What does 2 + 2 = 4 mean?

[Architeuthus]

Sounds like some of the uninteresting ones do. But yeah, if we ignore all the counterexamples to your claim the only claims left are ones which agree with you. Not sure what that's supposed to prove, but I guess it is one way to win an argument.

You're really not getting it, are you?

Every branch of mathematics I'm familiar with that includes "2", "4", "+", and "=" has the equality "2 + 2 = 4", and, as far as I know, no branch of mathematics whatsoever has the inequality "2 + 2 != 4".

Feel free to list any counterexamples you might have.

 

[KCfromNC]

You're really not getting it, are you?

Oh, how adorable, attempts to insult my intelligence. Typical sophisticated philosophy at work.

Every branch of mathematics I'm familiar with that includes "2", "4", "+", and "=" has the equality "2 + 2 = 4",

If you limit it to the systems of math where 2 + 2 = 4 , then yes, 2 + 2 always equals 4. Ignore all the cases where anything isn't true and it is always true. I'm really getting it now.

Feel free to list any counterexamples you might have.

Check my original response. For example, 2 + 2 = 2 if you define + the correct way. That's the whole point. These are arbitrary systems where the operators do whatever they're defined to do. Some of them are more or less useful, but none are some sort of essential truth that you seem to be hunting for.

 

[essentialsaltes]

If it "follows inexorably from his assumptions," then it is a true statement of the form A => B, not simply a fictional statement. Conversely, if it's fictional, it cannot "follow inexorably."

I don't see why not. If Holmes lives at 221B Baker Street, then it is inexorably true that his home is near Regents Park. The assumptions are the fictions. The Pythagorean Theorem is true, or it is false. It can be either depending on your assumptions. The Pythagorean Theorem is not inexorably true. It is only inexorably true if you are writing a fanfic in Euclid's world.

I'm having trouble reconciling those two statements.

I'm not sure why. If there were really numbers out there, we could prove some things with the certainty of mathematical truth (in some particular mathematical system, based on chosen assumptions). But in the real world, the best we have is provisional acceptance, unless and until someone writes a better story.

 

[essentialsaltes]

as far as I know, no branch of mathematics whatsoever has the inequality "2 + 2 != 4".

Feel free to list any counterexamples you might have.

 

[KCfromNC]

That example is "not very interesting" so we're supposed to pretend it doesn't exist. Duh. We're doing real philosophy here, get with the program.

 

[Architeuthus]

That table defines a set of equalities and implied inequalities on the set {0, 1, 2, 3}. Since the table doesn't contain 4, the inequality 2 + 2 != 4 is not included, as I said.

If you're going to include 4 by using the standard definition of Z4 = Z mod R, for a suitable equivalence relation R, then you will get 0 = 4.

For example, 2 + 2 = 2 if you define + the correct way.

I'm not aware of any branch of mathematics that has 2 + 2 = 2, except for the case where 2 + 2 = 0 = 2 = 4.

 

[Architeuthus]

I don't see why not. If Holmes lives at 221B Baker Street, then it is inexorably true that his home is near Regents Park.

Hardly, because in fiction we can rewrite the geography of London. Nothing in fiction is inexorably true.

The Pythagorean Theorem is true, or it is false.

You seem to be suggesting that it's an (always) true statement of the form A => B, where A contains the assumptions. But that contradicts the idea that it's purely fictional.

But in the real world, the best we have is provisional acceptance, unless and until someone writes a better story.

What is even provisional acceptance based on? And what makes one story better than another?

 

[KCfromNC]

That table defines a set of equalities and implied inequalities on the set {0, 1, 2, 3}. Since the table doesn't contain 4, the inequality 2 + 2 != 4 is not included, as I said.

So it's a case where 2+2=4 isn't true.

I'm not aware of any branch of mathematics that has 2 + 2 = 2

Now you do - it's an example of a bitwise logical operator. I mentioned that in my original reply.

 

[KCfromNC]

Hardly, because in fiction we can rewrite the geography of London. Nothing in fiction is inexorably true.

Just like we can create different systems of math on a whim. What's your point?

 

[Architeuthus]

So it's a case where 2+2=4 isn't true.

Which is not quite the same as 2 + 2 != 4 being true.

