Friday, November 23, 2007

Friday Puzzler

For the theoretically minded I offer this nugget: The Universe is explained by a geometry with 248 dimensions.

You can read it if you wish, it is mostly Greek to me. But this quote seems a little puzzling:
ONE of the mysteries of the universe is why it should speak the language of mathematics.

Isn't that like being struck by the improbability that Lou Gehrig would succumb to Lou Gehrig's disease?

43 Comments:

Blogger David said...

Isn't that like being struck by the improbability that Lou Gehrig would succumb to Lou Gehrig's disease?

Not completely, I don't think. There are two questions: first, why should the universe be predictable and, second, how come we can start with math and then make predictions about the universe? The first is necesary for math to work at all. The second might mean that math is not just descriptive, but in some sense True.

November 23, 2007 9:19 AM  
Blogger Susan's Husband said...

Not at all. Math "works" regardless of physical law. An unpredictable universe doesn't affect math as math at all, it only impacts the usefulness of math.

Second, we can make predictions about the universe using certain types of math because those types of math were selected on the basis of being useful for such predictions (mathematical evolution!). That's what the Gehrig's quote is about -- having selected certain mathematics for their utility, you are surprised that those mathematics are useful.

November 23, 2007 6:15 PM  
Blogger Harry Eagar said...

Hmmm. If the universe weren't in some broad sense predictable, we couldn't function.

If we ran from mice but not from lions, we would not reproduce very well, even if, sometimes, the mouse is more of a threat.

To me, the profounder question might be, what is the ratio of predictability to unpredictability that allows something like life to happen?

November 23, 2007 11:42 PM  
Blogger Peter Burnet said...

SH:

That is mind-boggling. It sounds like you are coming close to saying that the predictability of the universe is a function of the math we invented to explain it-- that the predicability is an invention, not a discovery. Wouldn't that turn biology on its head? Surely it isn't the usefulness of the math that is striking here.

I think that the fallacy in Duck's analogy is that he is confounding happenstance with its opposite. We may be moved, but we are not really "struck by the improbability" of anyone being felled by a disease or winning the loto, however we would be if we learned such had been foretold long ago, or even better, seeded in the big bang.

It is important to realize that the modern materialist/scientific mind often has a big psychological hang-up with mega-order. Just as religious folks are prone to gravitate instinctively towards evidence of purpose and predictability, sometimes absurdly, so the modern secularist gets nervous when they suggest themselves at a macro-level, which is funny because they are always boasting about proving it at the micro-level. Too many historical dragon-slaying stories about valiantly turning the "old certainties" on their heads, not to mention a bit of a threat to the buffered individual judging everything for himself. D'Sousa (actually a pretty good book)has some fascinating quotes from prominent scientists admitting how upset they were about the big bang theory and how hard they resisted it as a matter of will, not science. Too close to You Know What. This is quite different from what impelled most people for most of history and explains why we are so incredibly clumsy at analysing or pronouncing on, for example, the Middle Ages, which were defined in large part by a drive to build order and safety out of chaos. Trying to tell a modern secularist that there is an overarching order to the universe is a bit like trying to tell a modern poet that if it doesn't rhyme, it ain't poetry. Tends to put them off their biscuits.

I have absolutely no capacity to judge the science and math behind the anthropic principle, but I think I can see that the theoretical ripostes are pretty lame. "It just is" is the face of astronomical odds or wild conjectures about an infinite number of unseen parallel universes are clearly "Hail Mary" passes, no?

November 24, 2007 2:39 AM  
Blogger Duck said...

One problem with saying that math is the language of the universe is that it isn't, not totally. Otherwise we would have a Theory of Everything, which we still don't.

Another problem, which is demonstrated by this article, is that potential explanations are built upon mathematical structures that are immensely complex and convoluted. In that sense, I wonder if there is any possible universe, orderly or not, that couldn't be explained by some mathematical formula at some level. Math is like a builder's kit of tools, one that includes such neat gadgets as infinity and irrational numbers. What can't you explain with it?

I was struck, reading this article, of how cosmology is starting to resemble theology, in the manner of slapping together these contrived edifices of logic. The theologian has his own toolbox of gadgets, in the form of Platonic Idealism, or "Realism" as they like to call it, which holds ideal objects of immaterial perfection, which means they can pull them out of the air at will. Using 248 dimensions to explain the universe is the equivalent of giving a crafty rhetoritician an unlimited budget of dubious presuppositions with which to craft a proof that white is black, or that dogs are fish, or as one former president and master rhetoritician proved, having sex and not having sex are the same thing.

November 24, 2007 7:17 AM  
Blogger Peter Burnet said...

Yes, on second thought, I may be underestimating the danger of those wild and crazy mathematicians. Talk about a case for forbidden knowledge!

November 24, 2007 8:07 AM  
Blogger Harry Eagar said...

