Thursday, September 23, 2010

Spaces

Love for mathematics divides people like nothing else.

Sure, people can develop a fascination for the most bizarre things - knitting, collecting Hard-Rock-Cafe shirts, or photoshopping their wifes into shape - and leave others puzzled about their obsession. But when it comes to mathematics, indifference and puzzlement is replaced with plain rejection. There's those for who mathematics is the essence of everything, it's the language in which the book of Nature is written and the secrets of the universe are encoded in. And then there's those who believe that the lover of mathematics is narrow-minded, and that the world is so much more, so vastly more complex than what mathematics can possibly capture, that anybody who thinks incomprehensible, abstract symbols capture elementary truths must be seriously disturbed.

And sure, people can argue furiously about politics or who has the best pizza in town, but in no case I can think of do you find a comparable utter lack of understanding for the other side than when it comes to the power of mathematics. The lack of understanding is probably so complete, one can't even argue about it. Over and over I have found people who reject the notion of mathematics being a universal language, and who discard it as insufficient for reality. They are dead wrong to do so of course, but since I've encountered this attitude over and over again, I want to dedicate some paragraphs to what I believe is the origin of this divide.

At the very beginning is, of course, school education. Unfortunately, what's called mathematics in school has little to do with mathematics. It should more aptly be called calculation. Don't get me wrong, it is essential knowledge to be able to multiply fractions and calculate percentage rates, but it has about as much to do with mathematics as spreading your arms has with being a pilot. Problem is, that's about all most people ever get to know of mathematics. The actual heart of math however is not number crunching or solving quadratic equations, it's the abstraction, the development of an entirely self-referential, logically consistent language, detached from the burden of reality.

Let me focus on an example that those of you with high school education will have met: vector spaces. A vector space is basically a set with elements that have a structure allowing for an operation called addition and a second operation that's multiplication with a scalar. These operations have to fulfill certain criteria which you can look up somewhere if you've forgotten, but it's not so relevant for the following. What's relevant is how abstract this notion of a vector space already is. The vector space really is that definition, and nothing else. And it's taught in school! Of course, at the time pupils come across a definition for a vector space most know examples already and have a mental picture. My math teacher used pens to visualize vectors. But nothing in the definition of a vectorspace tells you it ought to be three-dimensional, or the elements be coordinate-vectors (pens).

Given that vector spaces are such a simple concept that is introduced even in school, I was surprised to learn how late in the history of science it came along. The phase space in physics is essentially a higher dimensional vectorspace whose elements aren't only coordinate vectors but also momenta (and, by virtue of this, has some additional "symplectic" structure). Knowing what you know today this sounds hardly like a revolutionary concept. But in the middle of the 19th century it was. In his (highly readable) Physics Today article on the history of the Phase-space, David Nolte writes:

Today it is natural for us to assign each variable its own axis in a generalized multidimensional space. But in the 1700s it was not natural. [...] Cayley in his 1843 paper titled "Chapters in the Analytical Geometry of (n) Dimensions" was the first to take the bold step of referring to a geometry of more than 3 dimensions. After that, the stage was set for the "invention" of multiple dimensions when Grassmann developed the concept of an n-dimensional vectorspace in 1844.

The vector space that you've heard of in school is a result of many generations of abstractions. Yet, it is still an extremely special case of what the mathematician considers a "space."

If you go out on the street and ask random passers-by what they associate with "space," you might hear office space, or the space they don't have in their living room after they bought the drum set for the eldest son. You might hear parking space, or the space between two letters in a sentence, or maybe outer space. That's the real world, and at first sight it seems indeed like a selection of complex and very different notions of space. But in fact all these spaces that you encounter in the real world are highly specific. They are three or lower-dimensional. They are to excellent precision flat. They come with a distance measure that allows you to tell if the new couch will fit next to the drum set. The general space in mathematics however may do away with all of these properties that we are so used to. Imagine an infinite dimensional space. Imagine one without distances. Imagine what it would be to try to park your car in one.

In physics one does encounter more general spaces than the standard 3-dimensional vector space. The best known example is probably the Hilbertspace of quantum mechanics, which can easily be infinite dimensional. But also in physics, the realm that we deal with is only a tiny part of all that mathematics has to offer. The functions we deal with are typically nicely differentiable, so are the manifolds we put them on, blessing us with plenty of additional structure. The differential equations we have are typically not higher than 2nd order, spaces are hausdorffian and almost all of the pathological examples you come across in mathematics the physicist never has to bother with.

This of course then brings one to the question, if the world of mathematics contains so much more, then where is it? Does it exist, somewhere, that space without distances, that module, that left-invariant subgroup? I have some sympathy for Tegmark's Mathematical Universe, which posits that all of mathematics must exist somehow, somewhere in the multiverse. My central objection to Tegmark's idea just is that it's not insightful and plain useless.

If you've scrolled down to this last paragraph, shame on you. The point of all the words above was that during the history of science we have come to realize it is the world of mathematics that is vastly larger than what the real world has to offer, not the other way round.

59 comments:

Reece Dunn said...

