# To Infinity and Beyond!March 17, 2010 6:24 AM   Subscribe

Sure, big numbers are fine. But infinity (in the set theoretic sense) is where the fun really starts. Developed almost entirely by one man in the late 19th century, set theory now forms the foundation of modern mathematics. Cantor showed that not all infinite sets are the same size. Notably, while there are just as many integers as rational numbers, there are more real numbers than integers. These results, along with others that soon followed like the axiom of choice, led to several fascinating consequences:

• The Cantor Set is a fractal constructed by repeatedly removing the middle third of a line segment. You end up with a set that has no length, yet as many points as the reals.
• The Hilbert Hotel was used by David Hilbert to demonstrate how an infinite hotel could be completely full yet still accommodate infinitely more guests. (Another retelling of Hilbert's Hotel.)
• The continuum hypothesis, which deals with whether there's a set that has a size between that of the integers and reals, has been shown to be independent of the basic mathematical model. Both it and its negation are consistent within the standard logical framework math is built on.
• The Banach-Tarski Paradox showed that it's possible to cut a sphere into a finite number of pieces and put the pieces back together again so that you end up with two solid spheres of the same size as the original!
posted by kmz (161 comments total) 75 users marked this as a favorite

Q: What is an anagram of "Banach-Tarski"?
A: "Banach-Tarski Banach-Tarski"
posted by Plutor at 6:28 AM on March 17, 2010 [62 favorites]

Something I've occasionally wondered, usually while drunk at conferences, is whether the axiom of countable choice gets us all the "nice" results of choice without unfortunate things like Banach-Tarski or the well-ordering principle. Wikipedia suggests, to my reading, that for trichotomy of cardinality you need full choice, so perhaps not. Anyone know of any kind of collection of results depending on full choice?
posted by PMdixon at 6:33 AM on March 17, 2010 [1 favorite]

David Foster Wallace's Everything and More is a fun little treatment of infinity from the greeks through Cantor. DFW was a huge math wonk, it turns out.
posted by logicpunk at 6:37 AM on March 17, 2010 [3 favorites]

1 bottles of beer on the wall....
posted by eriko at 6:50 AM on March 17, 2010 [5 favorites]

Oh FFS, I just spent 20 minutes searching for a kids' book version of the Hilbert Hotel that I read not more than a year ago and it just doesn't exist anymore. Grrrr.

No wait..here it is! Kids loved it.
posted by DU at 6:50 AM on March 17, 2010 [6 favorites]

Thanks for posting these. My degree is in math but your Banach-Tarski link was illuminating -- and I think it would be accessible to those without much math background, too.
posted by telegraph at 6:51 AM on March 17, 2010

My favourite set is the Russell set: A set which contains sets that do not have themselves as members.

Now the question remains is the Russell set a member of the Russell set? If it is a member of the Russell set then by definition it does not contain itself as a member and is therefore not a member of the Russell set.

If it isn't a member of the Russell set then by definition it does contain itself as a member and is there a member of the Russell set.
posted by PenDevil at 6:52 AM on March 17, 2010 [1 favorite]

But are numbers real?
posted by oddman at 6:54 AM on March 17, 2010 [2 favorites]

It's funny to think that the first and last math classes that I took both started with set theory. The first was Mrs Petlinski's first grade class at Alexander Hamilton Elementary and the last was advanced discrete math in Grad School thirty-two years later.
posted by octothorpe at 6:54 AM on March 17, 2010 [4 favorites]

Yeah, the Russell paradox is great, but I didn't feel it really fit into talking about infinite sets. I always like the barber formulation. In a certain small town, the town barber shaves all people who don't shave themselves and only people who don't shave themselves. So does he shave himself?

Similarly:

This sentence is a lie.
posted by kmz at 6:55 AM on March 17, 2010 [2 favorites]

Fun read The part when he goes to visit Kurt Goedel is hilarious and wonderful.
posted by wobh at 6:56 AM on March 17, 2010 [1 favorite]

It's funny to think that the first and last math classes that I took both started with set theory. The first was Mrs Petlinski's first grade class at Alexander Hamilton Elementary and the last was advanced discrete math in Grad School thirty-two years later.

Wow, your first grade class had set theory? I would have loved that.

I still remember one semester at UT when 4 out of the 5 classes I was taking were covering countable/uncountable sets at the same time. The other class was History of Rock and Roll.
posted by kmz at 6:57 AM on March 17, 2010 [2 favorites]

David Foster Wallace's Everything and More is a fun little treatment of infinity from the greeks through Cantor. DFW was a huge math wonk, it turns out.
posted by logicpunk at 2:37 PM

Oh man, please avoid this book.
"His book shows that he has done a ton of reading on the subject. He's even consulted the orginal mathematical papers. He seems to be familiar with some advanced mathematical topics. And yet, despite all of this, he is terrible at math. " - Review
posted by vacapinta at 7:04 AM on March 17, 2010 [2 favorites]

David Foster Wallace's Everything and More is a fun little treatment of infinity from the greeks through Cantor. DFW was a huge math wonk, it turns out.

there is a kind of funny takedown of that DFW book in the Notices of the AMS (American Mathematical Society)...
posted by ennui.bz at 7:08 AM on March 17, 2010 [1 favorite]

I don't think it'd be right to call the axiom of choice a 'result', would it?
posted by edd at 7:09 AM on March 17, 2010 [2 favorites]

PMdixon: Anyone know of any kind of collection of results depending on full choice?

I believe that the (usual) explicit construction of a set which is not Lebesgue measurable requires more than countable choice, since one "chooses" an element from each equivalence class of the real numbers modulo the rationals, which requires an uncountable number of choices.
posted by milestogo at 7:10 AM on March 17, 2010

I don't think it'd be right to call the axiom of choice a 'result', would it?

Yeah, I wasn't sure what the best word to use there would be. Maybe 'development' or something would have been better.
posted by kmz at 7:12 AM on March 17, 2010

I still remember one semester at UT when 4 out of the 5 classes I was taking were covering countable/uncountable sets at the same time. The other class was History of Rock and Roll.

Dude, your History of Rock and Roll class was weak.
posted by milestogo at 7:12 AM on March 17, 2010 [5 favorites]

Thank you all very much. It was hard work!
posted by georg_cantor at 7:27 AM on March 17, 2010 [3 favorites]

Yeah, how are you gonna talk about rock and roll shows without bringing up set lists? I'm pretty sure the Grateful Dead had a few uncountable sets.
posted by nicepersonality at 7:34 AM on March 17, 2010 [9 favorites]

kmz: "Wow, your first grade class had set theory? I would have loved that."

Well it wasn't very deep, plastic apples and oranges in different baskets and such.
posted by octothorpe at 7:35 AM on March 17, 2010

Nobody's come to talk about Logicomix? If you liked this post, get the book. It's a graphic novel about the life of Bertrand Russell, and set theory is a supporting character.
posted by BuddhaInABucket at 7:42 AM on March 17, 2010 [1 favorite]

My favourite set is the Russell set

Fun fact: that paradox is the problem that brought Wittgenstein to the world. He responded to Russell's publication of the problem and Russell was so impressed by Wittgenstein's answer that he invited him to Oxford.
posted by shen1138 at 7:49 AM on March 17, 2010 [5 favorites]

kmz: I don't know if you have been inspired by the recent BBC Horizon documentary 'To Infinity and Beyond' but it covered just this subject (with the excellent Steven Berkoff at the helm to boot).
posted by numberstation at 7:54 AM on March 17, 2010 [1 favorite]

As the author of a book on Zero -- in many ways, a dual concept to that of infinity (the original title of my book was "Infinity's Twin") -- I was asked review DFW's book. (For the Wilson Quarterly, if I recall correctly. Can't find it online, unfortunately.)

I've always been a fan of DFW's nonfiction; even when I don't *enjoy* his writing, it's always interesting to see him grappling with the subject matter. In my view, Everything and More was not an enjoyable book and not a successful book. But it was fascinating to watch DFW fail.

My recollection is that the book starts out reasonably enough, but it begins to bog down as DFW gets into more complex material. He begins to struggle, using all of his prodigious skill to try to get the narrative moving again -- but as the end of the book looms (the book, as with all of the others in the series, had to be really short), he begins to panic. It was like watching a powerful and elegant sabre-toothed tiger entrapped in a tar pit.

If DFW's editor had given him double or even triple the wordcount he was allowed, I think that the struggle would have had a very different outcome.
posted by cgs06 at 7:58 AM on March 17, 2010 [10 favorites]

Infinite zombies, you complete me.
posted by infinitefloatingbrains at 8:01 AM on March 17, 2010 [3 favorites]

Nobody's come to talk about Logicomix?

That post made me put Logicomix on my gift list and I got that for Bmas a few weeks ago. Still haven't opened it, but my current book is really slowing down so maybe I'll switch.

If DFW's editor had given him double or even triple the wordcount he was allowed...

If there's one thing DFW should not have been allowed to have, it was higher wordcounts.

To mashup-paraphrase Mark Twain and Einstein: I'm sorry this explanation is so long difficult, I didn't have time to make it shorter and explainable to a kindergartner.
posted by DU at 8:05 AM on March 17, 2010

Some people, when confronted with a sphere, think "I know, I'll cut it into a finite number of pieces and put the pieces back together again." Now they have two spheres.
posted by Combustible Edison Lighthouse at 8:13 AM on March 17, 2010 [18 favorites]

Some people, when confronted with a car, think "yo dawg, I herd you liked bowling alleys so I put a bowling alley in your car so you can bowl while you drive!" Now they have a car and a bowling alley.
posted by DU at 8:15 AM on March 17, 2010 [6 favorites]

And if you drive at just the right speed, the ball keeps rolling but never gets to the pins.
posted by weapons-grade pandemonium at 8:31 AM on March 17, 2010 [2 favorites]

"This sentence is a lie" is fun for a few minutes, but I prefer contemplating "This sentence is true."

It's a lie.

Also: the number that mathematicians conventionally refer to as i, as in e + 1 = 0, is generally held to be the same as that which electrical engineers conventionally refer to as j, as in this definition of the Fourier transform. It's my personal belief that in fact j = -i and nobody has noticed.
posted by flabdablet at 8:34 AM on March 17, 2010 [3 favorites]

HILBERT'S GRAND HOTEL

"Great service, but rooms a bit small, other guests rude"
•••oo

We got stuck in Hilbert, WI because of the massive snow storm. It was after 10pm when we got to the hotel. The concierge said that they were full, but then quickly found us some empty rooms. There was even a busload of seniors that came in after us, and were able to get rooms! At that point I thought I would try to use my points for an upgrade, but apparently all the rooms are identical (no suites!). The small rooms are all non smoking, but I was repeatedly harassed by one of the other guests for cigars for him and his buddies. There is also a Polish/Jewish deli restaurant on site – Banach & Tarski's – but it was already closed. Fortunately the chef was still in the kitchen and was kind enough to heat up a single entree of matzo balls, which incredibly fed both of us once the chef sliced them up. We'll definitely have to go back for a proper dinner some time!
posted by Kabanos at 8:36 AM on March 17, 2010 [16 favorites]

Fun read The part when he goes to visit Kurt Goedel is hilarious and wonderful.

Also by Rucker, White Light. Really fun book.
posted by brundlefly at 8:40 AM on March 17, 2010 [1 favorite]

In this universe, it's provable that, even in an infinite number of alternate universes, Nick and Nora's infinite playlists remain countable.
posted by Obscure Reference at 8:48 AM on March 17, 2010

The BBC did a program called Dangerous Knowledge.

High-level summary: math can drive you insane.
posted by Pirate-Bartender-Zombie-Monkey at 8:48 AM on March 17, 2010 [1 favorite]

What's sour, yellow and equivalent to the axiom of choice?

Zorn's Lemma.
posted by honest knave at 9:03 AM on March 17, 2010

Some people, when confronted with a car, think "yo dawg, I herd you liked bowling alleys so I put a bowling alley in your car so you can bowl while you drive!" Now they have a car and a bowling alley.

Somewhere, Xzibit sheds a single tear because his dream of a Pimp My Sphere spinoff will never be realized.
posted by Copronymus at 9:26 AM on March 17, 2010 [1 favorite]

David Foster Wallace had a word count? That is not as mind boggling as Russell's paradox but still I am surprised at reading that.
posted by bukvich at 9:32 AM on March 17, 2010

there is a kind of funny takedown of that DFW book in the Notices of the AMS (American Mathematical Society)...

That review is shattering, thanks. I have enjoyed that book twice through without having the mathematical background to know how badly he's fucking up-the metaphysics is pretty good, so I just assumed that the math was, and I quite like the way he writes. Shit.

It's yet another data point about how hard it is so hard to know things: things that I 'learned' in that book, I didn't understand at all-and I didn't know it. It's terrifying-and a reason why I quit philosophy. You can't be sure that everything you work so hard on and think you understand isn't worthless and wrong and just waiting for the right genius to give it a once-over. It's also why I can't stomach intellectual academic arrogance-you think you have it figured out? You're just a footnote-if you're lucky! I can't live on that knife-edge.
posted by Kwine at 9:49 AM on March 17, 2010

fucking i

…it is why I dropped my "beginning algebra" class is college. It made me nuts to think in my art school brain "wait, but arnt all numbers imaginary, show me 3 in your hand"
My analytical chem girlfriend, would just shake her head and walk away.

(and yes I know they dont mention i but man higher math is kooky)
posted by ShawnString at 9:55 AM on March 17, 2010 [1 favorite]

It's my personal belief that in fact j = -i and nobody has noticed.
Actually, j = ki.

I just learned about Logicomix at lunch, interesting.
posted by fantabulous timewaster at 9:59 AM on March 17, 2010

Infinitely awesome -- thanks for the links...
posted by Alexandra Kitty at 10:09 AM on March 17, 2010

Also: the number that mathematicians conventionally refer to as i, as in eiπ + 1 = 0, is generally held to be the same as that which electrical engineers conventionally refer to as j, as in this definition of the Fourier transform. It's my personal belief that in fact j = -i and nobody has noticed.

for some reason the British commonwealth has from time immemorial preferred left-handed coordinate systems.
posted by ennui.bz at 10:13 AM on March 17, 2010

Richard Feynman (PBUH) once contested the variant on Banach & Tarski that cuts up the sphere and reassembles it into a sphere as big as the universe, asserting that in the real world the slices would be thinner than the diameter of an atom.

He was forever doing stuff like that.
posted by tspae at 10:22 AM on March 17, 2010

Just a small, finite number of comments on this thread:

1) David Foster Wallace was a great author, but Everything and More has to be his worst book. It's a confusing mess, I agree with the posters here who encourage people to avoid this book if you want to actually learn anything about math.

2) The claim that set theory is the foundation of all mathematics is an early-20th century view. The theory of categories is becoming increasingly important and can avoid a lot of the unpleasant consequences of set theory.

