Back to mathematical ruminations about the relation between language and reality...
In math, there is something called an isomorphism. The point behinds an isomorphism is this: you take two systems, and show that at some level the only difference is a change in symbols (at least, within the context of a given type of mathematical structure). So, for example, I can take the integers (all positive & negative whole numbers + 0), and I can work with them under everyday, ordinary addition. At the same time, I can take the set of doubled integers (..., -6, -4, -2, 0, 2, 4, ...), again under addition. Now, if I take 1 + 1 = 2 from the normal set of integers, I can map 1 onto 2 and 2 onto 4 to get 2 + 2 = 4 in the set of doubled integers. This is again a true statement. I can, in fact, take any statement about addition in the normal integers, double the numbers, and get a true statement about addition in the doubled integers. The structure of the systems, with regard to addition, is the same.
Another set of relations are called homomorphisms. Before, I could take my doubled integers, halve them, and get equivalent statements about normal integers; the isomorphism works both ways. Homomorphisms which are not also isomorphisms only work one way, and so they preserve important structural elements while losing much as well. Let us take again our normal integers with addition, and then let us take the numbers 0 and 1. Every odd integer we will map to 1, and every even integer we will map to 0. Further, in our new set, 1 + 1 = 0, with other relations staying the same. Essentially, 1 = "odd" and 0 = "even", so a statement like 1 + 0 = 1 could be said to mean "odd" + "even" = "odd". Again, for any true statement in our normal integers, we get a true statement in our even-and-odd system. 3 + 4 = 7 becomes 1 + 0 = 1, and both are fine. Let us call our new system Mod-2.
We can have multiple systems like this. Let us take the numbers 0, 1, and 2, where 1+2 = 2+1 = 0 and 2+2 = 1. This system is Mod-3. The normal integers will map like such: anything which is divisible by 3 becomes 0, anything 1 more than such a number becomes 1, and anything 2 more than such a number becomes 2. Again, true statements in the integers stay true statements in Mod-3.
Let us take Sally, who is a fluent speaker of Mod-2. Sam, on the other hand, speaks Mod-3. Now, Sally makes the casual statement that 1+1=0. Sam is flabbergasted; every kindergartner knows that 1+1=2! Sally can't understand such nonsense; not only is 1+1 clearly 0, 2 is simply gibberish.
Each of these statements is perfectly correct within its own system. Therefore, the two speakers are both right. These statements are truly incorrect inside the other system. There is also more than a chance resemblance between their terms; it is not as if "1" means something completely different within the two systems, although its relations within the systems are different. Therefore, the two speakers are genuinely contradicting each other. Not only are both speakers correct within their own systems, both systems do genuinely reflect the structure of the integers ("reality," as it were), albeit in a pallid fashion. Finally, there are genuinely wrong statements; 0 + 1 = 0 is false in any Mod system, as it cannot possibly be part of a system which mirrors the structure of the integers.
Also, every speaker can only master a finite set, but reality is an infinite one. Therefore, although speakers can master larger and larger sets, they can never hold out on the basis that a given set doesn't fully capture reality; the limited versions are all that they can access. Bigger sets do capture reality's structure better, though (they participate in it more fully?), so there is a point to continuing to search for ways of expressing the world. Also, while if each speaker could be a pure individual, cut adrift from others, she may be able to mistake her own Mod-language to be whole in itself. However, once she must interact with speakers of other Mod-languages, through their differences she realizes that her own Mod-language must relate to something beyond itself.
Finally, does this help to explain the God-world relation? The Mod sets are not the integers; they are different sets with different properties. On the other hand, they are nothing, except insofar as they are finite structures within the integers and borrow terminology from the integers; they have no being over and above that. The integers, also, must contain all of the structures presented in the Mod sets, and so is the "perfect," that is complete, set. With regard to language, the sets only analogously represent the structure of the integers, although they do genuinely do such. The integers themselves are ineffable, in a sense, as no Mod-statement really captures them; though we can still talk about them in another sense, as every Mod-statement refers to them.
4 comments:
This analogy seems spot-on. What it reminds me of, at least at first, is the relationship between the Catholic and Orthodox on the one side, and the so-called "Oriental Orthodox" on the other. As far as I can tell, it now seems to be pretty widely agreed that much of the schism was (at least at first) as much linguistic and cultural as genuinely theological. So our language can't really do justice the fact that Jesus is fully God and fully man simultaneously, much less the manner in which this is so? Big surprise. And then when we try to translate these concepts across unrelated language families, disaster awaits. I can certainly affirm that "one and the same Christ, Lord, and only-begotten Son is ... in two natures without confusion, change, division, or separation ... in one person and one hypostasis." But I can also understand why if you try to translate this into Coptic, maybe it sounds like 2+3=5 sounds to a "speaker" of Mod 3, and likewise, why the language around this that sounds reasonable and (maybe?) orthodox in Syriac would sound lacking in English, Latin, or Greek. At the same time, I think these linguistic differences do produce genuine theological differences, because our words only vaguely, "in a mirror dimly", describe the ineffable God. Although that doesn't mean there aren't some 1+0=0 statements on both sides of any theological controversy, either.
