I agree that the downvoting of this comment was overly harsh. My theory on why it occurred is different, and best illustrated by an example: if someone posted a comment saying “2+2=4 is only true in some contexts; in arithmetic modulo 3, 2+2=1”, that comment would have been similarly downvoted.
However, let me be so bold as to say a word in defense of even that hypothetical commenter. Anyone mathematically sophisticated (including our downvoters) will agree that it is possible to construct a mathematical system in which 2+2 equals anything you like—or, more precisely, for any symbol x, a system can be constructed in which the formula (string of symbols) “2+2 = x” is given the label “TRUE”. Mod 3 arithmetic is an example for x = “1”.
Now, it is at this point that the downvoters protest: “But this is not the same thing as saying 2+2=1! All you’ve done is change the meaning of the symbols in the formula, such as ‘2’ and ‘1’. Two plus two is still four, for the original meaning of those words. You’re confusing the map and the territory. Downvoted!”
Well, the downvoters do have a point. But, at the same time, let me suggest that they’re also making the same mistake as our poor beleaguered commenter!
What they’ve done, you see, is to make a leap from “Ordinary (i.e. non mod-3, etc.) Arithmetic accurately models certain physical phenomena” to something like “Ordinary Arithmetic is true in (or of) the physical world”. Instead of saying what they mean, which is “the physical world is best modeled by a system that has ‘2+2=4’ as a ‘TRUE’ formula”, they say “2+2 is in fact equal to 4″.
Small wonder that confusion arises about whether mathematical statements are “emprical” or not! “The physical world is best modeled by a system that has ‘2+2=4’ as a ‘TRUE’ formula” is clearly an empirical claim. But what about 2+2 = 4, all by itself? When a mathematician at a blackboard proves that 2+2=4 in Ordinary Arithmetic (or, for Eliezer’s benefit, that infinite sets exist in standard set theory), has he or she made a claim about physics? No! Not without the additional assumption that the formal system being used is in fact an accurate map of the territory! But the mathematician makes no such assumption; he or she (acting as a mathematician) is interested only in the properties of formal systems. (Yes, that’s right: I’m advocating the view known as formalism here. The other well-known positions in the philosophy of mathematics, namely Platonism and intuitionism, suffer from map-territory confusion!)
Mathematical systems, like Ordinary Arithmetic or Mod-3 Arithmetic, are part of the map, not the territory. The facts of mathematics are, so to speak, cartographic, rather than geographic.
In the OB post tautologies have to be empirically observed somehow, Eliezer writes about waking up one day and discovering all sorts of evidence that 2+2=3. This wouldn’t be evidence that 2+2=3 in Peano arithmetic, it would be evidence that Peano arithmetic just doesn’t apply for some reason. In my down-voted comment, I was just giving an example of how there can be different kinds of arithmetic if you are willing to be flexible about what arithmetic is. (If you are not willing to be flexible, then you are not willing to allow the observation that 2+2=3 as an observation about arithmetic, because this is not possibly true in standard arithmetic. Well, the observations are possible but you’d have to account for it as some kind of grand delusion.) My point is that 2+2=4 in Peano Arithmetic independent of observation, but observation tells you if Peano arithmetic applies or not.
This wouldn’t be evidence that 2+2=3 in Peano arithmetic, it would be evidence that Peano arithmetic just doesn’t apply for some reason.
Exactly.
My point is that 2+2=4 in Peano Arithmetic independent of observation, but observation tells you if Peano arithmetic applies or not.
It is worth emphasizing that to claim that “2+2=4 in Peano Arithmetic independent of observation” is not to claim that our knowledge of this fact about Peano Arithmetic is independent of observation. (The former claim is about our map of the territory; the latter is about our map of our map of the territory.)
It is worth emphasizing that to claim that “2+2=4 in Peano Arithmetic independent of observation” is not to claim that our knowledge of this fact about Peano Arithmetic is independent of observation. (The former claim is about our map of the territory; the latter is about our map of our map of the territory.)
Could you elaborate? It sounds to me like the former claim is about the territory, and the latter is just hard for me to parse.