3) What's sour, yellow and equivalent to the axiom of choice?
Zorn's Lemma.

I've always heard the punchline of this joke as "Zorn's Lemon"

(What's purple and commutes?)
posted by crazy_yeti at 10:31 AM on March 17, 2010

PS: I can't believe I actually found a citation for this stupid joke:
http://www.ams.org/notices/200501/fea-dundes.pdf
posted by crazy_yeti at 10:36 AM on March 17, 2010 [1 favorite]

Crazy_Yeti: An Abelian grape.

What's purple, commutes, and is worshipped by a small number of people?
posted by valrus at 10:49 AM on March 17, 2010 [1 favorite]

What do mathematicians do when they're constipated?
posted by kenko at 10:50 AM on March 17, 2010

Oh hell, I posted that before I saw that you'd put that link up. Damnit.
posted by valrus at 10:50 AM on March 17, 2010

Infinity, division, and sets have been with us as intellectual problems for millennia. These are some of the most profound and productive concepts humanity has grappled with.

For a deep connection, see the original Zeno's paradoxes, as well as later variations. Everything old is new again - we've been grappling with these forever. I first heard of these when I was a child of six - from my grandpa. My little mind was blown. I believe that ultimately it was this that set me off to eventually get my degrees in philosophy. And I've always retained warm feelings for old Zeno.

"Achilles and the tortoise", "The Arrow Paradox" ----> "The Quantum Zeno Effect"

From wikipedia:

"Infinite processes remained theoretically troublesome in mathematics until the early 20th century. The epsilon-delta version of Weierstrass and Cauchy (or the equivalent and equally rigorous differential/infinitesimal version by Abraham Robinson), have developed a rigorous formulation of the logic and calculus involved. These works have resolved the mathematics involving infinite processes, including Zeno's, and the paradoxes no longer present any mathematical problems.[21][4]

While mathematics can be used to calculate where and when the moving Achilles will overtake the Tortoise of Zeno's paradox, some philosophers[5][6] say that mathematics does not address the central point in Zeno's argument, and that solving the mathematical issues does not solve every issue the paradoxes present. Philosophers[who?] also remark that Zeno's arguments are often misrepresented in the popular literature. That is, Zeno is often said to have argued that the sum of an infinite number of terms must itself be infinite - that both the distance and the time to be travelled are infinite. However, Zeno's problem was not with finding the sum of an infinite sequence, but rather with finishing an infinite number of tasks: how can one ever get from A to B, if an infinite number of events can be identified that need to precede the arrival at B, and one cannot reach even the beginning of a "last event"? Philosophers[5][4][6][7][22] say that calculus does not address that question, and hence a solution to Zeno's paradoxes must be found elsewhere. Philosophers[5][6][7] do not see how mathematics takes anything away from Zeno's reasoning that there are problems trying to explain how motion is not impossible."

posted by VikingSword at 12:08 PM on March 17, 2010

Cf. Zima's Paradox, which holds that you can never actually finish the bottle off because, seriously, it's fucking Zima.
posted by cortex at 12:10 PM on March 17, 2010 [1 favorite]

What's so paradoxical about Banach-Tarski?

I mean, the universe is expanding; space itself is expanding.

Banach-Tarski is acting itself out right before our very eyes.

I can't decide whether it would be more interesting to look at the physics of how space is expanding and use that to try for a constructive example of a Banach-Tarski decomposition, or to try for such a decomposition by mathematical means and use that to attempt to make a model for space expanding.

Maybe both at once, each guiding the other.
posted by jamjam at 12:23 PM on March 17, 2010

Awesome! I just finished reading Logicomix, which touches on a lot of this same territory. Can't wait to dig into this stuff.
posted by saulgoodman at 12:23 PM on March 17, 2010

But are numbers real?

Answering this comes down to figuring out what you mean by "real". A related question: do we create mathematics, or are we discovering it? Would mathematics "exist" (for some meaning of that word) if we did not exist?
posted by Chocolate Pickle at 12:44 PM on March 17, 2010 [1 favorite]

David Foster Wallace was a great author, but Everything and More has to be his worst book. It's a confusing mess, I agree with the posters here who encourage people to avoid this book if you want to actually learn anything about math.

While this may be true, fans of DFW should definitely read it, for just the reason cgs06 describes: it's agonizing but sort of fantastic to watch him struggle to write this book you're reading and ultimately all but explicitly concede that it's a failure.

Hey cgs06, I read "Zero" a couple of times. It was very good. Thanks for writing it!
posted by rusty at 12:47 PM on March 17, 2010 [1 favorite]

If you're looking for math jokes this list on mathoverflow is great.

Here's the (tied for) first place joke "a comathematician is a device for turning cotheorems into ffee."

---

Oh, and about Russell's paradox:
My favourite set is the Russell set: A set which contains sets that do not have themselves as members.

Now the question remains is the Russell set a member of the Russell set? If it is a member of the Russell set then by definition it does not contain itself as a member and is therefore not a member of the Russell set.
Well, isn't the whole point of formal set theory to make Russell's paradox impossible? So, for example the "set of sets that do not contain themselves" isn't possible. So for example:
In 1908, Ernst Zermelo proposed an axiomatization of set theory that avoided the paradoxes of naive set theory by replacing arbitrary set comprehension with weaker existence axioms ... ZFC does not assume that, for every property, there is a set of all things satisfying that property. Rather, it asserts that given any set X, any subset of X definable using first-order logic exists.
...
In ZFC, given a set A, it is possible to define a set B that consists of exactly the sets in A that are not members of themselves. B cannot be in A by the same reasoning in Russell's Paradox. This variation of Russell's paradox shows that no set contains everything.
Which means, I think that there is no 'set of all sets that don't contain themselves'. Rather (I think) you can only look at some other pre-defined set and then say "of all of these, I want the ones that do not contain themselves"
posted by delmoi at 12:54 PM on March 17, 2010

Cantorian and post-Cantorian set theory has also provided the basis for Alain Badiou's interesting (but massive and exhausting) philosophical system. Badiou insists that mathematicians are doing the actual work of ontology (as a study of being), while philosophers are at best working at the level of meta-ontology.

In any case, if you want a relatively readable introduction to Badiou's system, DON'T read Being and Event, which will make you take your life; instead, read the translator's preface to Infinite Thought.
posted by LMGM at 1:00 PM on March 17, 2010

Some more math jokes I like:
A poet, a priest, and a lawyer are discussing whether it's better to have a wife or a mistress.

The poet argues that it's better to have a mistress because love should be free and spontaneous.

The priest argues that it's better to have a wife because love should be sanctified by God.

The mathematician says, "I think it's better to have both. That way, when each of them thinks you're with the other, you can do some mathematics."
--
An engineer, a physicist and a mathematician are driving through the high country in Scotland. Atop a hill, they see a black sheep.

The engineer says: "All sheep are black!" The physicist says: "No, no, some sheep are black." The mathematician: "At least one sheep is black on at least one side."
--
An engineer, a physicist, and a mathematician find themselves in an anecdote, indeed an anecdote quite similar to many that you have no doubt already heard. After some observations and rough calculations the engineer realizes the situation and starts laughing. A few minutes later the physicist understands too and chuckles to himself happily, as he now has enough experimental evidence to publish a paper. This leaves the mathematician somewhat perplexed, as he had observed right away that he was the subject of an anecdote and deduced quite rapidly the presence of humor from similar anecdotes, but considers this anecdote to be too trivial a corollary to be significant, let alone funny.
Hmm... Most of the ones I find funny are of the "occupation X, Y and Z=mathematician variety
posted by delmoi at 1:02 PM on March 17, 2010 [2 favorites]

A related question: do we create mathematics, or are we discovering it? Would mathematics "exist" (for some meaning of that word) if we did not exist?

Yes, it would. At least, that's one of the things Godel's Incompleteness Theorem was intended to prove, and the proof as I understand it seems pretty convincing.

Among other things, Godel proved that all sufficiently complex formal systems necessarily yield true statements that cannot be proven within those systems (basically, any formal system worth its salt is incomplete--it can yield truths that aren't consistent with the formal axioms of the system. So, to put it really crudely, no formal system is powerful enough to explain all the truths it necessarily implies. Those necessary but incompatible truths, therefore, aren't products of the formal systems, but exist independently of them. The accepted interpretation of the proof is that formal systems describe an independent mathematical reality.

The BBC did a program called Dangerous Knowledge.

High-level summary: math can drive you insane.

I like the Logicomix take on this phenomenon better: People with preexisting mental instabilities may be drawn to the study of mathematics and logic because these fields seem to offer such an appealing promise of putting order to the universe.
posted by saulgoodman at 1:03 PM on March 17, 2010

I reviewed Everything and More for SEED but the review doesn't seem to be online anymore. The consensus above that it's an interesting failure is correct. Fans of DFW should NOT definitely read it unless they're completists, and even then they should probably read his book about rap first.

Anyway, since I can't post my review of the DFW book, here's "Power Set," my song about Cantor's theorem, from MeFi Music.
posted by escabeche at 1:10 PM on March 17, 2010

BTW, cgs06: I loved your book so much it inspired a song I made.
posted by saulgoodman at 1:10 PM on March 17, 2010

Whoa! Jinx, escabeche.
posted by saulgoodman at 1:12 PM on March 17, 2010

CONCEPT ALBUM!!
posted by escabeche at 1:13 PM on March 17, 2010 [1 favorite]

About the Banach-Tarski Paradox: Some might say that it shows that mathematics is broken, but it probably just casts doubt on the axiom of choice.
posted by JoddEHaa at 1:15 PM on March 17, 2010

But most people just say it means you can take a sphere apart and put the pieces together to make two spheres the same size.
posted by escabeche at 1:39 PM on March 17, 2010

cgs06 - i also loved your book. glad to see you here, and thank you.
posted by radiosilents at 1:46 PM on March 17, 2010

What, no link to the previous John Gabriel crackpottery thread?
posted by Rhomboid at 3:27 PM on March 17, 2010

do we create mathematics, or are we discovering it? Would mathematics "exist" (for some meaning of that word) if we did not exist?

Mathematics is a method for describing patterns and modeling the consequences of their relationships. We created the mathematics we use, and we've quite often gone on to discover features of reality whose relationships are well described by some of the mathematics we create. This is the only connection that mathematics has with the world outside our minds. No minds -> no mathematics (they don't have to be our minds - those of any sufficiently sentient being would do).

This seems perfectly clear to me, which is why I find myself getting a little exasperated when well-regarded thinkers like Stephen Hawking and Paul Davies go on about what "breathes fire into the equations" as if mathematics were in fact somehow prior to existence. Apparently there's money to be made in deliberately mistaking the map for the territory and torturing the idea of "existence" until it gives up and means anything you want it to.
posted by flabdablet at 3:32 PM on March 17, 2010

Thanks for the compliments, everyone... and, geez, a song! I'll listen as soon as I get home.

Love this crowd. *grin*
posted by cgs06 at 3:38 PM on March 17, 2010

Also by Rucker, White Light. Really fun book.

Seconded. But give its full title, "White Light: Or, What Is Cantor's Continuum Problem?" There's a particularly nice Banach-Tarski scene at the end. I recall Rucker saying the whole point of this book is that people can visualize, even "feel" infinities, even with their finite brains.

BTW, allow me to make you jealous: I have a first edition of White Light (the rare Virgin Books paperback edition). It's a special first edition, it includes a xeroxed letter from John Shirley, who sent copies of the book to reviewers at his own expense. I recall paying like \$5 at a going-out-of-business sale at a SF bookstore, the store owner clearly indicated he knew exactly what he had, but sold it to me because he knew I'd appreciate it, I'd pestered him for months to find me a copy (any copy).
posted by charlie don't surf at 5:03 PM on March 17, 2010

Everything that is a failure... is also a victory.

It's all about what size scale you are judging/allowing people to judge you on.

Also infinit-N'th-ing the sentiment of 'very much enjoying yr' gift to the world cgs06'.
posted by infinite intimation at 5:56 PM on March 17, 2010

This is the only connection that mathematics has with the world outside our minds. No minds -> no mathematics (they don't have to be our minds - those of any sufficiently sentient being would do).

This seems perfectly clear to me

Like many things about mathematics that seem perfectly clear, this is not perfectly clear.
posted by escabeche at 6:05 PM on March 17, 2010 [3 favorites]

Like many things about mathematics that seem perfectly clear, this is not perfectly clear.

Correct. Stanislaw Ulam said, "Insofar as mathematics accurately describes reality, it ceases to be interesting."
posted by charlie don't surf at 7:58 PM on March 17, 2010

I don't suppose you'd care to attempt to destroy my present perfect clarity on this matter? Because I'm still not seeing where the alleged difficulty lies.
posted by flabdablet at 10:41 PM on March 17, 2010

"no minds -> no mathematics" is perfectly correct if you mean that the subject wouldn't exist without its practitioners (which seems to correspond somewhat to your comment), but that's blindingly obvious, and true of every subject ever.

This statement: "We created the mathematics we use, and we've quite often gone on to discover features of reality whose relationships are well described by some of the mathematics we create."—what does it mean? Features of reality like the infinitude of the primes? What does mathematics describe? "We created the mathematics we use" has, again, a boring and true interpretation. No zeroes in Roman mathematics! No irrational numbers in Greek mathematics until the, what, the discovery of the incommensurability of a square's diagonal? Or the creation thereof?

Here's a paper on Benacerraf's Dilemma.
posted by kenko at 11:14 PM on March 17, 2010

that's blindingly obvious, and true of every subject ever

which precisely my point, which is why the idea that there must exist some mysterious universal process responsible for "breathing fire into the equations" has always struck me as weird. Mathematics is so good at describing patterns in what we see that I think many of its practitioners forget that mathematics is map, not territory; the equations describing the Big Bang are not, it seems to me, more fundamental than the Bang itself.

Features of reality like the infinitude of the primes?

The infinitude of the primes is a mathematical statement about and within mathematics, not something obviously applicable to anything outside mathematics. It's a map feature, and only occurs in the territory to the extent that the map is itself a part - and a very small part, all things considered - of the territory.

I'm not attempting to be controversial or subtle here, merely objecting to statements that appear to reveal the loss of a grasp on the bleedin' obvious.
posted by flabdablet at 11:41 PM on March 17, 2010

Like many things about mathematics that seem perfectly clear, this is not perfectly clear.

Like many things like 'like many things about mathematics that seem perfectly clear, this is not perfectly clear', 'like many things about mathematics that seem perfectly clear, this is not perfectly clear', is not perfectly clear.

To me.
posted by jamjam at 12:18 AM on March 18, 2010

flabdablet: There are (at least) two ways to read "no minds -> no mathematics". There's the "blindingly obvious" way -- mathematicians are people, so if there were no people there'd be no mathematicians. In that sense, it's equally true to say "no minds -> no geology". Without people there would, trivially, be no study of anything.

