that “2+2=4″.
Now, back in the days of the Babylonians, Egyptians, etc., before an independent discipline of “mathematics” existed, the first view was correct. Then, statements of arithmetic such as “2+2 = 4“ referred directly to abstracted physics: they were generalized propositions about apples and cats and dogs and so on. Subsequently, however, the Greeks came along and invented this new thing called “mathematics”. Ever since then, civilization has had a specific discipline where what you do is study the behavior of formal systems, and this discipline is now the appropriate context for the interpretation of statements such as “2+2=4”.
“2+2=4” is not a physical claim; a physical claim would be something like “apples behave according to the laws of integer addition as defined in ZFC set theory”. If the behavior of apples were to suddenly change, that would not change the truth-value of “2+2=4”; instead what would (or might) happen would be that funding agencies would demand that mathematicians stop studying ZFC set theory so much and start studying some other system that better modeled the new behavior of apples.
That does not mean, however, that “2+2=4” is “independent of the physical world”. The dependence simply occurs at a different logical level: the level where the instruments of formal verification (brains, computers, pencils, papers) are considered to be physical objects.
A note on language: “arithmetic” in English is singular. When I started reading your post I thought you were using the plural “arithmetics” to refer to distinct formalizations of arithmetic (there’s Peano Arithmetic, and so presumably other people have their own “Arithmetics”, or versions of arithmetic); but it soon became apparent that you simply thought the word was plural, analogously to “mathematics”. This is not the case.
(Also note that even though “mathematics” is plural in form, it acts grammatically as singular: “mathematics has”, not “mathematics have”, in your last paragraph.)
That does not mean, however, that “2+2=4” is “independent of the physical world”. The dependence simply occurs at a different logical level: the level where the instruments of formal verification (brains, computers, pencils, papers) are considered to be physical objects.
It doesn’t seem to me that the levels can be easily separated. Even if you verify that “2+2=4” using your brain, pencil or computer, I can always object that you have not verified the proposition itself, but that the system of brain, pencil or the specific computer program behaves according to the laws of integer addition as defined in ZFC set theory. Now the computer program was probably written to simulate the said laws of integer addition, while the apples were added before mathematics was formalised, but that’s a remark of merely historical interest.
Even if you verify that “2+2=4” using your brain, pencil or computer, I can always object that you have not verified the proposition itself, but that the system of brain, pencil or the specific computer program behaves according to the laws of integer addition as defined in ZFC set theory.
Actually, no: if you wanted to make an “objection” like this, you’d have to pass to the next level up. You’d have to say something like “we’ve just verified that the brain/pencil/computer behaves according to the laws of proof-verification as defined in [some formalization of metamathematics]”. This is not the same—and, crucially, is not incompatible with also having verified “2+2=4”. The two notions of verification are different: verifying “2+2 = 4″ has to do specifically with producing a proof, while verifying that “X behaves according to laws Y” involves empirically observing X in general, in a way that directly depends on what Y is.
(To help illustrate the distinction, note for example that to verify that “2+2 =4”, you only need to produce a single proof, in a single medium; whereas verifying that “X behaves according to Y” would usually require observing a multitude of instances.)
So this “objection” wouldn’t support the first view, which holds that “2+2=4″ is verified in the latter way, by observing apples (etc.), and denies the necessity of a formal proof intervening for verification to take place. (To the holder of the first view, a formal proof would just be a kind of instrument for predicting the behavior of apples and the like, and is not necessary to the meaning of the proposition.)
Of course I couldn’t say “we’ve just verified that the brain/pencil/computer behaves according to the laws of proof-verification as defined in [some formalization of metamathematics]” after producing a proof of “2+2=4″. To be able to say that, I would need to produce many different proofs and ascertain that they are correct from an independent source.
The idea of my “objection” was that I have to trust the medium of verification that it behaves according to the laws of proof-verification as defined in some formalization of metamathematics in order to use it for proving the theorem. But the medium itself is always part of the physical world, and there is no fundamental difference between proving the theorem using apples and proving it by drawing squiggles on a paper.
But the medium itself is always part of the physical world, and there is no fundamental difference between proving the theorem using apples and proving it by drawing squiggles on a paper.
The second statement does not follow from the first.
There is a difference between empirically observing that combining two apples with two apples yields four apples, and observing a sequence of squiggles on paper that constitutes a formal proof that 2+2=4. Yes, I’ll grant you that the difference isn’t written on the atoms making up the apples or the paper; rather it’s a matter of semantics, i.e. how these observations are interpreted by human minds. (You could presumably write out a formal proof using apples too—and then it would be just as different from the observation about combining pairs of apples as the squiggles on paper are.)
There is only one “level” of reality, but our model of reality can be organized into distinct levels. In particular, we use some parts of the physical world to model others; and when we do so, we have to be careful to distinguish discourse about the model from discourse within the model (i.e not to “confuse the map and the territory”).