Although, as I said, the standard treatment of Z4 (especially when written using 0, 1, 2, and 3) defines Z4 as Z modulo an equivalence relation, so that the "0" in the table refers to the equivalence class {..., -4, 0, 4, ...} and hence 2 + 2 = 0 = 4.

Now you do - it's an example of a bitwise logical operator. I mentioned that in my original reply.

That would be a logical "or" or "and" then, which are not usually written with a "+".

Just like we can create different systems of math on a whim. What's your point?

With respect, that's the entire point of the thread. The fictionalist account says that we can create different systems of mathematics on a whim. The Platonist, logicist, and empiricist accounts say that we are constrained by some form of external reality.

What makes you think that it's possible to create different systems of mathematics on a whim? And if we could, why on earth would you expect anything about the resulting system to be "inexorably true"?

 

[FrumiousBandersnatch]

Ayer would have it that in the context of decimal arithmetic, 2 + 2 = 4 is a factually significant statement in that it is verifiable (it has a truth value); however (like all such mathematical statements) it is tautological, as it is true by definition. I find it interesting that we can learn so much from identifying tautologies, but I suppose the same is true of metaphors.

As I understand it, Wittgenstein would say it is part of the language game of mathematics, and - in general - its meaning will depend on its usage (which this thread seems to support).

 

[Architeuthus]

Ayer would have it that in the context of decimal arithmetic, 2 + 2 = 4 is a factually significant statement in that it is verifiable (it has a truth value); however (like all such mathematical statements) it is tautological, as it is true by definition.

(my bolding)

That reminds me of the old joke:

"Professor Smith is working through a long and complicated proof on the blackboard. At one point he says, 'It is obvious that this follows from equation 17.'

A student towards the back of the lecture theatre raises his hand and asks, 'How is that obvious?'

Professor Smith starts to respond, but then looks back at the blackboard with a look of consternation. Then he springs into action, erasing all the other blackboards and filling them with densely packed equations and derivations. Finally, having derived the desired result, he looks up at the student and says, 'Yes, as I thought, it is obvious.' "

 

[KCfromNC]

Which is not quite the same as 2 + 2 != 4 being true.

So? You were looking for examples of cases where 2+2=4 weren't true. Moving the goalposts to try and weasel your way into techincally being maybe not wrong isn't all that convincing that you have a strong position.

That would be a logical "or" or "and" then, which are not usually written with a "+".

Proof of this claim? Let's see some statistics to back it up.

Even if you could, though, you'd have to prove "never" to support your position. Given the wealth of examples of it actually happening here in reality, you're out of luck. Search for "sum of products form" to see tons of examples from university courses using this very notation.

With respect, that's the entire point of the thread. The fictionalist account says that we can create different systems of mathematics on a whim. The Platonist, logicist, and empiricist accounts say that we are constrained by some form of external reality.

Then the latter camps are obviously wrong, assuming you're describing them correctly. But I guess this is philosophy, so instead of accepting empirical evidence one can pretend that word games somehow let us ignore the obvious.

What makes you think that it's possible to create different systems of mathematics on a whim?

The fact that people do exactly that.

And if we could, why on earth would you expect anything about the resulting system to be "inexorably true"?

Depends on what you mean by true. Many are internally consistent, which is one form of truth. Others are useful for mapping to explanations about how reality behaves, another form of truth. But if you mean given to us as a magical gift from god, no.

 

[Architeuthus]

So? You were looking for examples of cases where 2+2=4 weren't true. Moving the goalposts to try and weasel your way

It would be helpful if you responded to what I actually said, and not what you think I said.

Proof of this claim?

The onus of proof is on you. You're the one claiming that the mathematics community uses "+" to mean bitwise "and" or "or."

Then the latter camps are obviously wrong, assuming you're describing them correctly.

Really? And you say that based on what, exactly?

And you keep using this word "obvious." I do not think it means what you think it means.

The fact that people do exactly that.

Create different systems of mathematics on a whim? Hardly. In fact, when new mathematics is invented/discovered, the constraints imposed by mathematical reality leave precious little room for "whims." For example, if we "invent" an associative binary operator for a finite set, having inverses and an identity element, there is a limited list of options.

But if you mean given to us as a magical gift from god, no.

I can't remember saying that, either.

And I note that you're now starting to back away from the fictionalist account to having mathematics be "true" in some sense, such as being "useful for mapping to explanations about how reality behaves."