My physics adviser recommends a book called 'Mathematics: Queen of the Sciences' by his math teacher, Eric Temple Bell.

Unfortunately, I've never found a copy, but the message I was supposed to carry away was that there are so many maths, it is not surprising that some of them match observation; or that some created without any evident connection to observation later turn out to be useful in descibing an observation; while many others don't -- or haven't so far.

Take infinities. As some scholasticus should have said but didn't, nature abhors an infinity.

As I understand it, one of the big jobs of theoretical physicists is to cancel out all the infinities.

So is the math of infinities a descriptor or a non-descriptor of 'reality'?

November 24, 2007 9:03 AM  
Blogger Harry Eagar said...

My physics adviser recommends a book called 'Mathematics: Queen of the Sciences' by his math teacher, Eric Temple Bell.

Unfortunately, I've never found a copy, but the message I was supposed to carry away was that there are so many maths, it is not surprising that some of them match observation; or that some created without any evident connection to observation later turn out to be useful in descibing an observation; while many others don't -- or haven't so far.

Take infinities. As some scholasticus should have said but didn't, nature abhors an infinity.

As I understand it, one of the big jobs of theoretical physicists is to cancel out all the infinities.

So is the math of infinities a descriptor or a non-descriptor of 'reality'?

November 24, 2007 9:03 AM  
Blogger Susan's Husband said...

Mr. Burnet;

It is as Mr. Eager's book says, there are so many mathematics that it would be surprising if none of them corresponded to reality, not that some of them do.

As for mathematic vs. predictability, a mathematical system exists completely independently to any physical reality — there is no necessary relationship. What I am saying is that because the universe is predictable, we humans were able to invent mathematics that describe it out of the unending multitude of possible mathematics.

One must be careful to distinguish between models and mathematics. A model is an isomorphism from a mathematical system to physical reality. The inverse square law, for instance, is a bit of mathematics that exists regardless of how gravity works. Newton created a model (his Law of Gravitation) which claims that the inverse square law is isomorphic to the behavior of gravity. Even that isn't "true" or "false", but more or less accurate.

As for pure math, asking if it's "true" is exactly like asking if the English language is "true".

November 24, 2007 12:03 PM  
Blogger Susan's Husband said...

P.S. Note that the author original article cited doesn't understand the subject. He claims E(8) has 248 dimensions, and later 248 points, neither of which is true. It is more like a set with 248 elements. It is considered special because it's the equivalent of a prime number. You can have sets of that type of any number of elements, but most can be described as the product of other, simpler sets. E(8) is one that cannot, there is no combination of simpler sets that can be combined to yield it.

November 24, 2007 12:07 PM  
Blogger Mike Beversluis said...

Not to be overly dismissive, but the surfer-dude theory is moving through the popular media instead of through the normal journal channels. There's an romantic aspect of the lone-outsider living in his yurt and confounding the main community which seems to be a large reason for its popularity. But it's not a good sign for a scientific idea, and time will tell.

As to mathematics, I'm reminded of a Niels Bohr quote: "It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature."

Maybe SH can correct me, but it seems like the task of mathematics is to find out what is thinkable. Math is a formal language, but to decide the truth of any formal statement, we have to assign it meaning. So, at the foundation of mathematics are several axioms whose meaning and truth we assume is necessary. To deny them is thought to be unthinkable.

In that sense, the surprising thing is that the universe seems to be thinkable at all. It may be deeply counter-intuitive, but common sense and intuition depend a lot on where you hang out. Either way, you are able to reduce enormously the amount of knowledge about the the position and motion of nearly everything we see into a few extremely concise rules, and given the number of particles and number of forces they act and react on each other with, any theory is already such a model of brevity that physicists must be very witty.

Duck commented on the theory of everything's unfashionable lateness, but I'm not sure what sets that timetable. Y2K? The universe is big and hbar is small; it's technically difficult to make measurements on those length scales and with the associated energies.

Anyway, yes, with three exponentials you can curve fit an elephant, but we abhors an adhoc theory. Theories are like fortune cookies, in that you can always add "in bed" to them to make them funny, but people have sought to minimize their axioms. What generally determines the choice of a theory is that it needs to account for our observations with as few free parameters as possible, which leads to Occam's razor and a desire for maximum symmetry and elegance.

Furthermore, it's good if the theory can predict outcomes which we didn't build into the model.

For example, the wave-particle debate about light went on for a long time. Newton had enough suction to bury Huygens's competing wave theory, and through the early 19th century, light was thought of as rays. Fresnel was still working on wave diffraction theory, when one of his critics (Poisson) pointed out an absurd result: If you place a metal sphere into a collimated beam of light, wave theory predicts that there will be a bright spot in the center of the disc's shadow. So another guy went out and checked, and sure enough, there was the spot (ref)

Whether or not you find that spot surprising or not depends, I guess, on how much of the wave theory Kool-aide you have drunk. The surprising thing about Lou Gehrig is that there was a Lou Gehrig at all.

