What does 2 + 2 = 4 mean?

Architeuthus · May 11, 2015

This is a spinoff from the "what is 2?" thread. I've tried to include the main options, which are:

Platonism: numbers are abstract entities in a Platonic world which mathematicians explore, and 2 + 2 = 4 is a timeless, necessary, non-empirical truth about those entities (to quote G.H. Hardy, "mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations', are simply our notes of our observations.").

Logicism: similar to Platonism, but what numbers really are is shorthand for certain expressions in set theory and logic, as in the work of Russell.

Fictionalism: mathematics is purely a human invention with no connection whatsoever to the real world, and "2 + 2 = 4" is simply a statement within that fiction, just as "to be or not to be" is a statement in Shakespeare. Mathematics seem to work in science only because scientists filter reality through a mathematical lens.

Empiricism: numbers represent sets of real things (infinities and very large numbers have no meaning), and 2 + 2 = 4 generalises the empirical fact that + =

and ditto for atoms and apples. In this sense it is essentially a law of physics.

Boredom: the question is uninteresting/ill-posed/meaningless.

Other (please explain).

TillICollapse · May 11, 2015

Well, I'd probably go the Wittgenstein private language route again, wherever that would fall in the above options I don't know It wasn't a very popular or fruitful route to go in the other thread, so no worries.

I would like to toss this into the mix however, if anyone would want to respond: to see if the following type of thinking would play a role in the way anyone (believers probably) would conceptualize of what "2+2=4" may or may not be:

http://www.youtube.com/watch?v=Ysecinv367w

Architeuthus · May 11, 2015

TillICollapse said:
Well, I'd probably go the Wittgenstein private language route again, wherever that would fall in the above options I don't know It wasn't a very popular or fruitful route to go in the other thread, so no worries.

Well, could you explain it? Does your private language describe anything?

TillICollapse · May 11, 2015

Architeuthus said:
Well, could you explain it? Does your private language describe anything?

I thought I did explain it in the other thread, but it didn't seem like anyone wanted to really go down that route, so I wasn't going to go through the effort to explain it once again lol I was more or less tossing that video out there just to add to the mix, in case anyone wanted to address any thoughts it brought up

But anyways, briefly: imo, language is basically a construct that serves a social purpose, not a private one. When one tries to map the word being used to it's referent ... whether we're talking about dogs or numbers for that matter ... ultimately the referent is one's own private qualia. Socially speaking, it may be something we can "point to" (like a dog), or it may not (like the number "2" or "2+2=4"). But privately, we end up pointing right back at ourselves ... revealing that our language ultimately has no meaning on that level. The referent is diminished and not objective. "It is what it is" and that's that. To try and describe it with words, diminishes it objectively and turns it into something we've attempted to identify via the social tool of language.

So the answer to the question "What does 2+2=4 ?" ... to answer it with language involves social input. To answer it definitively to a reduced foundation, is beyond the scope of language ... so one can only go so far down that rabbit hole before they hit fluff and air, because you cannot put it into words or symbols. You will end up coming short of what the thing ACTUALLY is.

quatona · May 11, 2015

"2+2=4" is a correct statement made within an entirely abstract formal system created by humans and doesn´t refer to reality.
"Two apples and two apples are four apples" is meant to describe empirical reality. The statement is still made within a human made formal system (the English language), but "apple" is supposed to have a referent in reality. If we accept the idea that an apple exists, the statement is empirically verifiable.

Then again, whether an apple exists (or whether "apple" is just a human made category to slice reality into useable bits and pieces) is an entirely different question. (I´d vote for the latter.)

In any case, there´s a huge gap between the entirely abstract "2+2=4" and the "Two [insert category of choice] and two [insert category of choice] are four [insert category of choice]."

Architeuthus · May 11, 2015

TillICollapse said:
But anyways, briefly: imo, language is basically a construct that serves a social purpose, not a private one. When one tries to map the word being used to it's referent ... whether we're talking about dogs or numbers for that matter ... ultimately the referent is one's own private qualia.

Thanks for sharing that.

I guess that, on your view, communication happens when there is a correlation between our private qualia (e.g. we both are looking at the same dog). What I'm asking is: is there such a correlation with "2" and "4"? Are there objective referents for "2" and "4"? And what are they?

TillICollapse said:
So the answer to the question "What does 2+2=4 ?" ... to answer it with language involves social input. To answer it definitively to a reduced foundation, is beyond the scope of language ... so one can only go so far down that rabbit hole before they hit fluff and air, because you cannot put it into words or symbols. You will end up coming short of what the thing ACTUALLY is.

Or are you saying that it's impossible to answer my question?

Architeuthus · May 11, 2015

quatona said:
"2+2=4" is a correct statement made within an entirely abstract formal system created by humans and doesn´t refer to reality.

Thanks for sharing your view.

TillICollapse · May 11, 2015

Architeuthus said:
Thanks for sharing that.

I guess that, on your view, communication happens when there is a correlation between our private qualia (e.g. we both are looking at the same dog). What I'm asking is: is there such a correlation with "2" and "4"? Are there objective referents for "2" and "4"? And what are they?

Yes, communication via language happens when there is a correlation between our private qualia. When we both look at the same dog, we can point to the dog, and that gives us an "objective" referent.

Although when we use the word "dog" it may take on different meanings. Consider "All men are dogs" or "Dog days of summer" or "Sup dawg ?" etc. The IDEA of "dog" can be ascribed to other things, feelings, other ideas, etc. If I point at a cow and we both agree it's a cow, yet one person also thinks the cow is his reincarnated ancestor or a god ... the nature of the referent arguably doesn't change, where as what we mean when we communicate via language changes. So language doesn't set the standard. "Reality" *is* the standard, so to speak.