The interesting question is whether the things that are studied by mathematicians would be there without minds to study them. In the case of geology, I assume you'd agree that even if no geologist had ever lived, nonetheless, for example, the Rainbow Basin would still contain strata. The objects of study of geology would still exist.

Now, it's temping to say that, since the objects of study in mathematics are abstract -- things like numbers and primeness -- then it's obvious that they could not exist without the minds that imagined them. But that's too quick, I think. I think it can be argued that the most basic concepts in mathematics, such as quantity, are present in nature. On this view, it's a fact of the world, for example, that I have as many fingers as toes -- that my fingers and toes can be put into one-to-one correspondence. Even if you think that the abstract entity that we call "the number ten" is a mere invention of the human mind, the more basic fact that particular collections of things are equinumerous is surely a mind-independent fact of the world.

If you'll grant that much, then it's not much more of a stretch to say that there is a mind-independent fact of the matter about how particular collections of things can and cannot be partitioned. The collection of fingers on my left hand can be partitioned into two parts, one containing two fingers and the other containing three. Is there any way of partitioning them such that each partition has the same number of fingers in it? Yes, but there's only one such way, namely a single finger in each partition.

The final step is to note that there are certain facts about collections of things, like the one stated above, that don't depend upon what specifically are the things in the collection. For example, the fact that the fingers of my left hand can't be evenly partitioned isn't a particular curiosity of my left hand; the same holds of any collection that's equinumerous with the fingers of my left hand.

Now, in the above three steps there was never any need to invoke abstract entities. I've only talked about collections of things, and facts of the world that hold of those collections. But still we end up with an observation that looks like the mathematicians' statement that "the natural number 5 is a prime number". That's why it's fair to say -- in simple cases like this, at least -- that mathematicians are discovering facts about the world, and that the facts they discover are features of the world that exist independently of human minds.
posted by logopetria at 3:00 AM on March 18, 2010 [2 favorites]

I think it can be argued that the most basic concepts in mathematics, such as quantity, are present in nature.

I see that idea as probably being very near the top of a very slippery slope into incoherence at best and contradiction at worst, and I am inclined to be extremely suspicious of it.

On this view, it's a fact of the world, for example, that I have as many fingers as toes -- that my fingers and toes can be put into one-to-one correspondence.

In my view, facts about the world amount to features within some sentience's map of it.

Even if you think that the abstract entity that we call "the number ten" is a mere invention of the human mind, the more basic fact that particular collections of things are equinumerous is surely a mind-independent fact of the world.

Granted, but it's not a minds-independent fact; in my view, there is no such thing. Facts are map entries we use to navigate within the territory we find ourselves occupying.

The observation that particular collections of things are equinumerous involves being able to label or recognize or identify those things, and it seems to me that those activities require some kind of representation engine.

It seems to me that as soon as you grant facts an existence independent of the territory that they are facts about and independent of the minds that notice, contain and propagate them, I think you've started stretching the idea of existence further than is useful or appropriate and you will soon find yourself pondering paradoxes where none need be.
posted by flabdablet at 3:39 AM on March 18, 2010 [1 favorite]

The objects of study of geology would still exist.

I'd agree that territory of interest to geologists has existed and will continue to exist for a hell of a lot longer than there have been and will be geologists; but it's my view that the labeling and identification of objects and processes within that territory is the act of a mind, and is not inherent in the territory itself.
posted by flabdablet at 3:46 AM on March 18, 2010

Here's a little evidence in support of that view.
posted by flabdablet at 4:10 AM on March 18, 2010

I'm inclined to agree to some extent, but taking that position too far risks sliding into a kind of skepticism that I'm likewise rather suspicious about.

"In my view, facts about the world amount to features within some sentience's map of it."

That seems awfully close to (but just short of) denying the existence of any objective reality at all. It's not full skepticism, in that it doesn't deny that the world itself exists, but it seems to render impossible any claim that the world has any objective features -- since to describe those features requires that we pick out objects and properties, and that (you seem to be saying) depends on our maps and representations.

What do you say to the claim that "The Earth's shape is more like a sphere than like a disc"? Whatever status you attribute to that assertion -- whether you call it a 'truth', or a 'fact of the world', or whatever -- that's the status I'm claiming for the assertion "I have as many fingers as toes", and the other assertions in my previous comment. All I'm claiming is that statements about equinumerosity and so on are no less 'factual' or 'objectively meaningful and true' than statements like "The Earth is not flat".
posted by logopetria at 4:12 AM on March 18, 2010 [1 favorite]

it seems to render impossible any claim that the world has any objective features -- since to describe those features requires that we pick out objects and properties, and that (you seem to be saying) depends on our maps and representations.

Correct. In fact the very act of picking out objects and properties is, for most minds, the first step in the construction of any such map.

However, it is a matter of testable observation that there's a very high degree of consensus about useful ways to label the territory, and that many of the relationships between the features so labelled are consistent enough to allow them to be represented in fairly concise ways - in many cases, to quite remarkable degrees of precision and accuracy.

What do you say to the claim that "The Earth's shape is more like a sphere than like a disc"?

I say that there is a well established consensus for the meaning of "The Earth" within contexts like that claim; that most educated people are familiar enough with geometry to understand the relevant properties of the geometrical ideas labelled "sphere" and "disc"; and that the claim itself is both testable and true.
posted by flabdablet at 5:05 AM on March 18, 2010 [1 favorite]

I don't suppose you'd care to attempt to destroy my present perfect clarity on this matter? Because I'm still not seeing where the alleged difficulty lies.

Well, it's not like I'm saying that what you say is wrong; only that it's not "perfectly clear." I can tell it's not perfectly clear because it's been consistently been a subject of controversy and argument among smart people who have thought deeply about it. I think the majority of contemporary philosophers would be on your side, but wouldn't say the problem is easy.

On the other hand, if I understand you rightly, you take mathematical objects to have about the same amount and type of reality as does the Grand Canyon -- I think that is a somewhat less popular claim (though it's one I tend to agree with!)
posted by escabeche at 6:41 AM on March 18, 2010

re: axiom of choice as a "result"
Clearly the axiom itself is not a result, but there are plenty of results associated with its discovery/invention, such as the result that both it and its negation are consistent with the axioms of ZFC set theory as well as the result that it is equivalent to Zorn's Lemo—I mean lema.

saulgoodman: Among other things, Godel proved that all sufficiently complex formal systems necessarily yield true statements that cannot be proven within those systems (basically, any formal system worth its salt is incomplete--it can yield truths that aren't consistent with the formal axioms of the system. So, to put it really crudely, no formal system is powerful enough to explain all the truths it necessarily implies. Those necessary but incompatible truths, therefore, aren't products of the formal systems, but exist independently of them. The accepted interpretation of the proof is that formal systems describe an independent mathematical reality.

You seem to be misunderstanding Gödel's Incompleteness Theorem here, or at least misstating it. The theorem says that any sufficiently complex formal system is either incomplete or inconsistent. "Inconsistent" here means that you can derive something and its contradiction, and therefore you can derive anything, rendering the system completely useless. "Incomplete" does not mean that the system yields truths that are not consistent with the axioms. it means that there are statements you can make in the system that are true but not provable. (If your idea of "true" means the same thing as provable, think of the theorem as saying that there are statements that can neither be proven nor disproven in the system.) These statements are not inconsistent with the axioms. They are simply unprovable. You can add them to the axioms directly, but then there will be other statements that are neither provable nor disprovable.

small_yeti: 2) The claim that set theory is the foundation of all mathematics is an early-20th century view. The theory of categories is becoming increasingly important and can avoid a lot of the unpleasant consequences of set theory.

This is technically true, but it's a seriously small group of mathematicians who use categories as foundational mathematics. The vast number of people who use category theory on a regular basis (myself included) are still using set theory as a foundation, defining their categories in terms of sets and classes (i.e. "a category is a class of objects, a class of morphisms for each pair of objects with a distinguished identity morphism from each object to itself, and a composition operation on morphisms such that ...") You can base your mathematics on topos theory, taking categories as fundamental and defining a set theory based on that, but while there are some papers showing that this is possible, I don't think I've ever read a paper that actually did use category theory as a basis without building it on some kind of set theory. This is not to say that they don't exist, but that they're not very commonplace. At best, I've seen some papers where you can't really tell what the fundamentals are. I have seen a number of papers that don't take standard set theory (i.e. well-founded ZFC + axiom of choice) as a basis, e.g. choosing to use anti-founded sets, opting out of the axiom of choice, deciding to work entirely within intuitionist logic, etc.
posted by ErWenn at 7:01 AM on March 18, 2010

re: flabdablet's map vs. territory reality-of-mathematics argument -- this is one of the very best kind of questions. It's one that I can easily argue both sides of to myself. Don't stop!
posted by rusty at 8:03 AM on March 18, 2010

Math is the study of relationships, which may or may not exist in the real world.

Note, as sentient beings, we can make near-arbitrary relationships exist in the real world. We do this with cryptography.

One school of thought says, it doesn't matter what relationships actually exist, we will simply study all possible relationships and see where this leads us.

Another school of thought says, we should understand well those interrelationships that actually exist.

The former says to the latter, "But how do you know that a solution to your problem, won't come from our pure work?"

The latter says to the former, "Because your work is batshit and useless and whenever you shoehorn it into things, it's hideous and crazy."

Then the former says to the latter, "But it's so beautiful. And anyway, you're not very good at math for my definition math. Why am I even talking to you?" And then it wanders off.
posted by effugas at 2:39 PM on March 18, 2010

if I understand you rightly, you take mathematical objects to have about the same amount and type of reality as does the Grand Canyon

Then I've been unclear, for which I apologize. My view is that mathematical objects are best considered as map features, not territory features.

Maps of the territory and the territory itself are, of course, in a part-whole relationship, so strictly speaking, dismissing mathematical objects as "not territory features" is loose talk. However, given the present state of neuroscience it's currently not feasible to point to a particular pattern of brain activation and say "look, there's an infinite set of prime numbers", which it seems to me would be what we'd have to do in order to locate a mathematical object, or indeed any other map feature, in the territory.

There's a line of philosophical thought that ascribes a more fundamental kind of reality to map features than to the territory. I think this line is badly flawed, and can only be supported by an almost wilful persistence in confusing the map for the territory it describes.

None of this should be taken to deny the idea that making map entries that refer only to other map entries, without any obvious or intended connection to territory outside the map, is a good and worthwhile pursuit. In fact the vast bulk of mathematics consists of doing precisely that, and one of the really neat things about mathematics is just how often that process can produce surprising insights that turn out to be applicable in unexpected ways.

So, my own opinions on Chocolate Pickle's questions are these:

1. I think we quite frequently invent new mathematics before we discover ways to apply it.

2. Mathematics would not exist without mathematicians, and to suggest that it would is to overload the idea of "existence" and possibly the idea of "mathematics" in ways that confuse rather than clarify.
posted by flabdablet at 4:38 PM on March 18, 2010

LMGM: thanks for the link to Badiou. He sounds like somebody I would like to have a beer with :-)
posted by flabdablet at 4:52 PM on March 18, 2010 [1 favorite]

1. I think we quite frequently invent new mathematics before we discover ways to apply it.

2. Mathematics would not exist without mathematicians, and to suggest that it would is to overload the idea of "existence" and possibly the idea of "mathematics" in ways that confuse rather than clarify.

These two statements are true, but perhaps interact in unexpected ways. Relationships exist in nature independently of our ability to describe or even perceive them. (Incidentally, this is why the greatest linguistic mistake in all of physics is the word 'observe' -- it implies the relevance of a conscious observer.) The patterns of these relationships may map well at certain scales to patterns described by mathematical reasoning.

Or they may not.

Put in mathematical terms, the set of mathematical systems that describe anything in the real world, is smaller than the set of things that mathematical systems can describe. Not all universes exist.
posted by effugas at 5:04 PM on March 18, 2010

Relationships exist in nature independently of our ability to describe or even perceive them.

I dispute this spurious claim.

"Relationships" is a map label, and its referent is itself part of the map; it's a high-order abstraction. I claim that relationships do not exist outside maps, and I defy you to show me one that does.

This is not to say that obviously or even trivially related things don't exist independent of the map entries that correspond to them; of course I have as many fingers as toes, and of course my fingers and toes are parts of my body. But unless somebody or something has identified and labeled fingers, toes and bodies as objects in their own right, and has then gone on to relate them, I can't see how to justify a claim that the relationships themselves exist.

It further seems to me that even if your claim is actually about potential relationships - map entries that might possibly be made by some sentience at some time in response to its having noticed some features of the territory - then it still fails. With enough effort, it seems to me that once I've labeled any feature of the territory (which I need to do, as an absolute minimum, in order to be able to reason about it) I could relate it to anything else, even if only in a kind of Six Degrees of Kevin Bacon manner, and call the result a relationship. And to say that all such potential relationships already exist is, in my view, an abuse of the idea of existence. If this strikes you as a straw man, please disregard.

Clearly something exists; as a sentient being, I notice before all else that something's going on. That vague and as yet undefined "something" is what I've been calling the territory, and as soon as I notice it, I've started making my first map.

As an embodied sentience, I fairly quickly work out that it's useful to make a distinction between those parts of whatever is going on that move when I want them to, and those that don't, and label the former "me" and the latter "everything else". I've just drawn the first borderline on my map, if you like.

My point is that until I've done at least that much map-making, I don't have two things that can be related. So, existence is prior to relationships.

the greatest linguistic mistake in all of physics is the word 'observe' -- it implies the relevance of a conscious observer.

Is this truly a claim that conscious observers are irrelevant to physics in general? Because I think that's over-bold. I also think that loose use of the word "relevance" is likely to start some fairly unproductive arguments unless you give it rather more context than you have.

Not all universes exist.

If we're going to talk about great linguistic mistakes, one of my own pet peeves is the fact that the word "universe" gets used to label so many things that have so little in common.

And if I were the boss of everything, I would outlaw the word "multiverse".

More beer!
posted by flabdablet at 8:19 PM on March 18, 2010

"Relationships" is a map label, and its referent is itself part of the map; it's a high-order abstraction. I claim that relationships do not exist outside maps, and I defy you to show me one that does.

The universe operates according to rules that can be defined mathematically. These rules predate the existence of mathematics. Stars agglomerate by the mathematics of gravity, are orbited by planets that follow the laws of orbital mechanics, explode according to equations that can be seen in fluid dynamics. The relationships are there -- the universe does not operate randomly -- and, as was said, they are discovered.

One of the things that fascinates me about math are isomorphisms -- the fact that entire classes of problems can be transformed into one another. For example, in cryptography, RSA (a problem of primes) can be transformed into DSA (a problem of discrete logarithms). The universe may be the output of precious few relationships, transformed upon one another like cellular automata -- this, of course, was the conceit of Wolfram's A New Kind Of Science. But the universe does have relationships, whether or not we're around to understand them.