*detail: Physics has since moved on from the wave theory of light, and now photons are modeled as "wave-packets" that are particle like excitations formed out of a sum of waves. The fact that waves and particles are antipodal ideas is apparently a problem for us, but not for the photons or the math that models them.

November 24, 2007 2:37 PM  
Blogger Susan's Husband said...

Math is a formal language, but to decide the truth of any formal statement, we have to assign it meaning. So, at the foundation of mathematics are several axioms whose meaning and truth we assume is necessary.

No! Mathematics is a ever growing collection of formal statements and systems of formal statements. Such statements only have a truth value in the context of their "home" system / language. Nothing there has any necessary relationship (nor needs one) with physical reality.

Just for example, consider the Law of Commutation. This is based on the fact that if you have 4 rows of 5 objects, that's the exact same physical arrangement as 5 rows of 4 objects. So obviously 4*5 = 5*4. Except when it doesn't in a wide variety of mathematical systems. So even that level of physical correspondence is easily discarded.

As for assigning truth, there are mathematics that are simply rules for manipulating strings which we interpret as mathematical statements. But the "truth" of any such statement is determined solely by whether it can be generated from one of a fixed set of starting strings via the rules. No correspondence to any material reality is involved.

November 24, 2007 3:44 PM  
Blogger Harry Eagar said...

Lubos Motl has some VERY harsh things to sat about Lisi.

I was amused by Lisi's CV, in which he brags that his classes at Maui Community College get the highest student rating. Not, perhaps, the toughest critics.

He has, so far, had one real world consulting job, and I know a little about that one -- a superduper vacuum cleaner invented by a local drywall installer. I have written about that a couple times.

November 25, 2007 11:10 AM  
Blogger Mike Beversluis said...

Harry - yeah, I saw that too.

SH: Okay, but where do you get the rules of logic from? Or set theory? And how can you have a language without them? When I think of the fundamental axioms of math, what comes to my mind are the concepts of element, sets, subsets, and the ability to compare or order them.

November 25, 2007 2:40 PM  
Blogger Susan's Husband said...

The rules of logic? They're made up. You don't need to have any rules of logic, you can have string generation algorithms instead. For instance, propositional calculus. Even the rules of logic can vary between different systems, see here. Even the Axiom of Choice is ultimately a disagreement about the rules of logic. Or consider constructivism, which is an attempt to change the accepted rules of mathematical logic.

Set theory goes the same way, as there is no "set theory". Instead, there are a plethora of set theories. One can think of a set theory as fundamental, but not all mathmetical systems need it (for instance, Euclidian geometry doesn't). It may be possible to use set theory as a basis, but since it's optional it's, well, optional. Not fundamental. And even if you need it, there's a bunch of different ones with different axioms and rules of logic from which to choose.

So, in the end, all of that is just made up by people and kept if found to be useful.

November 25, 2007 4:02 PM  
Blogger Mike Beversluis said...

Hmmm. You have something there, but a couple of things still bother me. One, we are having this conversation, and it seems predicated on a lot of these rules; how can you get outside of a language to discuss it when you are using that language? Doesn't a self-referential statement that it is false invalidate itself?

Besides that, it seems possible to come to agreement as to what's useful in what context, too. Arithmetic is useful for counting physical things, and the inferences that we make about infinite sets of numbers would probably be true about infinite sets of physical objects too. Certainly that holds for irrational and transcendental numbers too. So how does that recognition of utility arise without reference to some underlying logical basis in reality, even if we cannot prove it?

Also, Euclidean geometry doesn't need set theory? What is a line except a set of points?

November 25, 2007 5:14 PM  
Blogger Susan's Husband said...

Easy one first: a line is the shortest distance between two points. Look through the axioms of Euclidean geometry and see if you can find any reference to sets of anything, or any alternate definition of a line (segment). One might also note that set theory is powerful enough to be subject to Gödel's Incompleteness, but Euclidean geometry isn't.

As for your other question, entire forests have been destroyed discussing that subject, which is known as Hermeneutics. My personal view is based on philosophical research I did for my doctoral thesis. Basically, we have a mental horizon which is the limits of our knowledge / understanding. We can only communicate with others via shared areas in our horizons. Luckily, we all live in the same objective universe, so we can share our understanding of that, if nothing else. As humans, we also share quite a lot of perceptual machinery, which helps. From that basis, we can build up via metaphors (Jaynes is wonderful on this point) more abstract languages, such as the one we're using here.

So, yeah, we pick our mathematics to be useful in thinking about physical reality, but that's not required. Moreover, there's no mathematical definition of "utility", which gets back to my point that mathematical systems are not intrinsically related to objective reality. The causality goes the other way, we start with objective reality and pick mathematics for which we can create models to relate them to physical reality.