I realize you're asking what the referents are for "2+2=4" and I don't think there are any. Arguably, the idea of any number is built upon at least one number having a referent. Perhaps the most foundation number would be "1", and I don't know that "1" has a referent :

Or are you saying that it's impossible to answer my question?

I don't think it's ultimately possible to answer the question, but even if it had a referent I don't think it's ultimately possible. It's still only "so" possible.

Consider the dog ... we can point to a dog. But what is a dog ? "I'm a collection of molecules known as a dog." But a dog is made up of lots of things, which are in turn made up of lots of things, on down the line. We can keep asking, "But what is that made of ?" etc ... at which point I think our language can only go so far, as can our experience. Unless we can directly *be* a "2" or a "dog" or a single "atom", etc, then we cannot definitively define things. We can only ever define them in part, and language fails to define them conclusively in a private way as well as a public way. It's ultimately impossible. But practically speaking, it may be possible. If we both agree that "2+2=4" is the equivalent of a dog for example, then practically speaking it has a referent Even though it's superfluous or meaningless perhaps in other contexts.

Architeuthus · May 11, 2015

TillICollapse said:
Yes, communication via language happens when there is a correlation between our private qualia.

Cool. I'm glad I finally get you.

TillICollapse said:
I realize you're asking what the referents are for "2+2=4" and I don't think there are any. Arguably, the idea of any number is built upon at least one number having a referent. Perhaps the most foundation number would be "1", and I don't know that "1" has a referent :

I don't think it's ultimately possible to answer the question, but even if it had a referent I don't think it's ultimately possible.

That's cool. I can respect that.

Were you the guy who mentioned Peano axioms? Because one approach, not listed in the poll, is "we have no idea what numbers are, but they exist in some sense, and they behave like this."

TillICollapse · May 11, 2015

Architeuthus said:
Were you the guy who mentioned Peano axioms? Because one approach, not listed in the poll, is "we have no idea what numbers are, but they exist in some sense, and they behave like this."

Nope, different guy/gal

Architeuthus · May 12, 2015

Two votes for Fictionalism, and none for Empiricism is really not what I expected.

And there seem to be some unexplained "other" votes. I'd really appreciate it if people outlined their alternative view.

KCfromNC · May 12, 2015

I picked other - the meaning of 2 + 2 = 4 depends on which arbitrary system of math you're assuming when you write it. There are any number of different meanings depending on which you pick, and in many of those cases the sentence contains either undefined terms or is simply false.

Math is just a more formal language, better at communicating some ideas than natural language. Kinda like your fictionalism idea, except that it obviously has some connection to the real world since it can be used to describe things.

Architeuthus · May 12, 2015

KCfromNC said:
I picked other - the meaning of 2 + 2 = 4 depends on which arbitrary system of math you're assuming when you write it.

Thanks for that.

I'm cool with ab != ba, but I didn't think there were any branches of mathematics where 2 + 2 != 4.

KCfromNC said:
Kinda like your fictionalism idea, except that it obviously has some connection to the real world since it can be used to describe things.

And that's why I would have expected people to suggest that mathematics was a language for communicating empirical facts about the real world.

KCfromNC · May 12, 2015

Architeuthus said:
Thanks for that.

I'm cool with ab != ba, but I didn't think there were any branches of mathematics where 2 + 2 != 4.

In boolean logic, 2 + 2 = 2. In any base less than 4, 4 is undefined. In linear algebra, the whole sentence is meaningless since there's not a matrix to be found anywhere in what you wrote.

And that's why I would have expected people to suggest that mathematics was a language for communicating empirical facts about the real world.

It is a language, but your "for communicating ... " is only one thing of many it is used for so I'd disagree with your summary.

Architeuthus · May 12, 2015

KCfromNC said:
In boolean logic, 2 + 2 = 2. In any base less than 4, 4 is undefined.

Well, in Z2, 2 + 2 = 0 + 0 = 0 = 4

In Z3, 2 + 2 = 1 = 4

In Z4, 2 + 2 = 0 = 4

In linear algebra, one would interpret 2 and 4 as scalars in the field R (or Q, or C), and so 2 + 2 = 4 again.

So I stand by what I said.

essentialsaltes · May 12, 2015

While it's true that 4 = 1 (mod 3), I agree with KC that in Z4, there is no 4.

Tree of Life · May 12, 2015

TillICollapse said:
http://www.youtube.com/watch?v=Ysecinv367w

Hah! I actually know that guy.

TillICollapse · May 12, 2015

Tree of Life said:
Hah! I actually know that guy.

So is this the kind of reasoning he actually uses, or is he being quoted out of context for the video ?

Architeuthus · May 13, 2015

essentialsaltes said:
I agree with KC that in Z4, there is no 4.

If you define Z4 by taking Z mod 4, then 4 = 0, and 2 + 2 = 4 = 0.

If you define Z4 by the table, then 4 does indeed not exist, but this means that "2 + 2 = 4" cannot possibly be relating to Z4.

Certainly, it is not true that 2 + 2 != 4 in Z4.

GrimKingGrim · May 13, 2015

I said other because this thread is pointless.

Lot's of overthinking.

What does 2 + 2 = 4 mean?

What does 2 + 2 = 4 mean?

Platonism

Logicism

Fictionalism

Empiricism

The question is pointless or has no meaning

Other (please explain)

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