Is this truly a claim that conscious observers are irrelevant to physics in general? Because I think that's over-bold.

Well, there's only two possibilities.

1) Dinosaurs existed 65M years ago
2) Some kid found a dinosaur bone, and that sent a signal 65 million years in the past that, now that the magic human has Seen The Effect, the Cause Must Instantiate!

The former is science. The latter is anthrocentric woo. Consciousness is totally irrelevant. When physicists say "observe", what they mean is "destructively measure, and all measurements are destructive". We don't have a good word for this. "Collide" doesn't imply measurement, and "measure" implies a measurer. It's a huge problem though.
posted by effugas at 3:40 AM on March 19, 2010

Well, there's only two possibilities.

1) Dinosaurs existed 65M years ago
2) Some kid found a dinosaur bone, and that sent a signal 65 million years in the past that, now that the magic human has Seen The Effect, the Cause Must Instantiate!

The two are indistinguishable, though. Dinosaurs are a bad example because they affect the modern world, they are implicit, in so many ways even before the discovery of an actual skeleton.

A better example, might be a star or galaxy whose light has just reached the Earth for the first time. Did it exist before we observed it, before there were any effects from it whatsoever on our part of the Universe? In physics, this is known as Delayed-Choice.
posted by vacapinta at 3:55 AM on March 19, 2010

Jack and the Aktuals
posted by Eideteker at 4:50 AM on March 19, 2010

Mathematics is a method for describing patterns and modeling the consequences of their relationships. We created the mathematics we use, and we've quite often gone on to discover features of reality whose relationships are well described by some of the mathematics we create. This is the only connection that mathematics has with the world outside our minds. No minds -> no mathematics (they don't have to be our minds - those of any sufficiently sentient being would do).

This seems perfectly clear to me
Well, congratulations on solving a millennia old philosophical problem so tidily. I guess it must be easy being so much smarter then Stephen Hawking and all.
which precisely my point, which is why the idea that there must exist some mysterious universal process responsible for "breathing fire into the equations" has always struck me as weird.
Weird != false.

It's an interesting question. One with five actual answers 1) Math exists independently of our minds. 2) Math exists only in our minds 3) it's possible to know the answer, but unknown at this point, 4) It may or may not be possible, and unresolved at this point or 5) It's impossible to know the answer.

The idea that all those other scientists and mathematicians who wondered about this are just incompetent and the answer is, in fact, obvious is ridiculous.
posted by delmoi at 5:13 AM on March 19, 2010

The two are indistinguishable, though. Dinosaurs are a bad example because they affect the modern world, they are implicit, in so many ways even before the discovery of an actual skeleton.

Uh, they're totally different things. One describes a universe in which time is a unidirectional arrow, the other describes a universe where magic humans back-instantiate all of creation.

What does one sound like, and what does the other sound like?

A better example, might be a star or galaxy whose light has just reached the Earth for the first time. Did it exist before we observed it, before there were any effects from it whatsoever on our part of the Universe? In physics, this is known as Delayed-Choice.

See, this is a prime example of the sort of pseudo-mystical malarky that the word "observe" causes. Wavefronts collapse upon interaction, not upon comprehension. Outer space is not empty. There's a few atoms per kilometer of open space. Over the course of hundreds of millions of years, those atoms were hit. Interactions occurred.

You know what didn't happen? The star didn't flare on up millions of years ago because some kid decided to look at it.
posted by effugas at 5:19 AM on March 19, 2010

effugas, you're just repeating your previous comment.

Uh, they're totally different things. One describes a universe in which time is a unidirectional arrow, the other describes a universe where magic humans back-instantiate all of creation.

The point is quantum mechanical equations don't care about Time. Here's another link on Delayed Choice.

Outer space is not empty. There's a few atoms per kilometer of open space. Over the course of hundreds of millions of years, those atoms were hit. Interactions occurred.

It's an idealized experiment of course. Grant me that.

You know what didn't happen? The star didn't flare on up millions of years ago because some kid decided to look at it.

If you so say so, sure. But the latter statement has no real support other than a belief system. A careful mathematician, aware of the time-independent equations, wouldn't make it. Quoting from the link above:

"If what you say is true," he said (in effect), "then I may choose to know a property after the event should already have taken place." [1] Wheeler realized that in such a situation, the observer's choice would determine the outcome of the experiment – regardless of whether the outcome should logically have been determined long ago.

"Nonsense," said the reductionists. "Rubbish," said the materialists. "Completely absurd," said the naïve realists. "Yup," said the mathematicians.

posted by vacapinta at 5:37 AM on March 19, 2010

vacapinta,

So. Where does the energy come from?

I'm serious. If my conscious comprehension is to have an effect, one that creates a signal collapsing wavefronts back through time, establishing the existences of dinosaurs and the very stars...

Where did the energy come from, to send that signal? What is it, in human perception and meat-computation, that possesses this remarkable power of generating all that mass in the past? Do chimps have this ability? What about dolphins? Can insects instantiate reality?

Can an infant?

If an infant is the first human to look in a particular region of the sky, how does she know what a star is, so that she may observe it into being? Where is that information encoded? Is there a chromosome I don't know about?

Look. The *equations* may not care about time. But *time* does not care about the *equations*. Reality has its own relationships, independent of any particular set of meat-generated tautologies.

It is a reality that part of human nature is a deep, yearning need for importance and centrality. People have been burned at the stake for suggesting so much as the universe does not orbit humanity. You know, it doesn't. Geocentricity had some beautiful equations, some beautiful mathematics -- just ask Ptolemy. But it did not describe reality.

Ultimately, the universe was here long before we got here, and you know, it won't blink out the day we disappear. As long as there are particles that may interact, there will be wavefronts that will collapse. You can call that a "belief system", I suppose. I'm going to call it science.
posted by effugas at 6:05 AM on March 19, 2010

...and, as long as we're having fun:

At what point does this backwards-signal-in-time get sent?

When the photons hit my eye, to be processed by the ganglion cells?

When the signals run up my optic nerve, entering the Lateral Genticulate Nucleus, merging with the rest of my senses?

Perhaps, somewhere in the occipital lobe, where all the basic edge detection starts, there's an extra layer in there, and it collapses the wavefront?

You're right. I'm being pedantic. Perhaps the signal needs to hit our distributed conscious perception.

Do we have to remember it?

Like, if we get really drunk, and forget we saw that star twinkling, does it untwinkle? Does the wavefront decohere? I mean, nobody remembers the twinkle. It's like it never happened.

Is it? Is it like it never happened?

Or is it that that star, twinkling up in the sky, would be just as bright whether or not I cared to bathe in its light for a moment?

(Sorry for being blunt. But there is absolutely no science, none whatsoever, in the belief that consciousness creates the universe.)
posted by effugas at 6:23 AM on March 19, 2010

The universe operates according to rules that can be defined mathematically.

No it doesn't. It just does what it does. We can use mathematically defined rules to produce concise descriptions of some of what it does.

These rules predate the existence of mathematics.

No they don't. The behavior we invented them to describe does, but the rules themselves require rule-makers to host them.

Stars agglomerate by the mathematics of gravity, are orbited by planets that follow the laws of orbital mechanics, explode according to equations that can be seen in fluid dynamics.

Stars agglomerate, planets orbit, fluids flow, and we have invented mathematics that allows us to work with patterns that we have discovered are sufficiently isomorphic to some of the observable features of these processes to make us able to describe, understand, classify and make useful predictions about them.

The relationships are there

As you can probably tell by now, I emphatically disagree with that proposition.

the universe does not operate randomly

Fairly substantial portions of it appear to do just exactly that. Most of us generally do our best to ignore that fact, but unless quantum indeterminacy and statistical mechanics are simply wrong, a fact it remains. Nature is not predictable to an arbitrary degree of precision, no two objects are completely identical, and no experiment is absolutely and completely replicable. The territory is simply not that regular.

the universe does have relationships, whether or not we're around to understand them.

I agree that stuff happens whether or not we're around to distinguish and classify its parts and relate them to each other. I also agree that dinosaurs existed 65Mya.

Consciousness is totally irrelevant.

Relevance is totally subjective.

When physicists say "observe", what they mean is "destructively measure, and all measurements are destructive". We don't have a good word for this. "Collide" doesn't imply measurement, and "measure" implies a measurer. It's a huge problem though.

Not from where I sit; from my point of view, it's quite a small problem. I can see how it could appear huge from the point of view of one whose map has substantial overlap between the ideas of physical law (loosely: concise and reliable summaries of repeatably observable physical behaviors) and human law (loosely: classification of behaviors as permissible or impermissible).

If you're proceeding from the assumption that all physical behavior is lawful, in the sense of being in principle completely predictable, given a sufficiently comprehensive description of its underlying principles - then yes, you will have difficulty with this kind of issue, because that worldview predisposes people to multiplying entities beyond necessity.

One of the map entries we all make as quite small children is the idea of object permanence. As a result, we expect the objects we identify to exhibit permanence, and experience cognitive dissonance if they fail to do so. So if your intuitive picture of the electron is of a small object, you will be troubled - perhaps quite profoundly - by your inability to identify which slit it has really gone through on its way from emitter to detector, and your inability to categorize it satisfactorily as either a wave or a particle.

But if you're happy to categorize the electron as a map feature whose sole purpose is to take part in mathematical models built to make useful predictions about the outcomes of carefully constrained experiments, then the question of where it "really is" when it's not interacting detectably with something else becomes irrelevant - or, if you prefer: frivolous, anthropocentric woo.

The nature and scale of things it needs to interact with in order to be detected - that is, in order for its wave function to collapse - becomes similarly unproblematic, as does the cause of the collapse of that wave function. Wave functions collapse when we decide that it is useful that they do so.

Wave functions - and the various and sundry particles they apply to - are our functions, our models, our understandings, our predictive tools used to further our purposes. Map features, not independently existing bits of territory.

What, I hear you cry - is that idiot truly trying to make a case that electrons don't exist? Surely this is where I can just write him off as a handwaving bloviator with no useful insights to contribute. Why, if it weren't for electrons - and photons, come to that - we wouldn't even be having this argument.

And I reply that your attitude is so 19th century. Our ability to have this discussion depends not on electrons but on transistors and logic gates on silicon wafers embedded in epoxy resin and connected with copper and lead-free solder and all that stuff that does exhibit clear object permanence. We are justifiably confident that when we shape these substances in these ways and hook them all up thus and so then they will behave like that; and we do, and they do, and here we are. And we have that confidence because our models work and because the technology we construct on the basis of those models shares important features with the experiments that helped us develop and refine the models.

The machinery contains electron emitters and electron absorbers and structures designed to shape and direct electromagnetic fields. But we don't know whether it contains point unit charges. We know that its successful operation relies on electron emission events and electron detection events with carefully designed relationships. Those events happen and I would not attempt to argue otherwise. But electrons as objectively existing point unit charges between emission and detection events? Sorry, no evidence.

The fact the you and I are currently having this discussion by means of technology that is only constructable because we know about wave functions and electrons is evidence for the utility of those map features, not evidence for the objective existence of territory features isomorphic to them.

And you know what? Other ways of understanding the territory are quite often useful as well, even some of those that we hard headed rationalists are inclined to write off as woo. And that's why I'd quite like to have a beer with Alain Badiou.
posted by flabdablet at 6:49 AM on March 19, 2010

Well, congratulations on solving a millennia old philosophical problem so tidily. I guess it must be easy being so much smarter then Stephen Hawking and all.

I came here for an argument!

Oh! Oh, I'm sorry, but this is sarcasm.

Oh, I see, well, that explains it.

Ah yes, you want room 12A, just along the corridor.
posted by flabdablet at 6:53 AM on March 19, 2010

delmoi: It's an interesting question. One with five actual answers 1) Math exists independently of our minds. 2) Math exists only in our minds 3) it's possible to know the answer, but unknown at this point, 4) It may or may not be possible, and unresolved at this point or 5) It's impossible to know the answer.

Or, in fact, it's largely a semantic argument centering on the words "math" and "exist," with "independently" and "minds" playing supporting roles.
posted by PMdixon at 7:01 AM on March 19, 2010

...and, as long as we're having fun:

How long, roughly, in milliseconds, is now?
posted by flabdablet at 7:01 AM on March 19, 2010

Oh, that one's easy: "now" lasts for about a year.
posted by fantabulous timewaster at 7:14 AM on March 19, 2010

flabdablet,

I wonder if you're accurately understanding my position here:

Stars agglomerate, planets orbit, fluids flow, and we have invented mathematics that allows us to work with patterns that we have discovered are sufficiently isomorphic to some of the observable features of these processes to make us able to describe, understand, classify and make useful predictions about them.

Of course. My point is that our ability to invent and describe such isomorphic mathematics, is utterly orthogonal to the existence of agglomeration, orbiting, and flow.

Those things are going to happen whether or not we understand them. The world is filled with things we don't understand, but happen anyway according to rules we later learn.

This should not be controversial. It's sort of the basis of science itself.

the universe does not operate randomly

Fairly substantial portions of it appear to do just exactly that.

Depends entirely on scale. Chaos theory is the branch of math that attempts to capture the perturbations through order and chaos that many systems undergo. Halley's Comet manages to come by every 87 years, despite being composed of matter that is very random at the quantum scale.

When physicists say "observe", what they mean is "destructively measure, and all measurements are destructive". We don't have a good word for this. "Collide" doesn't imply measurement, and "measure" implies a measurer. It's a huge problem though.

Not from where I sit; from my point of view, it's quite a small problem. I can see how it could appear huge from the point of view of one whose map has substantial overlap between the ideas of physical law (loosely: concise and reliable summaries of repeatably observable physical behaviors) and human law (loosely: classification of behaviors as permissible or impermissible).

Ah. This is because you are a scientist, quibbling a bit about whether one can describe extant complex systems as having "relationships". You are not proposing that discovering a dinosaur bone back-creates the dinosaurs.

But, you know, there are people who really do argue that. That really is a problem, because the plain meaning of the word "observe" really does imply somehow that conscious comprehension of a fact makes that fact "real" -- not in some philosophical sense, but in an actual, maybe our perceptions cause a signal to go back in time and back create stuff.

After all, time is just another variable in quantum mechanical equations.

"Observe" is a problematic word. It just has too much wrapped up in it. We use it because we want to express that we just can't measure anything at that scale without deeply altering it -- that the very interaction that drives a measurement, creates a quantifiable state change in the thing we're measuring. But it's not our ultimate reading of that measurement that creates the state change, it's the interaction with anything other than empty space. Conscious observation requires measurement, and measurement at the quantum scale requires destructive interaction (this is surprising), and destructive interaction (unsurprisingly) changes the system being measured. But people go all transitive, and say conscious observation changes the system being measured, and that's when the truckloads of woo arrive.

