Can you define “do arithmetic” in such a manner that it is at least as easy to prove that arithmetic has been done as it is to prove that excavation has been done?
Is still somewhat important to the discussion. I can’t define arithmetic well enough to determine if it has occurred in all cases, but ‘changes on a display’ is clearly neither necessary nor sufficient.
Well, I’d say that a system is doing arithmetic if it has behavior that looks like it corresponds with the mathematical functions that define arithmetic. In other words, it takes as inputs things that are representations of such things as “2”, “3“, and “+” and returns an output that looks like “6”. In an arithmetic logic unit, the inputs and outputs that represent numbers and operations are voltages. It’s extremely difficult, but it is possible to use a microscopic probe to measure the internal voltages in an integrated circuit as it operates. (Mostly, we know what’s going on inside a chip by far more indirect means, such as the “changes on a screen” you mentioned.)
There is indeed a lot of wiggle room here; a sufficiently complicated scheme can make anything “represent” anything else, but that’s a problem beyond the scope of this comment. ;)
Note that neither an abacus nor a calculator in a vacuum satisfy that definition.
I’ll allow voltages and mental states to serve as evidence, even if they are not possible to measure directly.
Does a calculator with no labels on the buttons do arithmetic in the same sense that a standard one does?
Does the phrase “2+3=6” do arithmetic? What about the phrase “2*3=6″?
I will accept as obvious that arithmetic occurs in the case of a person using a calculator to perform arithmetic, but not obvious during precisely what periods arithmetic is occurring and not occurring.
Well, yeah. My question:
Is still somewhat important to the discussion. I can’t define arithmetic well enough to determine if it has occurred in all cases, but ‘changes on a display’ is clearly neither necessary nor sufficient.
Well, I’d say that a system is doing arithmetic if it has behavior that looks like it corresponds with the mathematical functions that define arithmetic. In other words, it takes as inputs things that are representations of such things as “2”, “3“, and “+” and returns an output that looks like “6”. In an arithmetic logic unit, the inputs and outputs that represent numbers and operations are voltages. It’s extremely difficult, but it is possible to use a microscopic probe to measure the internal voltages in an integrated circuit as it operates. (Mostly, we know what’s going on inside a chip by far more indirect means, such as the “changes on a screen” you mentioned.)
There is indeed a lot of wiggle room here; a sufficiently complicated scheme can make anything “represent” anything else, but that’s a problem beyond the scope of this comment. ;)
edit: I’m an idiot, 2 + 3 = 5. :(
Note that neither an abacus nor a calculator in a vacuum satisfy that definition.
I’ll allow voltages and mental states to serve as evidence, even if they are not possible to measure directly.
Does a calculator with no labels on the buttons do arithmetic in the same sense that a standard one does?
Does the phrase “2+3=6” do arithmetic? What about the phrase “2*3=6″?
I will accept as obvious that arithmetic occurs in the case of a person using a calculator to perform arithmetic, but not obvious during precisely what periods arithmetic is occurring and not occurring.