What does 2 + 2 = 4 mean?

[Architeuthus]

I said other because this thread is pointless.

Lot's of overthinking.

Well, thanks, but I stuck in an "it's pointless" option just for you.

And overthinking? In philosophy, there ain't no such thing.

 

[quatona]

And overthinking? In philosophy, there ain't no such thing.

Yes, maybe "overwording" (or: "getting lost in words") would be the better term.

 

[Architeuthus]

Hmmm. The poll results seem to have vanished in the upgrade.

 

[KCfromNC]

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

In Z3, 2 + 2 = 1 = 4

In Z4, 2 + 2 = 0 = 4

Nope, 4 is meaningless in systems with bases <= 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.

Depends on the notation, I guess.

So I stand by what I said.

That's what philosophy is for - coming up with arcane rationalization to keep believing what one always has.

 

[Tree of Life]

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

He's trying to make a particular point. I believe if pressed on the issue he wouldn't articulate it like that again. And I don't think he really believes that.

He's trying to say that God's word is our highest standard for truth and we should measure everything else by it. If God said something like "2 + 2 = 5" then we should take him at his word. But, of course, God would never say anything like this and neither would the Bible. So his point isn't well made.

 

[essentialsaltes]

He's trying to make a particular point. I believe if pressed on the issue he wouldn't articulate it like that again. And I don't think he really believes that.

He's trying to say that God's word is our highest standard for truth and we should measure everything else by it. If God said something like "2 + 2 = 5" then we should take him at his word. But, of course, God would never say anything like this and neither would the Bible. So his point isn't well made.

It does remind me of the exchange between Jefferson and John Adams:

Jefferson: "I observe a bill is now depending in parliament for the relief of Anti-Trinitarians. It is too late in the day for men of sincerity to pretend they believe in the Platonic mysticisms that three are one, and one is three; and yet that the one is not three, and the three are not one…"

Adams: "This revelation has made it certain that two and one make three, and that one is not three nor can three be one. We can never be so certain of any prophecy, or the fulfillment of any prophecy, or of any miracle, or the design of any miracle, as we are from the revelation of nature, i. e., Nature’s God, that two and two are equal to four. Miracles or prophecies might frighten us out of our wits; might scare us to death; might induce us to lie, to say that we believe that two and two make five. But we should not believe it. We should know the contrary.

Had you and I been forty days with Moses on Mount Sinai, and been admitted to behold the divine Shekinah, and there told that one was three and three one, we might not have had courage to deny it, but we could not have believed it."

 

[Architeuthus]

Nope, 4 is meaningless in systems with bases <= 4.

One must distinguish between (a) different bases, which are just different notation for the usual numbers in Z and R (and hence mathematically not very interesting), and (b) Z2, Z3, Z4, etc., which are very interesting.

As I said, no branch of mathematics claims that 2 + 2 != 4.

 

[Architeuthus]

Follow-up question for the people who believe that mathematics is fictional: Why is science (which is built on mathematics) any better than astrology?

 

[essentialsaltes]

Follow-up question for the people who believe that mathematics is fictional: Why is science (which is built on mathematics) any better than astrology?

We have many mathematical ways to potentially describe the universe. Science endeavors to choose the ways that are the best descriptors. We adapt our mathematics to describe the situation.

A sentence in a novel might read "Abraham Lincoln was a president." or it might read "Abraham Lincoln was a vampire hunter." One fiction is a better representation of reality than the other.

1 planetesimal + 1 planetesimal + 1 planetesimal + 1 planetesimal+ 1 planetesimal+ 1 planetesimal = 1 planet

We don't use addition to describe this situation in this way. It's obvious that would be wrong. But it's true that "Planetesimals coalesce to form planets."

A ten pound weight falls 10 times faster than a 1 pound weight.
A ten pound weight falls 37 times faster than a 1 pound weight.
A ten pound weight falls at the same rate as a 1 pound weight.
A ten pound weight falls slightly faster than a 1 pound weight, if you account for air resistance.
There are an infinite number of mathematical possibilities for completing that sentence. One of them is bound to be the best description. But it doesn't make the naked mathematics any more real than any of the other options.