I’ll emphasize with the following analogy: you need to observe the sun to know of it. However, you can nevertheless be certain—as certain as you are of anything at all—that the sun exists independently of observation. You need to define the Peano axioms and observe the deductions that lead to the tautologies to know of them, but they are mathematically true independent of your observation.
I agree that the downvoting of this comment was overly harsh. My theory on why it occurred is different, and best illustrated by an example: if someone posted a comment saying “2+2=4 is only true in some contexts; in arithmetic modulo 3, 2+2=1”, that comment would have been similarly downvoted.
However, let me be so bold as to say a word in defense of even that hypothetical commenter. Anyone mathematically sophisticated (including our downvoters) will agree that it is possible to construct a mathematical system in which 2+2 equals anything you like—or, more precisely, for any symbol x, a system can be constructed in which the formula (string of symbols) “2+2 = x” is given the label “TRUE”. Mod 3 arithmetic is an example for x = “1”.
Now, it is at this point that the downvoters protest: “But this is not the same thing as saying 2+2=1! All you’ve done is change the meaning of the symbols in the formula, such as ‘2’ and ‘1’. Two plus two is still four, for the original meaning of those words. You’re confusing the map and the territory. Downvoted!”
Well, the downvoters do have a point. But, at the same time, let me suggest that they’re also making the same mistake as our poor beleaguered commenter!
What they’ve done, you see, is to make a leap from “Ordinary (i.e. non mod-3, etc.) Arithmetic accurately models certain physical phenomena” to something like “Ordinary Arithmetic is true in (or of) the physical world”. Instead of saying what they mean, which is “the physical world is best modeled by a system that has ‘2+2=4’ as a ‘TRUE’ formula”, they say “2+2 is in fact equal to 4″.
Small wonder that confusion arises about whether mathematical statements are “emprical” or not! “The physical world is best modeled by a system that has ‘2+2=4’ as a ‘TRUE’ formula” is clearly an empirical claim. But what about 2+2 = 4, all by itself? When a mathematician at a blackboard proves that 2+2=4 in Ordinary Arithmetic (or, for Eliezer’s benefit, that infinite sets exist in standard set theory), has he or she made a claim about physics? No! Not without the additional assumption that the formal system being used is in fact an accurate map of the territory! But the mathematician makes no such assumption; he or she (acting as a mathematician) is interested only in the properties of formal systems. (Yes, that’s right: I’m advocating the view known as formalism here. The other well-known positions in the philosophy of mathematics, namely Platonism and intuitionism, suffer from map-territory confusion!)
Mathematical systems, like Ordinary Arithmetic or Mod-3 Arithmetic, are part of the map, not the territory. The facts of mathematics are, so to speak, cartographic, rather than geographic.
In the OB post tautologies have to be empirically observed somehow, Eliezer writes about waking up one day and discovering all sorts of evidence that 2+2=3. This wouldn’t be evidence that 2+2=3 in Peano arithmetic, it would be evidence that Peano arithmetic just doesn’t apply for some reason. In my down-voted comment, I was just giving an example of how there can be different kinds of arithmetic if you are willing to be flexible about what arithmetic is. (If you are not willing to be flexible, then you are not willing to allow the observation that 2+2=3 as an observation about arithmetic, because this is not possibly true in standard arithmetic. Well, the observations are possible but you’d have to account for it as some kind of grand delusion.) My point is that 2+2=4 in Peano Arithmetic independent of observation, but observation tells you if Peano arithmetic applies or not.
Exactly.
It is worth emphasizing that to claim that “2+2=4 in Peano Arithmetic independent of observation” is not to claim that our knowledge of this fact about Peano Arithmetic is independent of observation. (The former claim is about our map of the territory; the latter is about our map of our map of the territory.)
Could you elaborate? It sounds to me like the former claim is about the territory, and the latter is just hard for me to parse.
I’ll emphasize with the following analogy: you need to observe the sun to know of it. However, you can nevertheless be certain—as certain as you are of anything at all—that the sun exists independently of observation. You need to define the Peano axioms and observe the deductions that lead to the tautologies to know of them, but they are mathematically true independent of your observation.