 

[FrumiousBandersnatch]

That reminds me of the old joke:

"... Finally, having derived the desired result, he looks up at the student and says, 'Yes, as I thought, it is obvious.' "

Excellent!

Even the simplest, most intuitive tautologies can be hard to prove - it took Russell & Whitehead around 380 pages of Principia Mathematica to show logically that you can prove 1+1=2, before going on to prove it...

 

[Architeuthus]

On re-reading this thread, I may have been a little unfair to those giving the fictionalist answer.

Perhaps what they meant was: mathematics consists of true statements about logically consistent imaginary alternate universes (if I'm right, please tell me).

Thus Pythagoras' Theorem (PT) is true in logically consistent imaginary alternate universes where the 5th postulate (P5) holds.

But, if we take that approach, we can package the assumptions into the theorem. The theorem P5 => PT (indeed, also P5 <=> PT) holds in all logically consistent alternate universes (whether imaginary or the one we live in) -- i.e., it is necessarily true.

Having removed the empirical content, this necessarily true statement must be interpreted, it seems to me, in some Platonist or logicist way (e.g. as being tautological, as FrumiousBandersnatch suggests).

 

[KCfromNC]

The onus of proof is on you. You're the one claiming that the mathematics community uses "+" to mean bitwise "and" or "or."

Hey, you made the claim. If you can't back it up, don't get snippy with me.

But anyway, here's an example of using + as a bitwise logical operator :http://courses.cs.washington.edu/courses/cse370/97au/admin/Slides/Week2Lecture3/sld003.htm. Or this one, which explicitly explains what the + sign is used for : http://www.ee.surrey.ac.uk/Projects/CAL/digital-logic/formsdef/index.html. I told you how to find this stuff in my last post, but I'm happy to hand-hold you to more examples if you need them.

Create different systems of mathematics on a whim? Hardly.

Who cares what the mental state of mathematicians are at the time they're creating new math? Please explain how any of that matters to the discussion at hand.

I can't remember saying that, either.

Never said you did. I was simply proposing a number of different ideas of what truth might mean - I never attributed any particular one to you. It would be helpful if you responded to what I actually said, and not what you think I said.

And I note that you're now starting to back away from the fictionalist account to having mathematics be "true" in some sense, such as being "useful for mapping to explanations about how reality behaves."

Why would you bring this up? I answered "other" on your poll rather than "fictionalist". What's up with the philosopher types being so caught up with trying to stuff views into their preconceived boxes of what everyone else should believe?

But with that aside, me asking what you mean by true is no way making any sort of claim one way or the other about the nature of mathematics and truth.

 

[Architeuthus]

Hey, you made the claim.

Sorry, I thought it was yours. You said:

In boolean logic, 2 + 2 = 2.

Or this one, which explicitly explains what the + sign is used for : http://www.ee.surrey.ac.uk/Projects/CAL/digital-logic/formsdef/index.html. I told you how to find this stuff in my last post, but I'm happy to hand-hold you to more examples if you need them.

Those two slides use "+" as an operator on binary functions, so they're not actually examples of 2 + 2 = 2.

Who cares what the mental state of mathematicians are at the time they're creating new math? Please explain how any of that matters to the discussion at hand.

You tell me. What did you mean by:

Just like we can create different systems of math on a whim. What's your point?

I answered "other" on your poll rather than "fictionalist".

Sorry, my mistake. Some of what you were saying sounded very much like a defence of fictionalism, though, particularly your suggestion that systems of mathematics were "arbitrary."

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.

I would still appreciate some clarity on what you think 2 + 2 = 4 means for, let's say, the case where 2 and 4 are elements of Z, and "+" refers to addition in Z.

 

[KCfromNC]

New forum software is broken

 

[Crandaddy]

My views are probably best classified as a form of Aristotelian realism, so they fall into a somewhat odd middle ground between Platonism and empiricism.

I'd say that 2+2=4 follows upon the observation that whenever a grouping of any two objects is conjoined with a grouping of any other two objects, then a grouping of four objects altogether results. The numbers 2 and 4 would be concepts of quantity, or “manyness,” where 2 conceptually denotes 2-manyness, and 4 conceptually denotes 4-manyness, and concepts of quantity would be derived from instances of it observed to occur in the world.