2 + 2 = 5.
2 + 2 = 5 in a way we can’t understand.
I don’t think there’s any question 1) is inconsistent. But is 2)? I mean, is it inconsistent to state something that seems inconsistent on the surface but then address the inconsistency as a matter of our lack of understanding? Because if it’s really a matter of our lack of understanding, the statement may very well be consistent after all and the apparent inconsistency is just virtual.
Well, when we consider that 2+2=4 is just a definition, so we don’t know how to make sense of 2+2=5 because that’s not a definition of anything, and so is no part of any system of mathematics that we understand, I think it is pretty clear that there is no inconsistency. We can’t understand a system that we don’t use.
I don’t think you can say that. 2+2=4 is not just a definition. ‘2’ is a definition - it means one thing and another thing. Same with ‘4’ and ‘5’. ‘+’ is a definition. ‘=’ is a definition. ‘2+2=4’ is a statement . Based on the definitions given for ‘2’, ‘+’, ‘=’, and ‘4’, ‘2+2=4’ can be assessed for its internal coherence/consistency.
Of course, you could say that ‘5’ is the symbol we’ll use to represent the number of Xs in the following sequence: XXXX, but that’s not what I intend to symbolize with ‘5’ in the above. I intend for it to symbolize the number of Xs in the sequence: XXXXX.
So essentially, the statement ‘2+2=5’ means ‘The number of Xs in XX added to the number of Xs in XX results in the same number of Xs as you have in XXXXX’ - or - ‘There are just as many Xs in XXXX as in XXXXX’.
Part of the definition of a natural number is that the next number is arrived at by adding “1”. Because of this - succession, in other words, “2+2=4” can be taken as an extended definition of “2”. For the definition of any of the terms to work, 2+2 must equal 4. The terms define each other.
As mathematics is based on axioms, circularity is not a problem. Or it is, but it is then a problem for any mathematical statement.
1 isn’t necessarily inconsistent, it merely informs the reader that we aren’t dealing with a base 10 number system. If we are dealing with a base 10 system, then it is incoherent as opposed to inconsistent. 2 just means that the definitions we are using are incorrect.
Oh, okay. If it’s a different notation. But that’s rather obvious. Do you have a further point?
Nope. Just addressing the OP. You covered the axiom stuff, so what more was there to say? An axiom can’t be wrong, that’s all.
Wha?
Faust,
So what does that say about the two statements in the OP?
Oops. I thought I was responding to gib. Just didn’t see you sneak in, Xun.
gib - what does what say? What Xun said or what I did?
Gib - are they inconsistent with what? With each other?
Is 1) inherently inconsistent? Or should I say, in light of what Xun said, incoherent?
Is 2) inherently inconsistent (incoherent)?
If you are using base 10 notation, 1) is incorrect. It’s inconsistent with the relevant definitions of terms used in a base 10 system.
If the symbols used are understood by everyone, that is. But that is a problem (or not) with any statement.
I’m not sure I understand the purpose of your question.
Well, it’s a little hard to explain.
I’m thinking of statement 2) as though we could disregard the content of statement 1). That is, if S were some arbitrary statement, then the statement 'S is consistent in a way we can’t understand" certainly doesn’t seem inconsistent at face value - it may be wrong, depending on what we fill in for S, but that doesn’t necessarily make it inconsistent. I think 2) is obviously wrong, and it’s because 1) is inconsistent, but does that make 2) inconsistent? Should it matter what we fill in for S?
I don’t know if that makes it any clearer.
I dunno, gib. If we know we can’t understand it, consistency would be an odd characteristic to ask about.
Well, let’s say we do understand it. Forget all the above, and let’s start anew:
Say S is an arbitrary statement. Nevermind the content.
T is the statement “S is consistent”.
If S is consistent, then T is both true and consistent. If S is inconsistent, then T is false, but is it inconsistent? Or is it just false?
Let me use an example.
“We cannot know the thing-in-itself. But this thing-in-itself is inconsistent with our perceptions of it.”
That’s just nonsensical.
If we can’t know the TII, then we can’t know what is or is not consistent with it. “Know” and “understand” is the same thing here - any further slicing and dicing of definitions here results in something like Kantianism - that is, hopeless incoherence.
“We know/understand that we can’t know/understand the (posited) inconsistency of 2+2=4.”
What does that even mean?
You mean, if S is internally consistent? What does that mean? “Roses are red”? That “red” is consistent with “roses”? How? In that they are both English words?
Nevermind.
Sorry, dude. I’m just not getting this one.
It’s all right. Thanks for trying… in fact, I’m now thinking I’m making a mistake. Maybe one does have to consider the content of S in order to determine the consistency of T.