November 25, 2007 9:37 PM  
Blogger Harry Eagar said...

'Luckily, we all live in the same objective universe, so we can share our understanding of that, if nothing else'

The older I get, the less I think this is right.

More precisely, it looks like we filter the material universe according to cultural rules that are not universally compatible. Whorf may have been wrong on all his detailed arguments, but he may have been right overall.

In what sense does 2+2=4 to people whose number system includes only '1' and 'more than 1'?

November 26, 2007 10:04 AM  
Blogger Susan's Husband said...

Of course, we have access to objective reality only through the intermediation of our sensory apparatus (another subject of my dissertation), but the fact that the underlying reality is the same and that we've all survived it for many years leads to a non-trivial amount of shared horizon.

As for "2+2=4", didn't I just spend the previous endless comments arguing that mathematics is no part of objective reality?

November 26, 2007 1:21 PM  
Blogger Harry Eagar said...

OK, these 'more than one' objects and those 'more than one objects' are -- are what?

I mean, if I have 2 pencils in my left hand and 2 pencils in my right hand, and some guy whose language does not include the concept '2' has the same, do we perceive the same reality?

Not a rhetorical question. I don't know the answer (I don't personally know anybody whose language doesn't include 2), but my sense is, no, we don't.

But I'm not sure my language can express his reality either.

November 26, 2007 5:22 PM  
Blogger Mike Beversluis said...

SH: The concept of noncoincident points (null set + elements) and pairs of points (sets and subsets) is tacitly assumed in the definition of a line. Euclid's axioms implicitly contain set theory.

That aside, to put my question another way, what does it mean that we communicate by referring to shared horizon events except that there are certain axioms and concepts that are fundamental to communication and rational argument? Aren't they the vanishing points on our shared horizon?

And while axiomatic systems are limited, much like the physical laws we model, we are able to perceive their limitations and attempt to improve them. Admittedly, we know we are approximating things, but that's just it, we know we are approximating them.

And while people do say that that there is nothing real, do they really say it and mean it? Are they even able to? Honestly, it seems like they are making too strong a statement.


I feel like the thinking about subjectivity can be taken too far. Things are uncertain, but they aren't maximally so, or completely so. They are fuzzy. They are slippery, but there's something there to be grasped. It seems to be a range: Beauty is subjective, the time of someone's death is better defined but still imprecise, the statement that George Washington crossed the Delaware more precise still, and the arithmetical meaning of 2+2 even more precise. There are Abelian and non-Abelian algebras, but there isn't Feminist arithmetic.

Maybe I'm wrong here, but it seems to me that even though mathematical axioms are optional, saying that arithmetic has no bearing on reality is too strong.

November 26, 2007 5:50 PM  
Blogger Hey Skipper said...

Harry:

I mean, if I have 2 pencils in my left hand and 2 pencils in my right hand, and some guy whose language does not include the concept '2' has the same, do we perceive the same reality?

Well, the person certainly perceives the same "number" of pencils. If you asked him to point to each in turn until he ran out of ones to point to, and you counted the number of times he pointed, you would get to four.

But that is because you have internalized what is essentially set theory. That is, you can place each of those objects in your hand in the set "pencils", and assign an incrementing index pointer to each.

If I may be so bold, this is what SH means when he says mathematics is not part of any objective reality, but rather a human construct that can, but need not, be based upon patterns within that reality.

November 26, 2007 6:27 PM  
Blogger Susan's Husband said...

Mr. Eager;

Skipper is basically correct, although he should write "ordinal" for "incrementing index pointer". I haven't written the post I have intended to on this subject, but your error is in presuming a single layered reality. Perceiving objects in a hand, and counting them to two, then adding those to get four, all occur at different layers. In your scenario, you have overlap in the first two layers, but not in the third. Because you share those aspects, you could use them to teach your hypothetical savage the concept of "2+2=4" in the aspect he doesn't original perceive.

Mr. Beversluis;

I must disagree that Euclidean geometry implicit contains set theory. I don't see how any sets are tacitly assumed by the Euclidean concept of a line. I think your error is in presuming that EG has to explain or correspond to anything other than itself. As Skipper noted, EG was inspired by physical drawings, but that doesn't mean it has to contains all features of physical drawings.

"what does it mean that we communicate by referring to shared horizon events except that there are certain axioms and concepts that are fundamental to communication and rational argument?"

You are too universalist. Communication requires shared horizon between the communicants. That doesn't require any universally common horizon, just like physics doesn't require an absolute frame of reference. You also presume that communication requires rational argument, which isn't the case either. "Build my pyramid or I'll whack your head off" is communication, but hardly rational argument.

"while people do say that that there is nothing real, do they really say it and mean it?"