So am I reading your point correctly?
More or less. I was thinking along the lines of what would make philosophy (and other intellectual exercises with less certainty than math) worthwhile, possible, capable of progress, and needing to look at concrete situations. So, if Kant, Aquinas, Plato, and Aristotle all disagree (in substance if not in words), then what can any of us hope for? So, what I'm getting at is not merely languages, but entire conceptual schemes and ways of life.
The whole God-world bit actually popped into my head after I had written the rest. I don't think that it is accidentally related to the main point, though. God is reality (the ground of being for all beings), and our different ways of living are our languages (that is, patterned ways of symbolically manipulating material reality).
Clarification question about Mod-3:
You say:
"This system is Mod-3. The normal integers will map like such: anything which is divisible by 3 becomes 0, anything 1 more than such a number becomes 1, and anything 2 more than such a number becomes 2."
OK, this means 3 = 0, 4 = 1, 5 = 2, 6 = 0, and so on. (Sorry, I know I shouldn't use '=' for mapping normal integers into Mod-3, but I don't know what other symbol to use here. '::'?)
What about the two integers preceding 3? You haven't defined them, have you?
We're talking about integers, not natural numbers, so the normal integers before 3 include:
-5 -4 -3 -2 -1 0 1 2
Now, -3 is divisible by 3, right? So that's 0. So:
-3 = 0, -2 = 1, 0 = 2, 1 = ??, 2 = ??
Nope, we still don't have defined values for 1 & 2 in the normal integers.
Unless...0 is divisible by 3?
So, then:
0 = 0, 1 = 1, 2 = 2, 3 = 0.
OK, that's better. I can overlook the remaining problem for my response (for now).
(The remaining problem seems to be that 0 = 0 & 0 = 2 in Mod-3.)
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"Let us take Sally, who is a fluent speaker of Mod-2. Sam, on the other hand, speaks Mod-3. Now, Sally makes the casual statement that 1+1=0. Sam is flabbergasted; every kindergartner knows that 1+1=2! Sally can't understand such nonsense; not only is 1+1 clearly 0, 2 is simply gibberish.
"Each of these statements is perfectly correct within its own system. Therefore, the two speakers are both right. These statements are truly incorrect inside the other system. There is also more than a chance resemblance between their terms; it is not as if "1" means something completely different within the two systems, although its relations within the systems are different. Therefore, the two speakers are genuinely contradicting each other. ..."
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Wait a minute...!
In order to contradcit each other, Sally and Sam have to make statements with contradictory content. Let A & B be Sally & Sam's statements, respectively. The propositional contents of A & B are contradictory iff A=~B & ~B=A.
Let me attempt to accurately render the propositional content of A & B in a language I can more readily understand. I'm using the symbol '::' to indicate an equivalence of propositional content. I take it, then, that
A :: 1 + 1 = 0 :: "odd" + "odd" = "even" :: The sum of two odd integers is an even integer :: The sum of the successor of an integer divisible by two and the successor of an integer divisible by two is an integer disivible by two.
B :: 1 + 1 = 2 :: The sum of the successor of an integer divisible by three and the successor of an integer divisible by three is the successor of the successor of an integer divisible by three.
How are these two statements contradicting each other?
I guess you could argue that they don't in the language I'm translating them into, but they do in their own language. But if it matters what language you're speaking, then in Mod-2 they aren't contradictory sentences, as one of them (B) is non-sense, and while they are contradictory sentences in Mod-3 (A is false, B is true), it's no longer true that both sentences are correct.
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Let me know if I'm missing something obvious here, or if I need to have something non-obvious explained to me in order to understand why you're not off your rocker. :)
Lack of motivation stinks, doesn't it, Scott? But glad to hear that today's going well!
For the first part of your question:
We'll set 0 -> 0, 1 -> 1, and 2 -> 2 as you suggested. In addition, -1 -> 2, -2 -> 1, -3 -> 0, and so forth. That way, 1 + -1 = 0 maps to 1 + 2 = 0, and so 2 functionally "is" -1 (in fact, it would occasinally be written as such if we were algebraists).
For the second part:
You're right on. It is important to me that there is some way of resolving the contradiction, so that we have a consistent model. However, I don't think that because the situation is not a straightforward logical contradiction, that it falls into the category of non-contradiction.
At any rate, I don't think that the "terms" '1', '+', '0', and '=' are either the same or different between the systems (or perhaps they are both. You decide.). "1 + 1" is not meant by the speakers as either "odd + odd" or "thing with remainder 1 when divided by 3 + the same". It is meant as "1 + 1", with non-trivial relations to the same source which yield an essential similarity to the terms.
If nothing else, the speaker of Mod-2 and of Mod-3 may have nothing to distinguish a real contradiction from these types of contradictions, so maybe we can call them practical-phenomenological contradictions. The analogy would be that you and I could straightforwardly contradict each other, and still not know that one of us must be wrong (assuming that homomorphic rightness can be considered rightness; if it's all we've got, then that seems reasonable).
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