2 + 2 = 4

I think it is a sort the pro from amateur kind of trick question. Regardless of how "real" mathematicians see it amateurs like me do work with the "default" frames of reference and it takes changing that framing to make 2+2 not equal 4. I don't think that my defaulting to the default framing is the same as or as simple as me making an arbitrary assumption. Maybe it is the utility of it that makes that default framing special, ie the usefulness of using abstract ideas of pure integers having qualities like quantities that can add and subtract.

What I think won't matter a jot to the pro's but I don't think sticking with 2+2=4 as the default is wrong; other frames may have mathematical validity but until and unless specified otherwise the default framing should be counted as true. I expect most quants would agree. A bit like surveyors working with such Euclidian abstracts as flat planes, straight lines and triangles that add up to 180 degrees, despite explicitly working with the curved surface of a sphere and having to do mathematical "corrections" to achieve equivalence.

Calling 2+2=11 true in base 3 makes 11 the notational equivalent in integer terms to 4 in base 10. Changing the base doesn't change the quantity - there is equivalence (likely not correct terminology - amateur here) - not even with a no base system, where every integer has a unique identifier. 2+2 in base 3 does equal 4, just expresses it in different notation.

However, in modular arithmetic it looks to me like there is a lack of such equivalence; performing the + function does something but it is not adding quantities (could be expressed as changing orientiations maybe?). Lacking equivalence it won't work as an example of 2+2 =/= 4.
 
Ken Fabian said:
What I think won't matter a jot to the pro's but I don't think sticking with 2+2=4 as the default is wrong; other frames may have mathematical validity but until and unless specified otherwise the default framing should be counted as true.
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No one's saying the default framing isn't true.


Remember the initial claim was all-encompassing: "there is no circumstance..."

And, no matter how you try to equivocate, that's just not true.
 
Ken Fabian said:
Calling 2+2=11 true in base 3 makes 11 the notational equivalent in integer terms to 4 in base 10. Changing the base doesn't change the quantity - there is equivalence (likely not correct terminology - amateur here) - not even with a no base system, where every integer has a unique identifier. 2+2 in base 3 does equal 4, just expresses it in different notation.
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Ken captures the crux of the issue very nicely in the above paragraph. James, by contrast, simply engages and obfuscation and distortion in his attempted rescue of Dave from manifest silliness. Consider:


James R said:
If I recall correctly, axocanth claimed that there are no circumstances in which the statement "2+2=4" would not be true. (I'm sure that axocanth will correct me if my recollection about this is wrong.)
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You do not recall correctly. Write4U originally posted the assertion that there are no circumstances under which 2 + 2 ≠ 4. He/she was, I believe, quoting a comedian (Ricky something) making this claim, obviously with Write4U's personal endorsement thereof. You can double check with him/her, though it's irrelevant to the issue at hand.

Dave reacted to this assertion by saying it is not true (regardless of whoever asserted it), i.e. in Dave's view, there are circumstances under which 2 + 2 ≠ 4. Dave then offered three candidate counterexamples, all of the same form. I focused, therefore, only on the first.

My claim, then, is that Dave's proposed counterexamples do not stand up to scrutiny; they are not counterexamples at all. My claim is not that there are no circumstances under which 2 + 2 ≠ 4. On this I am silent, though if a genuine counterexample were to be adduced, this would be a stunning discovery indeed.


The situation can be summarized thus, as I already laid out in post #215 of the "MR Complains" thread:

Dave is reacting to a certain proposition, which he himself states as follows (MR Complains, post 213): "There are no circumstances in which 2+2 does not equal 4."

Dave continues, "This was written , in text, in the video, and was not qualified. It is an unconditional statement; it asserts to be true under all circumstances, and if not; it is false."


As I also pointed out in the same post, uninterpreted marks on paper do not constitute a statement, a fortiori cannot be appraised for truth or falsity until a meaning is assigned to these marks. Much as James tries to distort the situation, it's not me who is forcing an interpretation or axioms on anyone: I'm not the person saying that a certain statement is not necessarily true - Dave is.


Now, as Ken points out, and as common sense dictates, we normally understand 2 + 2 = 4 to express the proposition "two plus two equals four". This is obviously what Write4U had in mind, and which he/she subsequently confirmed as having in mind. He/she was using the standard base ten notation; not speaking base three, base seven, Swahili, or Vietnamese.

I'm not forcing any interpretation on Dave. What I have done, and what he has up till now evaded, is ask him to tell us how he understands the proposition 2 + 2 = 4 (or 2 + 2 ≠ 4). I want Dave to state in plain English the proposition/statement he is reacting to. What does it mean? He has already told us that a statement was made, moreover his talk of the truth and falsity thereof would be incoherent had a meaningful statement not been made.

So, the first question Dave needs to address is this: What statement are you reacting to?

How do you interpret the statement "2 + 2 = 4", the one you are reacting to? Two plus two equals four (like everyone else)? Nine minus nine equals infinity? Frank Sinatra plus Frank Sinatra equals a hippopotamus? Or something else?

State it in plain English for us.

However Dave interprets this statement, the second question he needs to address is this: How does your explicit Base Three assertion "2 + 2 = 11" (i.e. two plus two equals four) constitute a counterexample to the necessary truth of the first statement?


Perhaps Dave will be so kind as to do that now. To repeat:

Q1: What statement are you reacting to? (state it in English)

Q2: How is the statement "2 + 2 = 11" in Base Three (i.e. two plus two equals four) a counterexample to the above statement?
 
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This whole argument is a bit silly, in my opinion. I don't think there's any real disagreement on the substantive examples that have been put, for "2+2=4" vs "2+2=11", say.

The rest is just people insisting that (a) statements they make on the internet can never be wrong, as long as they are properly interpreted, and (b) that whenever somebody else is wrong on the internet, a 1000 post discussion thread is needed to ram the point home.

I'm out.
 
There may be legitimate mathematical systems where 2+2 doesn't equal 4 but the example in the OP - or the base 3 example- don't look like them to me.
 
I explained why previously - quantity as a quality; modular arithmetic doesn't appear to deal in quantities. The disagreement could well be a problem with nomenclature (amateur here), where numerals aren't quantities and + does something different to adding quantities. And if me thinking of it like that is technically incorrect it matters not at all.
 
Ken Fabian said:
I explained why previously - quantity as a quality; modular arithmetic doesn't appear to deal in quantities. The disagreement could well be a problem with nomenclature (amateur here), where numerals aren't quantities and + does something different to adding quantities. And if me thinking of it like that is technically incorrect it matters not at all.
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Sorry, I don't really follow. that, but that's OK.

I honestly don't understand how anyone can make a valid argument against it.

For example, you feel there's a circumstance I've offered that you think shouldn't count.

Once again: recall the original statement that stared all this is "There is no circumstance where 2+2≠4". That's an unconditional statement, claiming to speak for all circumstances. There's no wiggle room in the claim.

And yet, I've presented a mathemically true and valid circumstance where 2+2=11.

That pretty incontroveribly invalidates the claim.

I have yet to see any refutation of this. People keep trying to dodge and weave and say "it depends on how you intepret 'no circumstance'". There is no interpretation in an absolute claim.
 
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Sorry, I wasn't watching the time.
It has been more than 23.5 minutes since I have restated my argument and I fear some of you have have forgotten it.
Please stay tuned for the restating of my argument.
:biggrin:
 
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