I think some do, but it's because they haven't bothered to think through their position in a rigorous way. This is a point where, although I subscribe to things like Hermeneutics and Speech Act Theory, I diverge from much of post-modernism. As noted, I believe it all grounds out on a layer of objective reality.

"saying that arithmetic has no bearing on reality is too strong"

I don't think that's too strong at all. IMHO, to say anything weaker is to fall in to the post-modernist trap of thinking our symbolic utterances can affect the structure of reality. Are going along with OJ's view that if we changed how we did arithmetic, physical reality would behave differently as well? On the other hand, if not, then what bearing does arithmetic have on reality?

November 26, 2007 7:18 PM  
Blogger Harry Eagar said...

The savage is not hypothetical. Tnere really are languages with counting words that either don't go beyond 'one' and 'many'; or 'one,' 'two' and 'more than two.'

If I introduce my language to such a person, I have changed his reality. Or have I?

Pencils seem intractable, though.

We don't think we can take 2 pencils and 2 pencils and have 5 pencils.

In the case of objects, the universe seems to map onto arithmetic.

But if the universe doesn't map onto a mathematical construct (like infinity) then in what sense is it real?

I had a friend who was a missionary in Bolivia. In ancient times, Quechua had a count up to (at least) a million, but the campesinos no longer know how to count that high.

There was another missionary, also an American, who knew classic Quechua, and the campesinos called him 'the man who can count to one million.'

Yet we commonly say, when speaking about the government deficit in dollars or about astronomical distances in light-years, that no one really comprehends the big numbers.

So it looks as if there is an uncertain crossover point at which we become more like a 21st c. Bolivian campesino than maybe we would like to admit.

I'm no mathematician, and I don't doubt the reality of pencils, but it is my understanding that people who are mathematicians have some difficulties in explaining arithmetic, and, consequently, there is thought to be some arbitrariness in the additive principle.

November 26, 2007 10:31 PM  
Blogger Harry Eagar said...

The savage is not hypothetical. Tnere really are languages with counting words that either don't go beyond 'one' and 'many'; or 'one,' 'two' and 'more than two.'

If I introduce my language to such a person, I have changed his reality. Or have I?

Pencils seem intractable, though.

We don't think we can take 2 pencils and 2 pencils and have 5 pencils.

In the case of objects, the universe seems to map onto arithmetic.

But if the universe doesn't map onto a mathematical construct (like infinity) then in what sense is it real?

I had a friend who was a missionary in Bolivia. In ancient times, Quechua had a count up to (at least) a million, but the campesinos no longer know how to count that high.

There was another missionary, also an American, who knew classic Quechua, and the campesinos called him 'the man who can count to one million.'

Yet we commonly say, when speaking about the government deficit in dollars or about astronomical distances in light-years, that no one really comprehends the big numbers.

So it looks as if there is an uncertain crossover point at which we become more like a 21st c. Bolivian campesino than maybe we would like to admit.

I'm no mathematician, and I don't doubt the reality of pencils, but it is my understanding that people who are mathematicians have some difficulties in explaining arithmetic, and, consequently, there is thought to be some arbitrariness in the additive principle.

November 26, 2007 10:31 PM  
Blogger Susan's Husband said...

"But if the universe doesn't map onto a mathematical construct (like infinity) then in what sense is it real?"

Here's the root of your problem. It is mathematics that maps to (has a model for) the universe. As long as you take the map for the territory, none of this will be clear.

Mathematics is just a map, drawn to suit various purposes with various irrelevant details left out.

"In the case of objects, the universe seems to map onto arithmetic."

That's no different than expressing surprise that city streets seem to map onto the lines drawn on the piece of paper labeled "city map". Would you then expect the actual streets of a city to conform themselves to your map if you drew new lines on it? Or a new city to appear if you drew on a new piece of paper? Or, if there was a line on the map ("Infinity Street") that didn't correspond to an actual street, would you wonder "in what sense is this map real?"?

P.S. There is a lot of arbitrariness to addition. There's a whole field of study (abstract algebra) devoted to just that.

November 27, 2007 7:45 AM  
Blogger Harry Eagar said...

I didn't express that very well. Yes, I understand that the math has to map to the universe not the other way around.

November 27, 2007 8:48 AM  
Blogger Mike Beversluis said...

Well, there have to be at least two non-coincidental points to define a line. Euclid tried to hand-wave past this point, so we have to define a point as a primitive element and non-coincidence as a null intersection between points. Euclid's definitions and axioms lack this, as it's not necessary that there be any points or lines. This was made explicit with Tarski's axioms.

Here's Riemann's translated paper on the topic, http://tinyurl.com/354ywo
(the internet is amazing), which in
a lot of ways got the whole axiomatic ball rolling.

