Jach's personal blog

(Largely containing a mind-dump to myselves: past, present, and future)
Current favorite quote: "Supposedly smart people are weirdly ignorant of Bayes' Rule." William B Vogt, 2010

Pedantic moment: I'm sick of "the whole is greater than the sum of its parts"

I really am.

Okay, I guess I can't just leave it at that because apparently it's not obviously false.

Two other variations on this phrase: "the whole is more than the sum of its parts" and "the whole is not the sum of its parts". I bring these up because this makes the mathematical error twice as bad. $$whole > \sum parts$$ is the original expression, $$whole \neq \sum parts$$ is another.

To a philosophical reductionist this is absurd. A chair can only be the sum of its parts--there is nothing extra to the chair besides the fundamental particles and their vector fields that make it up. To further use that statement in a proud fashion, as if it contains knowledge, is further ridiculous. I wonder if any 20th century physicists thought, prior to Einstein's General Relativity, "Newton's equations aren't giving the right answer. The whole gravitational effect must be greater than the sum of the individual gravity effects!" No, you just didn't have the right equations.

(On the other hand, things like Dark "Matter" suggest we still don't. If I was a leading physicist with cred to throw around, I'd suggest the whole community of physicists go through the literature and start replacing terms like "Dark Matter" and "Dark Energy" and other such things that are placeholders for ignorance with the word "magic". Why is the universe expanding at an accelerating rate? Magic!)

"But Kevin, you don't get it, it's the macroscopic nature of the chair and the utility of it to us that make a completed Ikea chair more than just the chair's pieces!"

Huh, that's a good point, a constructed chair does seem like it's more than just the sum of its parts, that is, four legs and a square. (For a simple chair.) Is my pedantry so easily defeated?

No, because you're neglecting parts. The reason a completed chair feels "more" than just the four legs and square is because it is more. But that's because there are more parts! Those parts include, mostly, a human brain looking at the chair and comparing it to an imagined incomplete chair. The very notion that we humans have a sense of "a completed chair" vs. "an incomplete chair" suggests that we humans are parts of the whole when the whole being considered is "a completed chair" or "an incomplete chair". If you ever think the whole is more than the sum of the parts, you haven't accounted for all the parts. The key missing piece is likely your brain.

It's okay, this failure to notice yourself as a part of the whole plagued many geniuses of the early 20th century. It's the chief reason for confusion in quantum mechanics and continues to be the chief reason for confusion when modern students are taught it "in historical progression" instead of what the actual state-of-the-art-that-matters is. Everyone fails to notice themselves at some point.

"But Kevin, if you mix ketchup and mayo for a sauce, that sauce is more than the sum of its parts because it's more than just ketchup-flavor mixed with mayo-flavor, it has its own unique flavor!"

Another fine point. Not really. In this case it's not so much a failure of not accounting for all the parts--you've got ketchup, mayo, and a human tongue--but a failure of what it means to sum. We're not summing correctly. The hypothetical person whining at me is probably trying to sum up just the flavors, and that is a non-linear sum which is probably where the confusion is.

$$flavor(sauce) = flavor(ketchup + mayo) \neq flavor(ketchup) + flavor(mayo)$$

In contrast, the color sum is linear because red+white=pink and thus

$$color(ketchup) + color(mayo)=color(ketchup + mayo) = color(sauce)$$

But we're not summing individual flavors, we're not summing individual colors, we're summing ketchup and mayo.

$$ketchup + mayo = sauce$$

Which implies that for all properties that ketchup and mayo share, we sum $$property(ketchup+mayo)$$ and not $$property(ketchup) + property(mayo)$$. The exception is if it's indeed linear like the color case, but then it's only useful as a potential optimization technique, not a general solution. We like linearity a lot though and so it seems a lot of people assume it when they shouldn't. The sauce isn't different from the sum of its parts, you just didn't add up the parts correctly.

It's like saying "Well, I have here $$25$$, which has two parts, $$20$$ and $$5$$. $$25$$ is bigger than the sum of its parts because $$\cos(25) > \cos(20) + \cos(5)$$."

No, you idiot, you're summing wrong. It's "$$\cos(25) = \cos(20 + 5)$$". Cosine isn't a linear function.

Even worse: "$$\cos(25) > \cos(2) + \cos(5)$$". Not only are you summing wrong, you're summing the wrong parts! You've committed both sins!

So in conclusion, this has been a pedantic moment. Stop using the phrase, it's incorrect and only proves you don't know what you're talking about. In order to help remove some confusion, look to see if you are summing correctly, and if the properties of something you are looking at form linear sums or not. If they do, you might not be accounting for some terms. If they aren't linear, then you might not be summing correctly.

Funnily enough, people who say the whole is more than the sum of its parts often do one of two things: they start reasoning about the whole as if they have knowledge of it anyway even though they know they lack knowledge that they don't think exists in their guess of the parts; alternatively, they start listing reasons the whole is greater than the parts without realizing that each reason is itself a part that should be added as well. Using the phrase should be a big flashing sign saying "I am ignorant of something." Which is fine, ignorance isn't a crime, but realize you're ignorant and if you can easily dispel that ignorance you probably should.

Posted on 2012-02-09 by Jach

Tags: rant


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