The second view is the correct one, but your characterization of it is not quite right:
It is not that the methods of determining mathematical truth “never defer to the external world”; it is that they do not depend on the external world on the object level. You do not verify that 2+2=4 by observing apples, and then earplugs, and then generalizing; instead, you verify it by writing down—on paper, or in your mind, or wherever and however, but yes, somewhere in the actual world—a formal proof within a formal system. You may then observe the behavior of apples to conclude that that formal system is a useful model of (some aspect of) the world—but that proposition is a distinct, meta-level proposition, and it is not the same as the object-level, within-the-formal-theory proposition that “2+2=4″.
Now, back in the days of the Babylonians, Egyptians, etc., before an independent discipline of “mathematics” existed, the first view was correct. Then, statements of arithmetic such as “2+2 = 4“ referred directly to abstracted physics: they were generalized propositions about apples and cats and dogs and so on. Subsequently, however, the Greeks came along and invented this new thing called “mathematics”. Ever since then, civilization has had a specific discipline where what you do is study the behavior of formal systems, and this discipline is now the appropriate context for the interpretation of statements such as “2+2=4”.
“2+2=4” is not a physical claim; a physical claim would be something like “apples behave according to the laws of integer addition as defined in ZFC set theory”. If the behavior of apples were to suddenly change, that would not change the truth-value of “2+2=4”; instead what would (or might) happen would be that funding agencies would demand that mathematicians stop studying ZFC set theory so much and start studying some other system that better modeled the new behavior of apples.
That does not mean, however, that “2+2=4” is “independent of the physical world”. The dependence simply occurs at a different logical level: the level where the instruments of formal verification (brains, computers, pencils, papers) are considered to be physical objects.
A note on language: “arithmetic” in English is singular. When I started reading your post I thought you were using the plural “arithmetics” to refer to distinct formalizations of arithmetic (there’s Peano Arithmetic, and so presumably other people have their own “Arithmetics”, or versions of arithmetic); but it soon became apparent that you simply thought the word was plural, analogously to “mathematics”. This is not the case.
(Also note that even though “mathematics” is plural in form, it acts grammatically as singular: “mathematics has”, not “mathematics have”, in your last paragraph.)
Corrected, thanks.
It doesn’t seem to me that the levels can be easily separated. Even if you verify that “2+2=4” using your brain, pencil or computer, I can always object that you have not verified the proposition itself, but that the system of brain, pencil or the specific computer program behaves according to the laws of integer addition as defined in ZFC set theory. Now the computer program was probably written to simulate the said laws of integer addition, while the apples were added before mathematics was formalised, but that’s a remark of merely historical interest.
Actually, no: if you wanted to make an “objection” like this, you’d have to pass to the next level up. You’d have to say something like “we’ve just verified that the brain/pencil/computer behaves according to the laws of proof-verification as defined in [some formalization of metamathematics]”. This is not the same—and, crucially, is not incompatible with also having verified “2+2=4”. The two notions of verification are different: verifying “2+2 = 4″ has to do specifically with producing a proof, while verifying that “X behaves according to laws Y” involves empirically observing X in general, in a way that directly depends on what Y is.
(To help illustrate the distinction, note for example that to verify that “2+2 =4”, you only need to produce a single proof, in a single medium; whereas verifying that “X behaves according to Y” would usually require observing a multitude of instances.)
So this “objection” wouldn’t support the first view, which holds that “2+2=4″ is verified in the latter way, by observing apples (etc.), and denies the necessity of a formal proof intervening for verification to take place. (To the holder of the first view, a formal proof would just be a kind of instrument for predicting the behavior of apples and the like, and is not necessary to the meaning of the proposition.)
Of course I couldn’t say “we’ve just verified that the brain/pencil/computer behaves according to the laws of proof-verification as defined in [some formalization of metamathematics]” after producing a proof of “2+2=4″. To be able to say that, I would need to produce many different proofs and ascertain that they are correct from an independent source.
The idea of my “objection” was that I have to trust the medium of verification that it behaves according to the laws of proof-verification as defined in some formalization of metamathematics in order to use it for proving the theorem. But the medium itself is always part of the physical world, and there is no fundamental difference between proving the theorem using apples and proving it by drawing squiggles on a paper.
The second statement does not follow from the first.
There is a difference between empirically observing that combining two apples with two apples yields four apples, and observing a sequence of squiggles on paper that constitutes a formal proof that 2+2=4. Yes, I’ll grant you that the difference isn’t written on the atoms making up the apples or the paper; rather it’s a matter of semantics, i.e. how these observations are interpreted by human minds. (You could presumably write out a formal proof using apples too—and then it would be just as different from the observation about combining pairs of apples as the squiggles on paper are.)
There is only one “level” of reality, but our model of reality can be organized into distinct levels. In particular, we use some parts of the physical world to model others; and when we do so, we have to be careful to distinguish discourse about the model from discourse within the model (i.e not to “confuse the map and the territory”).