To put it another way, analytic geometry does have well defined points (ordered pairs of numbers) and lines (equations), and it's primitives live are pushed downstairs into set theory. Since this connection does exist, I don't see how you can view algebra and geometry as anything but intimately related.

To respond to your points about the different kinds of algebra, there are many kinds of geometry, but they all have the concept of points and lines in common. Abstract algebra performs binary operations on set elements, and the types of objects are emergent: sets lead to groups, groups lead to rings, rings to fields and so forth. This isn't any different than saying that we can study the whole numbers, integers, prime numbers, rational numbers, real numbers, and complex numbers. They are manifestations of the fundamental properties of sets and the way we can order, partition, and reorganize them.

What I would suggest is that the concept of existence/non-existence leads to set theory, and this is the beginning and foundation of the many connections between reality and math.

Anyway, to get to Harry's language barrier, I can recall this happening to myself. I remember pretty clearly struggling with math when I was eight. I could do it, but it was a mechanical manipulation of symbols. But at a certain point it flashed, and not only did I understand the problems, I knew what the rest of the books were going to say and why they would say it, and why they would talk about those things in the order they would take them. This is how real analysis and calculus felt, too.

To use SH's map analogy, which I like, except that our mathematical systems are the map and the mathematical concepts, or truth, are the reality we are trying to convey. I always felt like a cartographer or maybe an archaeologist. I don't think that you can invent math any more than you can invent geography. You attempt to describe what's there to the best of your abilities, and while the language we use to describe is limited in its scope (and fundamentally so) the meaning of that language attempts to convey exists outside of our ability to formulate it.

To put it another way, some shared objective experience is necessary because we can argue about the realism of a map of NYC, but not so much a map of Narnia. It's actually pretty rational to threaten someone with death in order to build a pyramid, but it's not rational to hold a gun to their head and tell them they have to give you the right answer about a duckrabbit
Or that they have to give you this answer, and either way you will kill them.

But again, I might be wrong. I do have a tendency to universalize, and I think it's one that's inherent in scientists, which this NYT ed puts nicely, http://tinyurl.com/3dnw8m. (I have no idea what he's getting at in the last two paragraphs, it seems like a distinction without meaning). I don't think this is an error as much as it is a faith, in as you put it, grounds out on a layer of objective reality, that is circumstantially buttressed (and assailed). Honestly, I'm not sure what we are arguing about, since I think we agree.

November 27, 2007 8:22 PM  
Blogger Hey Skipper said...

Mike:

... so we have to define a point as a primitive element and non-coincidence as a null intersection between points.

I don't think so.

In a plane, there cannot be two coincident points.

November 27, 2007 8:56 PM  
Blogger Mike Beversluis said...

Hey Skipper: Sorry I haven't replied to your questions down in the metaphore post; I've been trying to come up with a good response.

Anyway, a "plane" is synonym for "set" (specifically a "dense" manifold of elements) and "cannot be coincident" is a synonym for "null intersection".

November 28, 2007 6:52 AM  
Blogger Susan's Husband said...

"Intimately related" and "necessary" are not the same thing. How did people manage to do Euclidean geometry for a millenia or more without set theory if the latter is required?

Set theory may illuminate much geometry, and let you move from Euclidean geometry to analytical geometry (not the same thing!) but necessary? No.

November 28, 2007 7:25 PM  
Blogger Mike Beversluis said...

Because they were making those assumptions, implicitly, all along. Subsequent theories just admitted to this. FWIW, I don't think this is controversial within the geometry community. Sorry if I seem recalcitrant.

Couple of other things from that previous thread:

SH: MW QM would need an uncountable number of universes, because there are operators with continuous eigenvalues. That doesn't fix the fact that QM still doesn't account for gravitation interactions, and so regardless of our metaphysical epistemology, there is still some work to do there.

Hey Skipper: In your last reply in that thread, it seemed like you said that the Incompleteness Theorem doesn't apply to false statements, ie, that we know false statements are false, but that there are true statements that we cannot decide on based on our original axioms. If that's what you meant, then that's not correct, because the negation of one of those undecidable true statements is an undecidable false statement. If the axiom of non-contradiction holds, then proving all the false statements gives you all of the true statements, which we know we can't have.

Also, I totally remember when SA went down the tubes too. I pretty much stopped subscribing my Freshman year. Trade journals seem to be better, and I still like reading Physics Today. They're meant to be read within the community, so there isn't that need to go into full evangelical mode; and they want to be interesting to a wide range of physicists, including those at an undergraduate level, so there are a lot of overview topics. It might be a little thin to justify the subscription if you're not using the other trade journal features. Among the professional journals, I really like Review of Modern Physics and Review of Scientific Instruments, for this, but you need to get to a university library to get at those journals (I think). The tendency is for a lot of this to go online and open.