You walk 5 miles an hour north on a train traveling 35 mph north. Your total northward speed is 40 mph with respect to the fixed earth. We do use addition for that. It's obvious that is right.

But then we discovered that with faster speeds, it's actually wrong. Galilean relativity is out, Einsteinian relativity is in. We use a different 'addition' formula. Did we compose a new formula to better describe reality, or did we discover a tiny formula carved into a subatomic particle? There are an infinite number of possible addition formulas, all of them mathematical fictions. Out of all of them, one appears to describe the real situation the best.

Getting to your direct question, science provides better descriptions (or predictions) than astrology does. Astrology is an unbelievable work of fiction.

 

[Architeuthus]

We have many mathematical ways to potentially describe the universe. Science endeavors to choose the ways that are the best descriptors. We adapt our mathematics to describe the situation.

Fair enough, but if you assume the truth of mathematics (as scientists do), and if mathematics is fictional, rather than "true" in some sense, what makes the mathematical descriptors "good"?

Getting to your direct question, science provides better descriptions (or predictions) than astrology does. Astrology is an unbelievable work of fiction.

Well, I'm glad we agree on that, at least.

 

[essentialsaltes]

Fair enough, but if you assume the truth of mathematics (as scientists do), and if mathematics is fictional, rather than "true" in some sense

It is true in a fictional sense. The same way that it is true that "Sherlock Holmes lives at 221B Baker Street." In the fictional world of Sherlock Holmes, this statement is true.

In the fictional world of Abraham Lincoln: Vampire Hunter, it is true that he is a vampire hunter.

In the fictional world of Euclidean geometry, the Pythagorean Theorem is true.

In the fictional world of non-Euclidean geometry, the Pythagorean Theorem is false.

Some fictional worlds are better descriptors of certain aspects of real things than others.

what makes the mathematical descriptors "good"?

Numbers are our fictions, that obey their fictional rules that tell us what's true and false.

And we've also built machines, with rotating dials with numerals on them to measure things in the real world (as best we can). We describe what's going on with reality by overlaying our chosen numerical system on top of it. If you pick the right mathematical fiction, you can get the numerals on the dials to express true statements (and that makes that particular fiction better (good) than others that produce false statements. This is not a bizarre coincidence; it is because we choose, out of our many potential mathematical fictions, the one that generates true statements.

 

[Architeuthus]

Numbers are our fictions, that obey their fictional rules that tell us what's true and false.

I get what you say, I think.

And we've also built machines, with rotating dials with numerals on them to measure things in the real world (as best we can). We describe what's going on with reality by overlaying our chosen numerical system on top of it. If you pick the right mathematical fiction, you can get the numerals on the dials to express true statements (and that makes that particular fiction better (good) than others that produce false statements. This is not a bizarre coincidence; it is because we choose, out of our many potential mathematical fictions, the one that generates true statements.

But how do you recognise true scientific statements? Typically we take a scientific theory (expressed in mathematical language) and a set of dial-measurements, as well as assuming the truth of a whole chunk of relevant mathematics (e.g. statistics), and use the mathematics to assess whether the dial-measurements support the theory (getting results like "five-sigma fit"). If the relevant chunk of mathematics is fictional, how does it provide reliable support for the truth of the scientific theory?

What, actually, is the difference between "we choose, out of our many potential mathematical fictions, the one that generates true statements" and "we choose, out of our many potential mathematical fictions, the one that makes our statements appear true"?

 

[essentialsaltes]

What, actually, is the difference between "we choose, out of our many potential mathematical fictions, the one that generates true statements" and "we choose, out of our many potential mathematical fictions, the one that makes our statements appear true"?

I'm not sure there is a difference. Our mathematical assumptions determine what is true. There is no 'appearance' of true. There is true in system X and true in system Y. We hunt for systems where the data are most in accord with truth. If the Pythagorean Theorem is false (by empirical data) then Euclidean geometry is not the best fiction.

Your point about statistics is potentially germane, but I don't know enough to talk about it. I understand there are raging arguments about "what it means" to say that a rolled die will fall on a 6 with a probability of 1 in 6.