Don't really know what to make of your big psychoanalysis. Reality is fuzzy but quantized -- we've got great evidence for it. Sure, electrons exist, but just as obviously they aren't particularly discrete particles as much as they are a cloud of goo with some quantum mechanical properties of discreteness. Don't look to me to surprised by the behaviors of quantum wells or quantum dots or even qubit systems; they're no longer novel, they're just part of how the universe works and we can build upon them.

(If you want to find something I _do_ consider controversial in my belief system, it's that I think quantum entanglement actually does prove hidden variables. Spooky action at a distance is a _hack_, and making excuses that it doesn't send information is positively Ptolemaic. But accepting hidden variables would throw out most of our models for how things actually work, and our models are really quite excellent at predicting the behavior of untangled systems. My suspicion is that the determinacy that does exist at the quantum scale really only becomes a relevant factor in very specific and rare situations, just like quantum mechanics itself is only very rarely relevant to orbital mechanics. Quite simply, you just normally don't need to worry about twinned particles. But they exist, and our present map does not account particularly well for this territory. At least, that's what I _suspect_. I don't have enough evidence to back that suspicion up.)
posted by effugas at 2:46 PM on March 19, 2010

(To clarify, "Not from where I sit..." in the previous post should be italicized.)
posted by effugas at 2:55 PM on March 19, 2010

I think many of its practitioners forget that mathematics is map, not territory ... [primes are] map feature, and only occurs in the territory ... Facts are map entries we use to navigate within the territory we find ourselves occupying. ... facts an existence independent of the territory that they are facts about and independent of the minds ... My view is that mathematical objects are best considered as map features, not territory features. ... map entries that might possibly be made by some sentience at some time in response to its having noticed some features of the territory - then it still fails. ... no two objects are completely identical, and no experiment is absolutely and completely replicable. The territory is simply not that regular ...
Confusing the map for the territory? I think you've confused an Analogy for Reality. Or metaphysics. All this reasoning about "Maps and territories" is also itself a "map" about something else, and therefore in your own model incapable of actually describing the "territory" of the interaction between mathematics and reality.

So the whole argument is self refuting.
(If you want to find something I _do_ consider controversial in my belief system, it's that I think quantum entanglement actually does prove hidden variables. Spooky action at a distance is a _hack_, and making excuses that it doesn't send information is positively Ptolemaic.
Google the Holographic Principle. There was a great video that someone linked too in some FPP talking that spent an hour explaining the entropy of black holes and how their surface contains a ton of information. So the speculation is that the universe exists as the information on some surface, and that gravity is a thermodynamic process, by which objects being together has a higher entropy then things being far apart. I can't find the damn thing now, though. Very annoying.
posted by delmoi at 3:20 PM on March 19, 2010

Delmoi,

I'm somewhat familiar with holographic math, and here's my semi-clueless question: In a system where information is equally distributed across the entire surface, how do you derive relativity? Effectively, for the system to be holographic, changes would have to propogate instantaneously. But, if they did, changes in the hologram would propogate instantaneously as well.

Put another way, reference frames, and spatially distributed information (even of a different dimensionality) seem to conflict. Has this been resolved?
posted by effugas at 4:01 PM on March 19, 2010

effugas: I don't know the math, but it would solve a lot of the "action at a distance" problems, for example, two 'entangled' particles could appear to be far apart in 3d space, but actually at the same point in 2d space. Or something.
posted by delmoi at 5:47 PM on March 19, 2010

My point is that our ability to invent and describe such isomorphic mathematics, is utterly orthogonal to the existence of agglomeration, orbiting, and flow.

And my point is that (a) our ability to reason about agglomeration, orbiting and flow is not a necessary precondition for the existence of that which agglomerates, that which orbits and that which flows and (b) agglomeration, orbit and flow are process descriptions - map features - not things with their own independent existence.

Those things are going to happen whether or not we understand them. The world is filled with things we don't understand, but happen anyway, [perhaps] according to rules we later learn.

Agreed, with the indicated small amendment.

This should not be controversial. It's sort of the basis of science itself.

Which is pretty much what I meant when I wrote This seems perfectly clear to me, which is why I find myself getting a little exasperated when well-regarded thinkers like Stephen Hawking and Paul Davies go on about what "breathes fire into the equations" as if mathematics were in fact somehow prior to existence.

I think we're on the same page; we're just looking at it through slightly different reading glasses.

Confusing the map for the territory? I think you've confused an Analogy for Reality. Or metaphysics.

I'm using the "map" and "territory" analogies for the purpose of concise exposition. I can readily understand how my idiosyncratic use of those terms could result in a lack of clarity in communicating my position. If you know of clearer words I could use instead, please do share.

All this reasoning about "Maps and territories" is also itself a "map"...

Agreed. Should be uncontroversial and obvious. But wait - there's more!

...about something else, and therefore in your own model incapable of actually describing the "territory" of the interaction between mathematics and reality.

So the whole argument is self refuting.

Sorry for being slow, but I don't see how you just got from one side of "therefore" to the other and I don't understand your conclusion. Could you elucidate?

At no time have I ever attempted to support a claim that my personal map, or any other person's map, or even the aggregate of all our maps (praise be unto Jimbo) is complete.

All I'm attempting to do is point out that classifying things as either map features or territory features is a useful confusion reduction exercise.

My central claim is about those equations that Stephen Hawking and Paul Davies have written of in terms that make it pretty clear that they consider the equations themselves to be more fundamental than the things they describe.

I simply claim that those equations, like all equations, are map features only; thus, the existence of the equations requires the existence of a mapper (e.g. us); thus, no such equations existed before mappers (e.g. us) did; thus, there is no requirement that they have "fire breathed into them".

In other words: the question of why reality should behave this way instead of that way has a simple, correct answer: because this way describes what we observe, while that way is just some shit you made up.

I claim that if that strikes you as a non-answer or an attempt to dodge the question, you are probably failing to recognize some parts of your map as descriptions of territory, and have instead come to think of them as if they themselves were the territory they describe.

Every time I see a writer waxing lyrical about the consequences of altering the fifth decimal place in the fine structure constant, I have to suppress an urge to write to them and ask for their opinion on seven sided pentagons.
posted by flabdablet at 5:50 PM on March 19, 2010

Spooky action at a distance is a _hack_, and making excuses that it doesn't send information is positively Ptolemaic.

I find it quite unsurprising that information can be shared between A and B without ever having been sent from A to B or B to A.

For example, you and I both know what turbulence is, but as far as I know, you've never explained it to me and I've never explained it to you.
posted by flabdablet at 5:55 PM on March 19, 2010

Sorry for being slow, but I don't see how you just got from one side of "therefore" to the other and I don't understand your conclusion. Could you elucidate?

Well, I assume when you say "map" and "Territory" you're talking about "what we're thinking about" and "What actually exists", right?

Then you spend a lot of time talking about how, for example relationships don't really "exist", they are just things we create. But isn't the relationship between the concept of "map" and the concept "territory" also a relationship, and therefore something that doesn't "exist", as opposed to something we've created? So for example when you say something is a "map feature only" aren't you in fact creating an artificial classification of "map features" vs. "territory features"

Anyway, it seems obvious to me that if you are talking about accurate real maps, then there are properties, or features that apply both to the actual map, and to the actual territory. So for example, if you have two mountains, the ratio of their height on the map, and the ratio of the height in reality are the same thing (+/- the margin of error). Or the surface area of a lake, relative to the scale. Those would be examples of "map features" that are also features of the territory. The surface area of the lake is a fundamental truth of the lake, assuming nothing has change since the map was made.

And furthermore, the mathematical rules of the universe is not the same thing as a chunk of land: The rules don't change as you move through it, while the features of the map do change as you consider a different location, or move about the territory.

And a territory is inactive. It can not create it's own maps. But the question is whether or not something fundamental about the mathematical rules of the universe therefore creates it's own map somehow. Or if perhaps there is some underlying map of the universe which was used to create it that we, by trying to map said universe are actually reconstructing.

So for example Someone built a turing machine that runs in a cellular automata universe (conway's game of life). You could then program that Turing machine to run another copy of the game of life. You could then argue that the "territory" created it's own "map"

Or going back to the non-metaphorical. A landscaper may have used a map to create a territory. Then, by mapping out the territory you are working to recreate that map. Which is not all that interesting for actual maps and actual territories but is interesting for the actual universe.
I'm using the "map" and "territory" analogies for the purpose of concise exposition. I can readily understand how my idiosyncratic use of those terms could result in a lack of clarity in communicating my position. If you know of clearer words I could use instead, please do share.
Well "math" and "universe" work. The relationship between math and the universe is not identical to a map and a territory because, as I said, a physical hunk of land cannot create it's own map without outside help. And ordinarily they have not been created from another map, certainly not without intelligent help. But perhaps the universe was created on the basis of underlying math.
posted by delmoi at 10:55 PM on March 19, 2010 [1 favorite]

flabdablet--

My central claim is about those equations that Stephen Hawking and Paul Davies have written of in terms that make it pretty clear that they consider the equations themselves to be more fundamental than the things they describe.

*shrugs*

Nature doesn't care about Stephen Hawkins' equations. The equation for a circle -- (pi)*r^2 -- is more beautiful, more pure than any natural circle. It is an idealized form. To the extent nature matches a perfect circle, the math is helpful. To the extent nature does not match a perfect circle, the math is what it is, and reality is what it is. They have diverged.

Look. There is something beautiful about reality matching a complex equation, at least within some degree of error (and there's always a degree). Something as simple as E=MC^2 informs us about the behavior of so much.

Reality came first, though. The stars were fusing long before we discovered fusion.

I'm not sure we actually disagree.

I find it quite unsurprising that information can be shared between A and B without ever having been sent from A to B or B to A.

In quantum entanglement, a particle is split, with one photon going one direction and one photon going the other. Theoretically, the photons *should* determine its polarization at time of measurement, and there should be no link between the polarization state of one versus the other.

In reality, the states are always opposite. One photon will always be aligned one way, and the other will always be the other. This means one of two things:

1) Quantum theory is wrong, in that the results of the "dice throw" aren't set at time of interaction, but are deterministically set by initial conditions
2) Somehow, the measurement of the first photon collapses the wavefront of the second photon, at a distance, at speeds greater than light. This is OK because the polarization can't be selected, therefore no information (other than, 'collapse now') can be sent.

I think #2 is a hack. But gosh, does the math backing up #1 hold up to experimental review.
posted by effugas at 1:33 AM on March 20, 2010

Well, I assume when you say "map" and "Territory" you're talking about "what we're thinking about" and "What actually exists", right?

Pretty much, except that some of the stuff we think about doesn't correspond at all to anything outside our own thoughts, so it doesn't really count as part of a map; invisible pink unicorns are the canonical example.

The territory is that which exists, considered as an undifferentiated whole: all that is, ever was, or ever will be. The map is an incomplete and highly compressed internal representation of territory, maintained and operated on by roughly the same bits of you and me that are responsible for carrying on this conversation.

Then you spend a lot of time talking about how, for example relationships don't really "exist", they are just things we create.

What I had intended to communicate was the idea that map features (such as relationships, or the notion of existence itself) cannot exist independent of their hosts because they are quite strictly parts of those hosts.

I am certainly not trying to support a claim that map features are inherently non-existent - merely drawing attention to where they exist: inside minds, which are themselves local processes within small regions of territory e.g. us.

But isn't the relationship between the concept of "map" and the concept "territory" also a relationship,

Of course.

and therefore something that doesn't "exist", as opposed to something we've created?

There's no opposition there. Things we have created have every bit as much existence as things we haven't. Some of the things we've created remain internal to our minds; that doesn't mean they don't exist, though it may well mean that they are more process-like than object-like in that they don't necessarily have object permanence.

So for example when you say something is a "map feature only" aren't you in fact creating an artificial classification of "map features" vs. "territory features"

Classification is one of the basic compression techniques we use to construct our maps. So if what you mean by "artificial" is "constructed by us" then yes, that's exactly what I'm doing.

Anyway, it seems obvious to me that if you are talking about accurate real maps, then there are properties, or features that apply both to the actual map, and to the actual territory.

Assuming that the map is isomorphic to the territory to some degree, then there will be statements we can make about the map that will have the same truth-value when applied to corresponding parts of the territory.

So for example, if you have two mountains, the ratio of their height on the map, and the ratio of the height in reality are the same thing (+/- the margin of error). Or the surface area of a lake, relative to the scale. Those would be examples of "map features" that are also features of the territory.

I would count those as map features that accurately represent features of the territory. I disagree that the map features and the territory features they represent can be conceptually interchanged without error.

Error, incidentally, is a map feature.

The surface area of the lake is a fundamental truth of the lake, assuming nothing has change since the map was made.

The surface area of the lake is observable and measurable and quite possibly something that ought to go in a good map. But in order to do that, you need to identify the lake as a lake - that is, you need to draw one of those artificial boundaries around that part of the territory that you are going to label "lake".

And in order to do that, you're going to need to ignore a certain amount of information. The more closely you examine the boundary of the lake, the more detail you will see, and the fuzzier will become your measurement of its surface area (that effect will be much more noticeable if what you're attempting to measure is its circumference, but it happens to area as well).

In fact the precise figure you end up with for the surface area of the lake depends on exactly where you choose to put its boundary in your map of it, which is a fairly clear indication that surface area is a map feature rather than something inherent in the territory.

That lake may well have been there for a million years, but its surface area has only existed for as long as people have been making mental maps that include the concepts of boundaries, surfaces and areas.

And a territory is inactive. It can not create it's own maps.

When I'm thinking about territory, I'm thinking about something that includes you and me and our internal maps and Old Uncle Tom Cobbleigh and all; something without distinctions. The fact that we need to make distinctions in order to render the territory comprehensible (i.e. map it) says more about the process of mapping than it does about the territory.

But the question is whether or not something fundamental about the mathematical rules of the universe therefore creates it's own map somehow.

And this, I think, is where your map and mine start seriously diverging.

If you're at all interested in exploring a map more like mine, I strongly urge you to give your "somehow" some careful scrutiny. Turn it into a "How?" And if you find, as I did, that none of the answers actually work for you, turn your attention to the idea of "mathematical rules of the universe" and see how the idea that such rules are things that exist only within our own minds and writings sits with you.

Or if perhaps there is some underlying map of the universe which was used to create it that we, by trying to map said universe are actually reconstructing.

Or perhaps there is an invisible pink unicorn watching me as I type this. My personal preference is for a less cluttered map, but whatever floats your boat.

Well "math" and "universe" work.

They don't really work for me, because they don't emphasize the one-way representational relationship between territory and map. There are vast amounts of math that are not about anything outside math itself. And "universe" has been so muddied by overloads (the universe, multiple universes/multiverses, universe of discourse, possible universes, alternative universes, parallel universes) that I find myself wasting a hell of a lot more time hacking through disagreements about what "universe" means than I do trying to communicate my own idiosyncratic notion of "territory". But thanks for the suggestions; if you have more, keep 'em coming.