Harry: I just wanted to reply to your comment about the difficulty in explaining sound, because when I took Math Methods, we had to work through that derivation. QED purports to be an exact theory, at least as far as the electromagnetic forces are concerned, and thus could explain the phenomenon of sound because it gives you a complete system of equations for the motion of every electron within every atom within every molecule in the room. To put it another way, nobody thinks anything beyond electrostatic forces are responsible for sound, and we can write those down.

Because of the large number of particles, solving these equations is intractable right now. Although the field of computational protenomics is essentially trying to do this in order to predict how the genetic sequence leads to specific protein structures, so we're currently up to the number of particles present in a 2-10 nm chunk of matter. Since these phenomenon are somewhat robust, we'd like to think that we can grossly approximate the theory and still arrive at the same qualitative result. The art is to arrive at more or less the same answer while throwing away 99.99999999...% of the starting information.

Macroscopic E&M does this, and the field of fluid dynamics, does this with extremely strong approximations because you are taking moments of properties averaged over 10^6 through 10^22 particles. Also, everything gets made linear. The place where this seems to be true are the linear harmonic oscillators in QED, essentially because photons are massless; matter is strictly anharmonic, but that's too hard to model much past the hydrogen atom.

This is all still useful, obviously, because these approximate theories allow a lot of qualitative predictions, and more exact predictions could always be worked out in principle.

November 29, 2007 9:06 AM  
Blogger Harry Eagar said...

Thanks.

I'm pretty sure sound happens, even if it cannot be modeled.

We could say the same, I suppose, about weather or national economies, to a near approximation.

November 29, 2007 9:47 AM  
Blogger Hey Skipper said...

Mike:

In your last reply in that thread, it seemed like you said that the Incompleteness Theorem doesn't apply to false statements, ie, that we know false statements are false, but that there are true statements that we cannot decide on based on our original axioms.

No, what I really meant to say is that within a non-trivial consistent formal system:

- Some statements are provably true
- Other statements are true for all known cases, but not provably true for all possible cases.: these statements, neither provably true or false, are undecidable.
- The rest are provably false.

Which is a direct mirror image of theological systems. Because they individually and mutually contain contradictions, no statement within theology is decidable. Consequently, all theological claims are worthless until assessed outside the theological realm. For example, it is impossible to distinguish between a cult and a religion, or "good" and "bad" religion, without resorting to material criteria.

It is worth noting that since all theological systems contain contradictions (and, since all theologies address the same problem space, no unique theology is without contradiction), the Incompleteness Theorem doesn't really apply to them, since it is based upon systems which cannot prove an internal contradiction. (Which is why I find criticisms of Dawkins failing to discuss the great theologians ridiculous -- wasting time on, as Harry put it, Random Noun Generators is a fool's errand.)

I don't wish to derail this thread, so I shall leave it there.

Also, I totally remember when SA went down the tubes too ...

I'm still waiting to find a replacement -- if there is a periodical that covers the scientific spectrum and is pitched at the well educated college graduate, I'd love to hear of it.

I have occasionally considered Nature, but balk at the $180 subscription cost.

November 29, 2007 11:20 AM  
Blogger Mike Beversluis said...

Pardon the hijacking, but I'll try to be brief:

Susan's Husband is going to, I'm guessing and also somewhat correctly, argue that I am trying to have it both ways when I've been arguing above that there is a universal truth that we can agree on and then say to you that there isn't a sharp line that delineates theology from the epistemology of physics. This is especially manifest in quantum mechanics, which is why he's so interested in the Many World's interpretation (again I'm guessing).

To put it another way, any sufficiently familiar magic is called science. I don't know why charges move when I move some other charges. And my ability to say what a charge is goes into quantum field theory, which punts the question up a level. But I can say, and fairly objectively, a lot about how charges move, which delineates certain aspects of the interaction. E.G. because of the inverse square fall-off of the power radiated, the effect happens in three dimensions. It has a propagation velocity, which is constant in every inertial frame, etc.

QED describes it like a Frisbee game played between two frat boys walking around a campus quad. Where one goes, the other follows (with virtual photons as the Frisbee), but why they are playing Frisbee, and how they got there is another question.

The reality science and mathematics attempt to describe has inaccessible aspects too, and besides our own existence, the evidence is circumstantial. The strength of a circumstantial case varies. This is, I think, were I differ from Susan's Husband, in that sometimes the evidence is so strong, that to deny them requires an OJ-jury like obliqueness, so to speak.

This also applies to theological statements, which are statements relating to the phenomenological causes and meanings of what we observe. Materialism with universal laws counts as a theology here, and I think, is essentially a Spinoza-like "the universe is a brute-fact" deity.

Quote: It is worth noting that since all theological systems contain contradictions (and, since all theologies address the same problem space, no unique theology is without contradiction)

That, imo, is way, way, too strong a statement. One: Many contain contradictions. Most do, but all is a leap. Perhaps ironically, a leap of faith. Two: Inconclusiveness is not a contradiction.