Or to take another tack. If we assume something has a linear relationship, we often compute a least squares fit. This is clearly a choice. We could compute a least |cubes| fit or a least quarts fit. Oh, if we assume our whatsits are normally distributed, then Gauss showed the least squares is our best bet, but this is still more assumptions. We're throwing our assumptions onto the data. A priori, we don't know that our data or variances are normally distributed.

If we write a fanfic in which our variables are, according to the fictional canon, normally distributed, then the least squares fit will get us closest to the best answer. But there is an if there. In other situations, other models would yield the 'maximum likelihood'.

 

[Architeuthus]

We hunt for systems where the data are most in accord with truth.

Yes, but what is "truth," and how do you recognise it -- that's my point. In the hard sciences, mathematical techniques of various kinds get used heavily in recognising scientific "truth."

Your point about statistics is potentially germane

Well, yes, that's why I mentioned it.

if we assume our whatsits are normally distributed, then Gauss showed the least squares is our best bet

You're also assuming that this bit of pure mathematics by Gauss is "true" in some objective sense, which is OK only from a Platonist, logicist, or empiricist perspective. On the fictionalist account, Gauss didn't "show" anything, he merely "wrote a work of fiction about the use of least squares."

And that is why I was so surprised to see pro-science people favouring the fictionalist account. I had expected such responses only from those on the post-modernist fringe who genuinely believe that science and astrology are "equally valid ways of knowing."

 

[essentialsaltes]

You're also assuming that this bit of pure mathematics by Gauss is "true" in some objective sense, which is OK only from a Platonist, logicist, or empiricist perspective. On the fictionalist account, Gauss didn't "show" anything, he merely "wrote a work of fiction about the use of least squares."

No, it follows inexorably from his assumptions. If Holmes lives at 221B Baker Street, then he is not far from Regents Park. If we decide to play Gauss' game, then we can't avoid his conclusions. I am not dogmatic about the normal distribution of every variable.

 

[essentialsaltes]

In the hard sciences, mathematical techniques of various kinds get used heavily in recognising scientific "truth."

The quotes are well-warranted. There is no proof or truth in science. Only well-substantiated theories. Our stories about what's going on.

 

[Architeuthus]

No, it follows inexorably from his assumptions.

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."

There is no proof or truth in science. Only [...] Our stories about what's going on

A CONSENSUS OF SCIENTISTS SAY IT ... THAT JUSTIFIES A PROVISIONAL ACCEPTANCE OF IT UNLESS AND UNTIL FURTHER EVIDENCE COMES TO LIGHT OR A COMPETING THEORY GAINS CREDENCE

I'm having trouble reconciling those two statements.

 

[Conscious Z]

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.

Forgive me if I'm wrong, but I don't believe you are characterizing the private language argument (I'll say PLA) correctly. Wittgenstein was not arguing that all language eventually has one's own qualia as a referent. Rather, he was arguing that a language with one's own qualia as a referent is necessarily impossible because it is inherently indecipherable by others. When we say things like "The dog is in the street," we are referring to things that are in the public domain. Things like dogs and streets are accessible to all people, and we can easily imagine scenarios in which we can perfectly clarify what we mean by such terms. However, this is not the case when we try to construct sentences that refer to our own qualia. To use his example, I can say "S" in referent to a sensation I am experiencing, but the listener is left to guess what I mean when I say "S." The problem here is that there is no method by which to verify whether "S" is correct or incorrect. As such, it isn't a language.

Wittgenstein's PLA has been interpreted many different ways, so it's possible you have a different understanding of it than I do. As I understand it, though, Wittgenstein isn't trying to paint a problem for all language, only language that has a necessarily internal referent.

 

[quatona]

Follow-up question for the people who believe that mathematics is fictional: Why is science (which is built on mathematics) any better than astrology?

1. Astrology is built on mathematics, too. Thus, if science is "any better" than astrology, the reason is probably not mathematics vs non-mathematics.
2. "Better" in doing what?

 

[KCfromNC]

One must distinguish between (a) different bases, which are just different notation for the usual numbers in Z and R (and hence mathematically not very interesting), and (b) Z2, Z3, Z4, etc., which are very interesting.

As I said, no branch of mathematics claims that 2 + 2 != 4.

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.