But perhaps the universe was created on the basis of underlying math.

The day we actually, truly reach the End of Physics is the day I might start giving that notion any serious attention.
posted by flabdablet at 2:26 AM on March 20, 2010

Ergh, need to stop doing that. Last sentence should read:

I think #2 is a hack. But we've built a whole lot of real world, working systems, based on the idea that quantum states collapse on destructive measurement. There's a bunch of math that doesn't work anymore if #2 is wrong. #1 may seem more likely but #2 is building stuff, so...
posted by effugas at 2:26 AM on March 20, 2010

Something as simple as E=MC^2 informs us about the behavior of so much.

Indeed. Finding substantial improvements to a map's compression ratio is intensely pleasurable.

I'm not sure we actually disagree.

Not on much, it seems. More about legitimate uses of language than anything else, I think.

In quantum entanglement, a particle is split, with one photon going one direction and one photon going the other. Theoretically, the photons *should* determine its polarization at time of measurement, and there should be no link between the polarization state of one versus the other.

It seems to me that if that's your intuitive prediction, you're probably thinking of photons as if they were objects in their own right, displaying object permanence and in-principle separability from their surroundings.

I don't think of them that way. I think photons are map features. I will go so far as to allow that photon emission and absorption/detection events are probably territory features.

In reality, the states are always opposite. One photon will always be aligned one way, and the other will always be the other.

So, let's quickly review what we have:

1. We create an entangled photon emitter.
2. We observe that for any given pair of absorption events that we can causally link to a given emission event, certain parameters of those absorption events are repeatably correlated.

This means one of two things:

1) Quantum theory is wrong, in that the results of the "dice throw" aren't set at time of interaction, but are deterministically set by initial conditions
2) Somehow, the measurement of the first photon collapses the wavefront of the second photon, at a distance, at speeds greater than light. This is OK because the polarization can't be selected, therefore no information (other than, 'collapse now') can be sent.

I think you've left out a possibility:

3) Attempts to apply quantum theory to parts of such an experiment in a manner that assumes that those parts are independent, as opposed to working out the consequences of the wave function for the entire emission and absorption set as an integrated whole, will yield predictions that disagree with observation.

Now, the consequences of (3) might make modelling more difficult, in that the complexity of actually evaluating the applicable QM equations explodes as the number of linked entities increases, but if that's the way it is, then we kind of need to just suck it up.
posted by flabdablet at 2:54 AM on March 20, 2010

It seems to me that if that's your intuitive prediction, you're probably thinking of photons as if they were objects in their own right, displaying object permanence and in-principle separability from their surroundings.

Please stop the psychology speak. It's unscientific, and vaguely insulting.

A photon is a quantized unit of energy with an uncontroversial particle/wave duality, that is apparently resolved during destructive interactions ("observation").

3) Attempts to apply quantum theory to parts of such an experiment in a manner that assumes that those parts are independent, as opposed to working out the consequences of the wave function for the entire emission and absorption set as an integrated whole, will yield predictions that disagree with observation.

No, that's pretty much 2), the assertion that the two twinned photons need to be viewed as members of the same system, which apparently are able to operate across long distances of time and space, at superluminal speeds, with no known methodology.

Sure, it sounds ridiculous. But, what's the alternative? That there's some sort of "Quantum Pseudorandom Polarity Generator", seeded with the precise position, velocity, or Planck-time clock cycle of a given destructive interaction, which betrays its shared seed when there are two outputs for a single input?

Such an alternative flies in the face of Heisenberg, of almost all established quantum theory, of significant experimental data, and working quantum systems. I mean, I believe wholeheartedly that this is what's happening, and I'm the first to say extraordinary claims require extraordinary evidence which I don't have.

There's no third option here. Either polarities are determined at destructive interaction, and the two twinned particles communicate "spookily", or polarity is determined at particle birth, and destructive interaction only yields a predestined polarity.

I'd love to see a way out of this dichotomy, but I don't.
posted by effugas at 3:21 AM on March 20, 2010 [1 favorite]

They don't really work for me, because they don't emphasize the one-way representational relationship between territory and map.
But those are the actual names of the actual things. And the whole point is: we don't actually know what the relationship is. You've decided it's a certain way, and then are arguing based on an analogy to your own idea. But all the flailing about "maps" and "territories" has nothing to do with the actual things. That's why you use the real words, rather then ones that (you feel) are analogous.

Anyway, the problem with your whole argument is that you're taking this "map/territory" thing as a given and reasoning from there. There's no reason why we should assume that the relationship between mathematics and the physical universe are like that of a map and region of land. And you haven't made any effort to explain why they should be, just going on and on about how obvious it all is -- going so far as to claim that scientists were "deliberately" confusing the two.

There is no reason claim that the two relationships are the same.
posted by delmoi at 3:28 AM on March 20, 2010

I don't think of them that way. I think photons are map features. I will go so far as to allow that photon emission and absorption/detection events are probably territory features.
See this? This is what I'm talking about. The "Map/Territory" stuff only exists in your head. But more importantly they don't necessarily exist in anyone Else's head. That's the important thing. If you want to thing that way, feel free, but no one else is obligated to think the same way just because you do. It's not at all obvious.
posted by delmoi at 3:56 AM on March 20, 2010

delmoi,

OK. I think I see the problem here:

flabdablet wants to speak in the language of psychology, philosophy, and so on. We don't.

Ultimately, I don't know how much I actually disagree with the map/territory point, given that it's roughly akin to my primary thesis. If you replace all instances of "map" with "derived mathematics" and all instances of territory with "how the universe actually works", I think we're actually on the same page here.

Humans make maps/equations. The territory/reality is quite independent of that.

flabdablet,

Your problem is that you seem to want to communicate directly in terms of philosophical constructs. This does not work outside of highly structured academic environments. Not, "this is invalid" or "you are right or wrong". Just, "you are communicating in the wrong language, not unlike as if you were suddenly speaking French."

In normal communication, you can basically say "The map is not the territory" and people will understand you. What you can't get into, and expect the rest of us to understand, are complex transformations across your semantic constructs. Given that I suspect you've contorted yourself into disagreeing with something you actually agree with, I'm not sure how useful this entire train of thought has been for even you.

Also, we're discussing an objective scientific reality, so (no offense) feelings don't matter.
posted by effugas at 4:56 AM on March 20, 2010

No, that's pretty much 2), the assertion that the two twinned photons need to be viewed as members of the same system, which apparently are able to operate across long distances of time and space, at superluminal speeds, with no known methodology.

But there are no superluminal speeds involved. The spacetime intervals between photon emission events and photon absorption events remain non-anomalous, as far as I've heard; causality is safe.

Either polarities are determined at destructive interaction, and the two twinned particles communicate "spookily", or polarity is determined at particle birth, and destructive interaction only yields a predestined polarity

...or modelling this experiment using the same conceptually separable particles that work so well elsewhere is simply not appropriate.

Really, all we know is that when we set up an experiment thus and so, we see correlations we would not expect to see in the light of present models. And all that means is that some of our assumptions about the entities and relationships involved are are unjustified.

Here's a testable prediction: if we do ever come to understand how to model these cases properly, the new model will reduce to QM for the cases where QM doesn't presently yield incorrect results, but the "polarity of a photon" will mean something different from what we think it means now.

Here's another: we'll be happy as clams for a while until somebody notices something else that doesn't quite fit right.

the problem with your whole argument is that you're taking this "map/territory" thing as a given and reasoning from there.

That's not a bug, that's a feature.

There's no reason why we should assume that the relationship between mathematics and the physical universe are like that of a map and region of land.

There's also no reason why we shouldn't entertain that possibility as a working hypothesis, and find out whether doing so raises fewer untestable questions than some other hypothetical relationship does. In my experience, for what it's worth, it does.

If you want to thing that way, feel free, but no one else is obligated to think the same way just because you do. It's not at all obvious.

If you go back and read the comment that opened our little side discussion, you will see that I wrote

This seems perfectly clear to me, which is why I find myself getting a little exasperated...

and I have never attempted to suggest that anybody else is obligated to think the same way I do. I am simply pointing out that this view is available, arguing that it's coherent and workable, and that its consequence is being able to experience very little distress over disagreements between how stuff actually behaves and how we think it ought to, while remaining quite open to the wonder of discovery. Experiencing a small amount of exasperation when people who I know are smarter than me seem hellbent on making mountains out of what I perceive to be pictures of molehills strikes me as a very low price to pay for that.

...on preview:

Also, we're discussing an objective scientific reality, so (no offense) feelings don't matter.

So you're over feeling vaguely insulted, then? I'm happy to know that :-)
posted by flabdablet at 5:20 AM on March 20, 2010

Given that I suspect you've contorted yourself into disagreeing with something you actually agree with

What might that be?
posted by flabdablet at 5:25 AM on March 20, 2010

The territory/reality is quite independent of that

except insofar as we are parts of it and tend to do stuff to other parts of it.
posted by flabdablet at 5:27 AM on March 20, 2010

If you replace all instances of "map" with "derived mathematics" and all instances of territory with "how the universe actually works", I think we're actually on the same page here.

Pretty much, except that deriving overtly mathematical relationships is only part of what we do to make our maps. Quite a lot of the map is pre-verbal. For example, simply experiencing ourselves as beings separate from our environments is exercising our mapping ability; in my opinion this is one of the first map entries most of us make, and it establishes the notions of boundaries and the separability of objects. These are wired in so deep that we tend not to think about them much; we simply experience ourselves and the objects around us as separate and/or separable.

However, I have yet to see any boundary in reality that close examination doesn't render fuzzy. Can you think of one? Do I simply need better reading glasses?
posted by flabdablet at 5:39 AM on March 20, 2010

But there are no superluminal speeds involved. The spacetime intervals between photon emission events and photon absorption events remain non-anomalous, as far as I've heard; causality is safe.

Causality is safe because no information is transferred. The concept is that the destructive interaction with one twin collapses the probability wave of the other, forcing its spin (not polarity) to be the opposite of whatever the first particle cohered into. The collapse indeed is supposed to happen instantaneously, no matter the distance between the particles.

This is, by definition and somewhat conveniently, an untestable hypothesis.

...or modelling this experiment using the same conceptually separable particles that work so well elsewhere is simply not appropriate.

That's the point, that effectively the two twinned particles are somehow acting as one.

Unless you want to argue that two entangled particles are just a possibly infinite size 1 dimensional particle that spreads out from the point of twinning -- OK, I'll grant this is a third possibility. Not much testable precedent here for anything like this though.

and I have never attempted to suggest that anybody else is obligated to think the same way I do. I am simply pointing out that this view is available, arguing that it's coherent and workable, and that its consequence is being able to experience very little distress over disagreements between how stuff actually behaves and how we think it ought to, while remaining quite open to the wonder of discovery.

"Observe" is not merely a mental model, as valid as any other. Either the universe depends on conscious input to function, or it does not.

Man, this is not philosophy class. This isn't about treating every perspective as a special snowflake. Some maps are more valid than others, in that they are more accurate engines of predictions.
posted by effugas at 5:45 AM on March 20, 2010

Pretty much, except that deriving overtly mathematical relationships is only part of what we do to make our maps. Quite a lot of the map is pre-verbal. For example, simply experiencing ourselves as beings separate from our environments is exercising our mapping ability; in my opinion this is one of the first map entries most of us make, and it establishes the notions of boundaries and the separability of objects. These are wired in so deep that we tend not to think about them much; we simply experience ourselves and the objects around us as separate and/or separable.

I'd go so far as to say the math flows from our pre-verbal perceptions. We throw an apple, it follows a Newtonian path, we eventually learn equations that describe the path of the apple.

Past that, there's really no relevance to what you're saying in this paragraph, to the rest of this discussion. Like, OK, humans have an me/not-me model. What does this have to do with photons?
posted by effugas at 5:49 AM on March 20, 2010

destructive interaction only yields a predestined polarity

How would positing that this is true of every destructive interaction that ever occurs anywhere break QM, provided that we don't concurrently posit the in-principle existence of a method for gaining information about that predestination?
posted by flabdablet at 5:50 AM on March 20, 2010

Like, OK, humans have an me/not-me model. What does this have to do with photons?

The me / not me model is readily generalized to a that / not that model, which we exhibit a natural tendency to apply to pretty much everything. It's implicit in the view that entangled photons A and B must somehow communicate with each other in order to achieve the observed correlations, and it's implicit in the view that this "action" is "spooky".
posted by flabdablet at 5:59 AM on March 20, 2010

The collapse indeed is supposed to happen instantaneously, no matter the distance between the particles.

This is, by definition and somewhat conveniently, an untestable hypothesis.

(In my opinion | it seems perfectly clear to me that | it's my personal view that | can we take this as read from now on please | ymmv), the reason it's untestable is that it's not about the objects that these experiments are set up to test. The collapse of the wave function is a modelling or mapping act; it's the point at which our state of knowledge about the system under observation loses a chunk of uncertainty. The system just does what it does; it wouldn't know a wave function from a cheese sandwich - wave functions are not necessarily isomorphic to the reality we use them to model. They're descriptions of what we know about that reality.

I'd be interested in hearing about any experimental result that makes such a position untenable.
posted by flabdablet at 6:08 AM on March 20, 2010

I'd go so far as to say the math flows from our pre-verbal perceptions. We throw an apple, it follows a Newtonian path, we eventually learn equations that describe the path of the apple.

Agreed.
posted by flabdablet at 6:09 AM on March 20, 2010

How would positing that this is true of every destructive interaction that ever occurs anywhere break QM, provided that we don't concurrently posit the in-principle existence of a method for gaining information about that predestination?

If it was proven true that "two bites at the apple" (two twinned particles) shared a predestination state, then the odds that a single "bite at the apple" (all other interactions) was predestined would be much higher. Essentially, there'd be a burden flip to prove the existence of two systems rather than one.

It's implicit in the view that entangled photons A and B must somehow communicate with each other in order to achieve the observed correlations, and it's implicit in the view that this "action" is "spooky".

flabdablet,

The particle is split.
A year passes.
One particle is measured.
Then the other is.
They oppose.

Now, one of two things have happened:

1) The first particle and the second particle shared a hidden but opposite variable from their birth, causing the opposite reading a year later.
2) The first particle communicated with the second that it was time to collapse its waveform in a particular manner that led to opposite readings

It's really only 1 or 2. Either the answer was determined on birth (my belief) or on "observation" (QM, with lots and lots of evidence). The answer was determined -- its just a question of T=0 or T=1yr.

We have no idea how #2 can be true, just that if it is, it must not transfer any information, but must move at instantaneous speeds. That's true even with your weird third variant where you just don't call the two particles, two particles, but change nothing else.

