I should be writing a blurb about the various 2017 mathy books that have passed my way the last few months, but instead the volume I just finished reading is an older classic, Roy Sorensen’s 2003 A Brief History of the Paradox. Toward the end comes a ‘paradox’ (perhaps known by some as 'the odd universe' paradox) I was unfamiliar with and frankly don’t quite understand, though it doesn't appear too difficult. Am passing it along because some of you may find it interesting (…or be able to explain it better to me!).
Verbatim from the book (I’ve bolded a few bits that I especially have difficulty following):
“Meanwhile, Nelson Goodman kept sharpening the knife of nominalism. In 1951 he published The Structure of Appearances. This book contains a logic of parts and wholes. Goodman denies that there are sets. Instead, there are fusions built up from smaller things. Unlike a set, a fusion has a position in space and time. You can touch a fusion. I’m a fusion. So are you. Goodman’s ‘calculus of individuals’ says that there are only finitely many atomic individuals and that any combination of atoms is an individual. Objects do not need to have all their parts connected, for instance, Alaska and Hawaii are parts of the United States of America. Goodman does not let human intuition dictate what counts as an object; he also thinks that there is the fusion of his ear and the moon. In a seminar Goodman taught at the University of Pennsylvania around 1965, John Robison pointed out that The Structure of Appearances implies an answer to ‘Is the number of individuals in the universe odd or even?’ Since there are only finitely many atoms and each individual is identical to a combination of atoms, there are exactly as many individuals as there are combinations of atoms. If there are n atoms, there are 2n - 1 combinations of individuals. No matter which number we choose for n, 2n - 1 is an odd number. Therefore, the number of individuals in the universe is odd! The exclamation point is not for the oddness per se. Aside from those who think the universe is infinite, people agree that the universe contains either an odd number of individuals or an even number of individuals. What they find absurd is that there could be a proof that the number of individuals is odd. ‘Is the number of individuals in the universe odd or even?’ illustrates the possibility of one good answer being too many. Our expectation is that this question is unanswerable. The lone good answer confounds beliefs about what arguments can accomplish.”Anyway, seems like an interesting thought exercise to play with.
(If you can explain it any more lucidly in the comments feel free to give it a go. The primary part I'm unclear about is, in the 2nd part that I've bolded, why does the 2nd sentence necessarily follow from the prior sentence?)
7 comments:
My own understanding runs as follows:
(a) there are a finite number of atoms
(b) we define an "individual" as a (non void) group of atoms
(c) therefore the total number of individuals must be odd.
Of course, if you define a void set of atoms as an individual, the total number becomes even :-)
Needless to say, an individual is not a person nor an object (like che combination "Goodman' ear + moon" shows), therefore there's not much of a paradox in stating this.
P.S.: I found out his surname is Sorensen, not Sorenson.
Ahh, thanks for the input and the spelling correction! (now changed)
Doesn't this argument assume that all possible combinations exist in the universe? If we are missing a few of the possibilities (and we must be, for how could there be more individuals than atoms?), then we can't draw any conclusion.
Since for a combination to exist under this assumption it is sufficient that its constituents exist, there's no problem.
With that definition, I'd quibble with using the word "individual." I can see counting you and me and the moon and the solar system as individuals. But the combination of my right thumb and your left foot? Nope.
Yeah, I’m still confused by the sentence, “If there are n atoms, there are 2^n - 1 combinations of individuals.” The sentence prior to that makes sense to me, but by their definition, an individual is a “combination of atoms,” so the 2nd sentence really translates to: “If there are n atoms, there are 2^n - 1 combinations of, combinations of atoms.” I can agree there are 2^n - 1 “combinations” (which IS an odd no.) but don’t see why that necessarily implies there are 2^n - 1 (or any other odd no. of) “individuals.” Just some murky semantics I guess, 'cuz feel like I’m missing something simple!
It feels to me like an argument along the lines of "Nothing is better than eternal happiness, and a ham sandwich is certainly better than nothing." An individual is a combination of atoms, so we'll count every possible combination of atoms as an "individual"...
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