Finally, physicstoday.org is $69, although that's not general science.

Pardon the rant to follow, but I'd avoid Science and Nature. The role they play in announcing break-throughs makes them incomplete (usually), unreliable (surprisingly often), and prone to overhype (always). Every result is "novel"; they actually banned that word, but since the push is the same, that will just result in new pejoratives. Cases of fraud are about as common as steroids are in sports.

Science funding is played a lot like a game where two teams are refereed by their division rivals. As you can imagine, there's conflict of interest, and this plays a big role in what gets published where, and has a distorting effect on how it's reported. The editors of Science magazine and Nature are actually professional reviewers, no longer directly engaged in grant writing, but I think their impact factor (which is a calculated number, and essentially translatable into grant dollars; an article in Nature is nominally worth $500k a year, and the big guys are playing with million dollar chips.) This makes it hard to avoid fraud and political maneuvering. Personally, I think peer-review could stand an overhaul, with some sort of guild of science eunuchs to keep watch.

November 29, 2007 2:44 PM  
Blogger Harry Eagar said...

As far as I am able to judge, Steve McIntyre and friends at Climate Audit are providing a satifactory model of how to review the reviewers (call in the accountants, the same theory as inspectors general in armies and bureaucracies), although they have not solved the funding mechanism yet.

November 29, 2007 6:27 PM  
Blogger Hey Skipper said...

Mike:

Thanks for the comments on Nature and Science; I guess I will have to continue looking for a SA replacement.

Why SA had to become the answer to Omni, when no one was asking the question, is way beyond me.

[no unique theology is without contradiction], is way, way, too strong a statement. One: Many contain contradictions. Most do, but all is a leap.

The statement is not too strong; moreover, it is unavoidable.

Even if one was to posit -- or find -- a unique theology that is not internally contradictory, that is no help whatsoever.

In order for that theology to be unique, it must make at least some statements no other theology makes.

But since those statements are within the universe of theological statements, there will be external contradictions.

Within the realm of theological statements, there is no way of determining which is true, because some must be false, but which are which is undecidable.

In other words, there is no theological argument that can decide between Islam and Christianity.

At the moment, there are any manner of material arguments that would create a preference.

That only proves my point.

November 29, 2007 7:21 PM  
Blogger Mike Beversluis said...

Re Nature and Science, I do have some personal biases, since I'm in that game.

Harry, Calling in the accountants is one part, although I'm sure the NSF's and NIH's et al budgets get audited. We're then left to decide who gets next year's money. Gross malfeasance gets shaken out reasonably soon. People are more stubborn about their metaphysical beliefs, and so usually the old generation has to die out before the evidence wins out. Hence Kuhn and his paradigms.

Skipper: I'll step aside on this after this statement, but no, when you say this:


Within the realm of theological statements, there is no way of determining which is true, because some must be false, but which are which is undecidable.


depends implicitly on whom has to do the determining. You? God? The Universe? And again, I think there is a hierarchy of questions: Number 1: Do you exist? Number 2: Do you have free will? Ramadan vs the Eucharist is down the list.

The fact that we cannot decide some theological questions doesn't rule out that they aren't decided. It just seems, to us, absurd or unfair.

November 30, 2007 6:02 AM  
Blogger Harry Eagar said...

McIntyre and CA aren't following the money. They're auditing the sources of data (with surprising findings that it isn't there) and auditing the statistics.

An interesting volunteer audit is an informal inspection of temperature stations in the U.S.

An amusing finding/revelation is that the station with the widest variation is in a paved parking lot in Tucson.

Some of this stuff is high school science project elementary, and some of it is way over my head.

What's revealing is that it hasn't been getting done.

McIntyre is the guy who exposed the Y2K hoax and also forced NOAA to acknowledge it had goofed by a large amount in its temperature series by melding two incommensurate series.

November 30, 2007 8:58 AM  
Blogger Hey Skipper said...

Mike:

Depends implicitly on whom has to do the determining. You? God? The Universe? And again, I think there is a hierarchy of questions: Number 1: Do you exist? Number 2: Do you have free will? Ramadan vs the Eucharist is down the list.

Speaking only up to the moment I am typing this, humans are doing the determining.

Could change in the next instant. Who knows?

Clearly, there is an objective reality regarding each of those questions. The further up the list, the less likely we can properly formulate the question, never mind obtain an answer.

Further down the list, where Ramadan and the Eucharist live, we can both formulate the question, and obtain some sort of an answer.

Materially, that is.

Within the human context, the only one available to us as of this writing, there is no such thing as knowledge within the realm of theology.

November 30, 2007 11:36 AM  
Blogger Harry Eagar said...

Shoulda said NASA, not NOAA

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