#2 is the standard dogma. #1 is what I still suspect.
posted by effugas at 6:14 AM on March 20, 2010

Either the universe depends on conscious input to function, or it does not.

Some relatively small bits of it do (e.g. this conversation); sometimes those small bits have butterfly-effect consequences (e.g. the bombing of Hiroshima). But most of it just does what it does.

I have previously engaged in friendly disputes with people who claim that the whole thing is conscious or is fundamentally made of pure consciousness or some other variation on that theme. That belief strikes me as likely to have been come to during a meditiative, drug, seizure or possibly stroke-induced brain state, where the mapper shuts down for a while and all perception of distinctions disappears.

It's a fascinating and exhilarating state to be in, and it truly does feel as if one has somehow become the entire universe and that it is manifestly conscious. But that feeling of universal consciousness is not compatible with the more feature-rich map that does include distinctions and/or the passage of time.

So while I'm sympathetic to the universal-consciousness view, I'm not convinced that it's capable of making any useful predictions about physical reality.

However, there's a baby that I don't think should be thrown out with that bathwater, and that's the idea that boundaries and distinctions are map features, not inherent in the territory, and often prove to be worth revising when we find ourselves faced with baffling map failure.
posted by flabdablet at 6:39 AM on March 20, 2010

What's wrong with the following re-wording, then, in your view?

A tangled photon pair emission event occurs.
A year passes.
A photon-absorption event occurs, yielding a spin measurement.
The experimenter applies some quite trivial mathematics to that measurement to make a prediction about the spin that will be measured when another photon-absorption event occurs at a specified time and place.
The other event occurs, yielding a spin measurement.
The experimenter's prediction is confirmed.

I seem to have mislaid both the spooky and the communication.
posted by flabdablet at 6:50 AM on March 20, 2010

So while I'm sympathetic to the universal-consciousness view, I'm not convinced that it's capable of making any useful predictions about physical reality.

Sorry, that's pretty sloppy wording. I'm trying to say that although I think I see where people who hold this view are coming from, and fully respect the experience underlying it, I think that consciousness is a local property, not a universal one. It seems to me that we don't function well in a state where we fail to distinguish part from whole, and that only in such a state does the statement "the universe is conscious" have any resemblance to truth.
posted by flabdablet at 6:58 AM on March 20, 2010

Really, all we know is that when we set up an experiment thus and so, we see correlations we would not expect to see in the light of present models. And all that means is that some of our assumptions about the entities and relationships involved are are unjustified.
What? If you're talking about quantum entanglement and stuff, then no, the models predict it perfectly. The experiments about those correlations were done to prove QM, not open up gaps in it.
Here's a testable prediction: if we do ever come to understand how to model these cases properly
Again. You're just wrong. We can model these things just fine using the standard model. The only "Gaps" in quantum mechanics are things like we don't understand where gravity comes from and things like the Hierarchy problem, where experimental results are different then what theory suggests but if you just plug those constants in then you get a working system that predicts everything perfectly.
and I have never attempted to suggest that anybody else is obligated to think the same way I do. I am simply pointing out that this view is available
That's not true at all, here's what you said at the start of the thread:
This seems perfectly clear to me, which is why I find myself getting a little exasperated when well-regarded thinkers like Stephen Hawking and Paul Davies go on about what "breathes fire into the equations" as if mathematics were in fact somehow prior to existence. Apparently there's money to be made in deliberately mistaking the map for the territory and torturing the idea of "existence" until it gives up and means anything you want it to.
See, you said they were deliberately getting it wrong. In order for something to be deliberate, it has to be done knowingly, which means that you apparently thought that these people do know of your idea, and know it's right, and are either lying about it or being "exasperatingly" pig-headed by not agreeing with you. Like it was so obviously true

the problem with your whole argument is that you're taking this "map/territory" thing as a given and reasoning from there.
That's not a bug, that's a feature.

WTF? It's not a feature, it means your entire thing is meaningless unless people already agree with you! If you agree that you're taking your view as a given, then all the stuff you've typed here is totally meaningless! Why are you even doing it? It's not any different then like a religious belief.
posted by delmoi at 6:26 PM on March 20, 2010

Was that exasperation?
posted by flabdablet at 6:47 PM on March 20, 2010

The only "Gaps" in quantum mechanics are things like we don't understand where gravity comes from and things like the Hierarchy problem, where experimental results are different then what theory suggests but if you just plug those constants in then you get a working system that predicts everything perfectly.

Have we even seen a mildly compelling explanation for how "spooky action at a distance" might work? Because that's a pretty big gap if not.
posted by effugas at 9:03 PM on March 20, 2010

The type of correlations that were given that derisive name by Einstein are completely described by nonrelativistic quantum mechanics. Two particles in a two-particle state give correlated results when you measure them both; two particles in independent one-particle states aren't correlated. If your model of a measurement is that some physical "wavefunction" is instantaneously altered by a "classical" device, then you have a problem with relativity --- but those are assumptions that are useful in nonrelativistic quantum mechanics. If instead you consider your system as being two particles and two detectors, and call the "measurement" the comparison of the two detectors, then you have lots of classical, relativistic information flowing between the detectors as part of the comparison.

My personal opinion is that there is some symmetry that keeps the entangled particles correlated even when their separation is spacelike. I haven't seen a good paper on this idea.

Was there a real connection between this question and the original topic of infinity, or is there a connection only in that both involve saying "whoa" in a Keanu Reeves voice?
posted by fantabulous timewaster at 11:29 PM on March 20, 2010

My personal opinion is that there is some symmetry that keeps the entangled particles correlated even when their separation is spacelike.

If I understand effugas correctly, that's near enough to his option #1.

I seem to recall reading somewhere that somebody published a proof that busted hidden-variable models. Anybody care to have a crack at explaining current thinking on that? Also, how is a symmetry different from a hidden variable?

Was there a real connection between this question and the original topic of infinity, or is there a connection only in that both involve saying "whoa" in a Keanu Reeves voice?

Ask oddman and Chocolate Pickle. I'm only sticking around making a spectacle of my ignorance because I got egged on by Rusty.

posted by flabdablet at 1:04 AM on March 21, 2010

fantabulous--

If instead you consider your system as being two particles and two detectors, and call the "measurement" the comparison of the two detectors, then you have lots of classical, relativistic information flowing between the detectors as part of the comparison.

See, this is why I hate the words "observe" and "measure". Me putting the output of two spatially distributed detectors into a spreadsheet, does not represent a flow of information between two detectors. Excel does not define the universe! That the terminology we happen to have fallen into, happens to allow us to describe effects wayyyyyyyyyyy down the causality chain as if they were causes and not distant effects, is an unscientific tragedy.

My personal opinion is that there is some symmetry that keeps the entangled particles correlated even when their separation is spacelike. I haven't seen a good paper on this idea.

I agree with this, except such a symmetry is expressly denied by QM. Functions are supposed to be in a superposition of states until destructive interaction forces them to choose/collapse into one state or another. A symmetry would effectively mean that the output of that collapse was predestined. That would be huge, and hugely problematic.

Was there a real connection between this question and the original topic of infinity, or is there a connection only in that both involve saying "whoa" in a Keanu Reeves voice?

We were discussing the question of whether math would exist without humans to have created it. I argued that the universe operates according to relationships that can be summarized mathematically, and that these relationships were there before humans arrived and will continue after we are all gone. We then got onto a pet peeve, which is the implication that the math of the universe somehow has human consciousness as a variable. And here we are.

flabdablet--

I seem to recall reading somewhere that somebody published a proof that busted hidden-variable models. Anybody care to have a crack at explaining current thinking on that? Also, how is a symmetry different from a hidden variable?

Bell's inequality is supposed to explain this. I don't find it wildly compelling, but I don't have the chops to really complain about it.

What seems to be happening is that the math is more than good enough to explain experimental data, and even to execute tremendous feats of engineering. The little thing about "we have no idea how this could possibly be happening in the real world" sort of pales.
posted by effugas at 2:29 AM on March 21, 2010

What's wrong with the following re-wording, then, in your view?

A tangled photon pair emission event occurs.
A year passes.
A photon-absorption event occurs, yielding a spin measurement.

There. It doesn't just yield a spin measurement. It's supposed to generate a spin -- at the location where the first photon was measured, at the time the measurement occurred.

The experimenter applies some quite trivial mathematics to that measurement to make a prediction about the spin that will be measured when another photon-absorption event occurs at a specified time and place.
The other event occurs, yielding a spin measurement.
The experimenter's prediction is confirmed.

How did the second photon "know" to have an opposite spin? It's as if I created two shoes, threw one in black paint, "predicted" that the other, a thousand miles away, would be coated in white paint...

...and was right!

Basically, either something happens at superluminal speeds to "paint the other photon", or the two photons were in sync. QM really doesn't want the latter, but it's just a big shrug for how the former might work.
posted by effugas at 2:40 AM on March 21, 2010

It doesn't just yield a spin measurement. It's supposed to generate a spin -- at the location where the first photon was measured, at the time the measurement occurred.

What's the difference?

It's as if I created two shoes, threw one in black paint, "predicted" that the other, a thousand miles away, would be coated in white paint...

Why should anything you know about shoes apply to photons?
posted by flabdablet at 3:51 AM on March 21, 2010

In other words: what if photons don't exist in anything like the way shoes do? What if they are nothing more than a mathematical convenience for keeping track of causal connections between photon-emission events and photon-absorption events?
posted by flabdablet at 3:54 AM on March 21, 2010

flabdablet,

Eh, photons are not shoes, but they are discrete objects that move through space. It's not like there's just emission and absorption; something is moved through that fiber optic cable, and takes a discrete amount of time to execute that move.

Photons and shoes are allowed to behave as differently as they like, as long as there's some proposed mechanism for that difference. What is the proposed mechanism for spooky action at a distance? As far as I can tell, there in none. There's just an out of place pile of hand wavey pseudoscience inviting woo. It's not even like gravity, where gravitons are proposed. The wave function just collapses, wherever it happens to be in the universe. It just occurs, instantaneously, regardless of distance.

In a universe where the speed of light is such a defining barrier, what the hell is that?
posted by effugas at 4:39 AM on March 21, 2010

photons are not shoes, but they are discrete objects that move through space.

Interesting assumption. How would you test it?
posted by flabdablet at 5:43 AM on March 21, 2010

In particular: what properties are shared by all discrete objects?
posted by flabdablet at 5:44 AM on March 21, 2010

I agree with this, except such a symmetry is expressly denied by QM.
Since you clearly don't know anything about quantum mechanics, why are you talking about it?
posted by delmoi at 10:21 AM on March 21, 2010

Since you clearly don't know anything about quantum mechanics, why are you talking about it?

Eh. I have to have a base level of knowlege of QM professionally, since quantum crypto is one of the things in our toolkit. And I think I've been pretty open about the limits of my understanding.

Symmetry is cool and all, but it's another hidden variable proposal. I LIKE hidden variable proposals. But:

1) The experimental evidence around Bell Inequalities continue argue against hidden variable theories in general, and the number of "loopholes" (possible problems with experiment) are running out
2) There really isn't any proposed mechanism for how wave function collapse can be coordinated across spacetime at superluminal speeds (apparently, at a minimum of 10,000c)

Remember how we're having a huge amount of trouble reconciling QM and Relativity? This is one of the big areas why.

We basically have a situation where we have a really good mathematical framework with strongly experimentally validated results at one scale, that utterly violates our basic understanding of the universe (speed of light, locality, reality) at others.

I'm OK with my suspicions continuing to be on the wrong side of experimental data. I could be wrong here.

I'm just not OK with consciousness woo, which I think is inserted into the discussion (by QM people, even) with "observe" and "measure". I do think I know enough to say that that needs to stop. Do you disagree?
posted by effugas at 4:48 PM on March 21, 2010

I agree with this, except such a symmetry is expressly denied by QM.
... Maybe. Measurements that show a violation of Bell's inequality are usually interpreted as ruling out "local hidden variables." Whether a global symmetry is subject to the same constraints is a question I don't know the answer to (not for want of trying).
Also, how is a symmetry different from a hidden variable?
That's a question with a long answer, maybe for another day.
See, this is why I hate the words "observe" and "measure". Me putting the output of two spatially distributed detectors into a spreadsheet, does not represent a flow of information between two detectors.
You're right, it doesn't. It's when you compare the results of the detectors that you have information flowing between them.

Note that this is a problem that's usually treated with nonrelativistic quantum mechanics, so it shouldn't be surprising if a solution comes out inconsistent with relativity. Note also that all of these theories are just "maps" (an excellent phrasing, flabdablet) and the universe is under no obligation to follow any of them exactly.

As an aside, I think of a "measurement" as a purely mechanical process, and any consciousness that inserts itself will end up with hurt feelings.
posted by fantabulous timewaster at 5:40 PM on March 21, 2010

Also, how is a symmetry different from a hidden variable?

That's a question with a long answer, maybe for another day.

I've been thinking about situations where probability distributions are inverted, i.e. you have 100% to 0% in one direction, and 0% to 100% in the other. That would, over a large number of samples, agree with test data -- it's something of a midpoint. Is this something akin to what you're thinking?

You're right, it doesn't. It's when you compare the results of the detectors that you have information flowing between them.

No, comparing the values of A1 and B1 in Excel does nothing to cause information transfer between subatomic particles.

I know this is the standard speak in QM. And, on some level, it's accurate -- the bits need to arrive at the same location, via classical information transfer, to be compared. But whether or not the bits are "compared", the events still happen.

It's the difference between "you can't compare these bits, without meeting these constraints" and "the comparison of bits creates the bits".
posted by effugas at 1:41 PM on March 22, 2010

I've been thinking about situations where probability distributions are inverted, i.e. you have 100% to 0% in one direction, and 0% to 100% in the other. That would, over a large number of samples, agree with test data -- it's something of a midpoint. Is this something akin to what you're thinking?
No, I don't think it is. Here's what I was thinking.

Just to be clear, I'm thinking about an entangled-polarization photon experiment. A photon can have one of two polarizations in any of three bases. (That's "base-ees," the plural of "basis.") Usually the bases have labels like horizontal/vertical, rising/falling diagonal, and left/right circular. It's possible to produce pairs of photons where the relationship between the two polarizations is fixed (for instance, "always different"), but the actual polarizations aren't known. If both photons are analyzed in the same basis, the relationship holds; if they are analyzed in orthogonal bases, there is no relationship at all. (And I don't mean "analyzed" in the sense of "examined and pondered by a thinking person"; I mean "interacts with an apparatus in a way that depends on its polarization.") If you think there is something magical about "measurements" that irrevocably chooses a photon's spin from its probability distribution, and you think that causally linked events must be separated by a timelike interval in spacetime, then the orientation where you can generate absolute correlations is spooky.

One promising idea for describing this process is that each photon has its own "hidden" polarization, or some set of local parameters that determine the way it interacts with its analyzer. That model correctly accounts for the experiment as I described it above: if both photons are analyzed in the same basis, the relationship holds; if they are analyzed in different bases, there is no relationship at all. But there's another experiment you can do, where the analyzer orientations are neither identical nor orthogonal: there is a continuum of orientations between horizontal and diagonal. When the analyzer orientations are "mixed" like this, the polarization measurements are only partly correlated. Bell showed that any theory where the polarizations are somehow hidden and local always predicts less correlation here, with mixed bases, than does quantum mechanics. And of course there have been thousands of Bell-inequality experiments with entangled photons, electrons, neutrons, atoms, etc. All so far give correlations that are consistent with the quantum-mechanical prediction and inconsistent with hidden-variable theories of varying complexity. So that's what I meant by "hidden variables."

Now, what does it mean to say that a system "has a symmetry"? That means there is some transformation you can apply to the system that leaves it unchanged. Symmetry is an important topic in modern physics and there are lots of examples of symmetric systems. Translational symmetry means that if I do an experiment sitting over here, I should get the same result as if I do it sitting over there. Relativity (sometimes called Lorentz symmetry) says that if I do an experiment while I'm stationary, I should get the same result as if I do it while I'm moving at a constant velocity. In quantum mechanics, all fundamental particles of a given type are identical, and so if I swap two identical particles I should get the same outcome from any measurement I'm doing. This is called exchange symmetry. But in the mathematical treatment of quantum mechanics, there is an intermediate quantity (the wavefunction) that gets squared in the process of making a prediction. There are two ways that a transformation can leave the square of a number unchanged: the transformation can leave the number itself unchanged, or the transformation can flip the number's sign. Which type of exchange symmetry the wavefunction has turns out to be what distinguishes matter particles (fermions) from force carriers (bosons), although in a discussion where we're wondering whether math exists independently from the mathematicians, this is assuming an awful lot of things from a theory.

Now as far as anybody knows, exchange symmetry is exact. Even though we've never met, my electrons are all antisymmetric with your electrons: if I traded one of mine for one of yours, the sign of our collective wavefunction would change. In relativistic quantum mechanics there is not a "mechanism" that enforces this symmetry, in the sense I think you mean above. But there is also no mechanism that breaks the symmetry. If your model of a system begins with antisymmetric electrons, the part of the model where I have my electrons will evolve in exactly the same way as the part of the model where I have your electrons, and the model is still antisymmetric under exchange at the end.

That's the sort of thing that I think is happening in entanglement experiments. It's not correct to say that the measurements are independent: the detectors are also quantum-mechanical and have correlated, non-local wavefunctions beginning before the experiment. Even if the "measurement" events are spacelike separated, the detectors remain correlated and aren't really independent. The apparent paradox comes from only considering part of the system.

I warned you that question had a long answer.
posted by fantabulous timewaster at 7:40 PM on March 22, 2010

Nice clear one, though. Well done, and thanks.
posted by flabdablet at 7:56 PM on March 22, 2010

The apparent paradox comes from only considering part of the system.

...which is a perfectly reasonable thing to do, if the intuitive parts of your map tell you that photons are in some useful way analogous to shoes :-)
posted by flabdablet at 8:33 PM on March 22, 2010

If not the Quantum timecube master; can it matter what anyone is or is not "ok" with... when the modern science and observations would seem to run counter, or at least parallel, or even cross-counter to the stated opinions.

I will disagree with that statement... Because all of our maps should have been updated to that one from the west wing... the upside-down one.

In the same way that "forest succession/evolution" is evolution of a greatly different scale and time frame from the evolution of a particular tree species...
Reality and Life will continue on unabated. Idea formation evolving. Universe self-conception doing likewise. Your opinion will not collapse the waveform that comprises other individual ideas thoughts and processes.

Those ideas, intentions and words will continue to be able to have measurable effects upon the material world. Making the very LAWS which operate the physical universe fundamentally different than if there were no such thing as "an observer" or, to add a layer, "an actor". Tell me Einsteins theories and ideas didn't create a massive explosion. Tell me Dr. Kings words didn't change physical realities for physical people, that his words didn't cause the movement of physical matter... (now where "are" those ideas, words and theories?)

In particular: what properties are shared by all discrete objects?

There are no discrete "objects" when you factor in the totality of our scientific explorations. All matter in the universe comes from the fires of the same bang... all particles and sub-particles are entangled in their dance from one to the next instant.

To show where the disconnection between any objects 'starts'... is nothing, when to consider that the atoms are themselves but abstractions of a lower layer; a lower layer we are just learning to measure, not to mention "see" and understand, and add as factors in our simple day to day conceptions of the Universe.

No part of our reality has been "injected" into our universe from an external source, or at a time "after" our Singularity; all "things" share one thing. They share a creation point. The Singularity.

We can stipulate that no energy has been added to our system (universe)... it all DID enter the system through the "breath of fire" of a singularity. All matter in all the Universe owes it's heritage to energy which entered our system at one particular event. All energy is tied together in it's point of origin (as the many quantum condensate experiments show); the information it subsequently forms shares an inherited "gene" expression through its connection.

Photons are not shoes... photons are working on a scale that shoes could only dream of, they also operate on 'branes between states, and connect the layers of our "conscious" reality that are "time" and "space", one day, after we have met other lifeforms, and are not so "sure" of all that is arrayed before us. Our tools for measuring and observing are so far just the first most basic tools. One day we will recognize this quantum mass that is reality as a fundamental life-form in our Universe. As integral to human existence as the cyanobacteria were in influencing the atmosphere of the planet Earth.

Shoes are an evolution of the sub-matter that was forged in the Singularity; just as stars, in their life cycle are an embodiment of the evolution of particle forms.

What Biologists call a "founder effect" is at play in the cosmic systems we observe... stars (& their planetary systems), Galaxies, Clusters, Super-Clusters... Stars are the same around the Universe due to the First Fire, and the particles that were forged from this; one example of this being, we have Particles, rather than Anti-Particles, that more anti-particles were destroyed had a cascade effect upon our Universe's evolution, and today we see Particles... not the opposite. The end of Stellar Evolution being the dispersal of vast quantities of "elements" (masses and flavors of "particles"... which themselves evolve into compounds... which eventually become me, eating ice cream.

Another example is the relative abundance of various "Elements" the abundance of elements in the Universe are intimately tied to the distribution of the Sub-Elemental... this distribution is tied to the physicality of the Universe, and the Information that Composes it. And ultimately we are constrained by the distribution of Quantum Information/Energy Potential.

But what is the speed of light stopping from happening? Nothing. It is just one of the 'physical' laws that applies to its own tiny section of Universal Evolution, at one particular scale of observation; that which applies to one group of evolved, but connected scales of observation of the universe. It applies to traveling... Moving Physically through space; not to transmission of Information between the totality of a Universe of an Entangled origin.

As a question, I would ask, do you think there is no difference between a pile of atoms, or even a molecule... and Life.

I would submit that whenever it was that "Life" first "entered the state which is opposite of 'not existing'" at that "branch in potentiality", where Life itself could well have failed immediately after beginning, but due to an automatic, or even basally intentional action; Life survived.
This world is different. It is not the same as a passive universe. At the very least it adds on one more layer of complexity... by no means can "life" breathe fire into the universe... rather the universe breathed life into...Life. Literally, each particle, and each point in time on the chart of times arrow that we call from our vantage point, 'the past' (as in cyanobacteria seeding the early atmosphere of Earth) you have examples of things which have been in the universe indefinitely.

Think of the NIToI* as the manner in which the Universe does a "check-sum" or whatever you want to call the process by which Physical&Quantum Laws are harmonized, and operational order is being maintained... it verifies the Data of all quantum states at every instant in the entire universe... Imagine the "steady beeping of a life support monitor"...
"all physical laws still working?- Keep continuing physical laws operating... Checking States... Measuring Size. If size is over limit begin compaction. If size = Smaller than limit... continue expansion. Particles operating on Wave/particle manner. Observers Present. Sol System>begin operation quantum state/non-locality. "
Without this "Near instant communication" between "spatially distant" "elements" of the Universe, it wouldn't exist. It is how "Universal Physical laws" are enforced... and it happens with things that we are not creating in Labs also.

Did the Universe "change" with the arrival of the first "observers"? Or did the Universe change to accommodate the first "observers" before they could evolve.
*Near Instantaneous Transmission of Information.

tl;dr... It is all in your head.

but your "head" is nonlocal.
I do greatly enjoy when you mention maps and territories, and discuss how, and how often we confuse them.
However, Maps and territories are themselves intensely human conceptions... conceptions which are themselves the subject of debate, and discourse. Let's call this the "political scale" of reality.

For this particular line of thinking...
What we should instead do is see the globe as it is from outside your currently limited frame of reference... suddenly it looks more like this... O ...a "whole globe." one body, with so very many parts... not some artificially sliced up hunk of colors and "allegiances" that are nations, which make up what we will term "political scale".
One total unit... because we are still measuring and learning about the universe on a scale where we see and can witness daily, the distinctions between "this shoe" and that "shoe" {this is political scale}... we certainly do not operate day to day thinking about the particles, and sub-particles... and pure quantum potential bubbling up from the lowest level..."shoes" themselves the product of an intensely active bundle of matter doesn't mean that we can listen so attentively to skeptics of a connected conception of the universe, and dismiss what we see when we have the tools to measure at the scale of the {whole globe}

The very hot and the very cold... connected.
"Shadows live in a simple world. They glide effortlessly across any sort of surface, oblivious to the higher dimension of space in which 3-D bodies move, collide and sometimes block the paths of rays of light.

Shadows have no idea how important that third dimension is, and how objects in it endow those very shadows with their quasi-physical existence. Indeed, the laws of shadow physics all depend on the third dimension’s presence. And just as the clueless inhabitants of the shadow world require an extra dimension to explain how they exist and interact, reality for humans may also depend on an invisible dimension or dimensions unknown."
...
"String math describing such ripples stems from an idea called the holographic principle, used by string theorists to describe certain kinds of black holes. A black hole’s entropy depends on its surface area — as though all the information in its three-dimensional interior is stored on its two-dimensional surface. (The “holographic” label is an allusion to ordinary holograms, where 3-D images are coated on a 2-D surface, like an emblem on a credit card.) The holographic principle has value because in some cases the math for a complex 3-D system (neglecting time) can be too hard to solve, but the equivalent 4-D math provides simpler equations to describe the same phenomena."

The very cold. Condensates.
How Nist entangled two mechanical oscillators...

Or if you would like a maps/territories/car analogy... It's like thinking you can use a hand drawn map of ancient Greece to navigate the modern road-ways thereof whilst driving a car... The driver would be so "in" their 'moments reality and physicality', that it is hard to "see" the rest of the picture is plainly incomplete. - yet technically it "is" a "map" to that "territory".
posted by infinite intimation at 8:58 PM on March 22, 2010

It applies to traveling... Moving Physically through space; not to transmission of Information between the totality of a Universe of an Entangled origin.

Could you please clarify the distinction you appear to be making between physical movement through space and information transmission? In particular, when you speak of information transmission, are you speaking of (1) establishing a causal pathway between A to B, or merely (2) observing that causal relationships operate near B in pretty much the same way they do near A because everything near B shares a common cause with everything near A?

If (1) and you're predicting that one day there might be a reasonable business case for offshoring all the iTunes Music Store servers to a cheap hosting provider in the Lesser Magellanic Cloud, I think you're probably wrong.

But if all you mean is that (2) I could take my iPod with me to the LMC and it would still work, we have no real beef*.

*Is beef real?
posted by flabdablet at 9:44 PM on March 22, 2010

I think we are firmly on the same page.
But No Beef...Just a little MadCow!
And Only if you find and recruit 1000 people who agree to switch to the Upside-down map instead of the normal one...

Except I have to add that I didn't really mean either of those particular scenarios, but they do help me explain what I was thinking... more that the "digital bits" you describe are being stored "in" containers which operate on the same old scale as we go about day to day...

I am finding that there is a difference between the bits that we use to reproduce a sound recorded ten minutes ago, which we have carefully ordered, and labeled, and captured, and prepared, and then stored ... v.s some energy which has not yet formed or been previously measured, "captured", or otherwise manipulated by me... The bits of information stored on the ipod are in a state that is evolved beyond the state of uncertainty. The particular quanta of energy that is "storing" a "copy" of the song has already bubbled up into 'our' scale... but the song is not dependent on that "particular" version... in fact other people have the "same" song... in different storage... I want to think about energy, Universal Information layer, BEFORE anything can be "encoded" by a user. The natural state of the energy, before it was captured, and encoded with a song... contains information (location, spin, other data that we don't/can't understand)

(I can know with 99.9999999999% sure certainty that that I-pod will play (for ex.) Chumbawumba's Tubthumping... because that is what the song is that is on that persons physical ipod... We can know this. This is essentially a knowable fact that is observable in standard classical physics... The contents are predictable, and must either live up to my expectation of being what is on there... or... In a "breakdown of physics" the information could NOT be what I expect to be on there... and this is where I was getting to the "universe using NIToI as 'checksum' method.

If there were to be a sudden breakdown of the laws of physics... there would be a serious problem, or at least an 'issue' with the 'laws' of the Universe... this would have a cascade effect. Physical Laws that we define being the interaction of energy, interactions of various colors, flavors and spins.
(It is hard to imagine a universe where a "local" breakdown in the laws of physics does not erupt to effect not only all surrounding space... but also the whole systems' structure.

What I am getting at is that we really don't have a device that uses the information I am thinking of as the backbone, or framework of the 'all'... Inherent in consciously "storing data" is knowing the very factual matter of what "is stored" - the information which is the totality of the Universe, which we cannot, or have not yet measured, on a scale below what we can comprehend and/or predict/ experience/ daily operate upon/ become accustomed to the laws operating as constraints on the contents thereof.

The "fluency" we all have as humans from our daily operating on the particular scale that we operate on... it is a "fluency" that we cannot (it seems) just "intuitively" jump into (or out of) understanding and "thinking in".
We are biased by what we live every-day into thinking that 'our' scale is the "true" scale... this is not necessarily true.
posted by infinite intimation at 11:11 PM on March 22, 2010

This thread has gone completely off the rails.
posted by delmoi at 11:12 PM on March 22, 2010

Well, you can't talk about infinity forever.
posted by fantabulous timewaster at 11:30 PM on March 22, 2010 [1 favorite]

You won't get anywhere trying to run a train of thought through lakes and up mountains with thread. What you need is superstring.
posted by flabdablet at 12:02 AM on March 23, 2010

Sure you can. Only thing is, you always just end up right back where you started everytime. :|
posted by infinite intimation at 12:22 AM on March 23, 2010

And there might be two of you.
posted by flabdablet at 12:29 AM on March 23, 2010

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