Nice article.

Mathematics is the foundation for everything we have developed, be it the car or computer.

For the simplified (nee abstract) representation of the world mathematics fits very well, for example in predicting the orbits of the planets around the sun or where and when solar eclipses occur. We can calculate these with a very high degree of accuracy using fairly simple mathematics.

These systems are such that they are predictable and deterministic. They are not affected by changes in the environment.

There are other systems that interact with and are affected by the world around them. There are systems such as the flight patterns of birds and the weather.

An interesting aspect of these systems is that they exhibit emergent behaviour -- behaviour that is not determined by the rules that the individual components are using (even if those rules are expressed mathematically).

Computational examples of this are Conway's Game of Life which has the ability to create complex and varied behaviours (including behaving like a computer!), Langton's Ant (http://en.wikipedia.org/wiki/Langton%27s_ant) and other cellular automata.

Another example in nature is the brain and the emergence of consciousness, language and social behaviour.

Peter Turney said...

My non-mathematical friends and acquaintances are completely baffled when they hear that vectors can capture the meaning of words (From Frequency to Meaning: Vector Space Models of Semantics).

Uncle Al said...

an entirely self-referential, logically consistent language, detached from the burden of reality. String theory is rigorous, gargantuan, and mostly irrelevant (re economics).

such a simple concept that is introduced even in school, I was surprised to learn how late in the history of science it came along. Euclid was infallible for 2000 years, until the early 1800s. Draw a triangle on a 2-sphere, deep sea navigate. Didn't anybody suspect?

almost all of the pathological examples you come across in mathematics the physicist never has to bother with. The universe may be whimsical. Fantastical contentions require fantastical validations (or, ask an organic chemist. Synthesis is not derived.)

Physics inserts symmetry breakings as patches when elegance is falsified by observation. Physics actively denies emergence. Reality could be fundamentally emergent and asymmetric. Somebody should look,

http://www.mazepath.com/uncleal/erotor1.jpg
Two parity Eotvos experiments. The worst it can do is succeed.

the world of mathematics that is vastly larger than what the real world has to offer, not the other way round. Reality may contain processes beyond understanding, or not. Consider the universal failure of religions.

The Tree of Knowledge was not an apple, Kazakhstan highlands and inedibly sour. It was a pomegranate, Iran through the Himalayas, including Mesopotamia. Eve's vastly unforgivable act was peeling the fruit and learning how to count.

Sin like you mean it.

Robert L. Oldershaw said...

Here is something worth considering that is on topic, and relevant to a certain recently revealed "delicate condition".

How could one possibly discuss the general observation that "Ontogeny Recapitulates Phylogeny" in a coherent manner if one's discussion were confined to "the universal language of mathematics"? Or try to discuss just the ontogeny, i.e., morphogenesis, of a fertilized cell into a fully developed human organism? We can apply some math to parts of the development, but a mathematical model of the whole process - No Way!

Personaly, I think it is important appreciate both the huge role that the approximate language of mathematics plays in quantitative science, especially physics, and the inevitable limitations of that approximate language.

If you treat something like Euclidean geometry as THE absolute, immutable and universal mathematics of nature, then look at what a prison you have put yourself in!

Mathematics is not perfect and immutable. It is imperfect and always only an approximation of nature. This approximate language evolves (think Euclidean --> Riemanian --> ?) like life.

Steven Colyer said...

Mathematics is the one truly infinite field of study. And if all the other fields depend on it, then ....

Thank goodness there are different-sized infinities. :-)

(The set of real numbers vs the set of integer fractions, e.g.)

Also, I am glad there are 4 different ways to prove things. 2 days ago I didn't know that. Now I'm walking around doing my
new mantra "Counterexample, Direct method, Indirect "reductio ad absurdum" method, Induction, ... " ... repeat.

Reece Dunn said...

Here is something to boggle the brain:

1. the set of integer fractions is infinite;
2. infinity is the largest number;
3. the set of real numbers is a strict superset of the set of integer fractions and is infinite;
4. the set of integer fractions consists of two numbers each of which are from an infinite series.

To make things more interesting, the number sets of reals, complex numbers, hamiltonians, octonions, etc. comprise of 2^n real numbers, each of which are on an infinite number line. This means that complex numbers consist of a pair of numbers each of which can be infinite resulting in 2 x infinity combinations. Not only this, but the series itself is infinite!

Michael F. Martin said...

There may be some universe that captures all mathematical truths. But the 20th century work on foundations shows that the converse cannot be true -- i.e., that even all of mathematical truth cannot answer some questions that nonetheless have answers.

The people who can move back and forth between abstract and particular are the ones for whom I reserve my highest admiration.

tspin said...

Language is even more fundamental then mathematics, without language mathematics would not exist (beyond the most basic notion of arithmetic).

Mathematics may be thought of as a part of language and just as with language I only find it interesting as long as it describes the Word we live in.

andrewg said...

Tegmark's mathematical universe is interesting not because it contains no extra meaning, but because its negation contains no meaning.

I remember having an argument with a university friend around 1993 about the "physicality" of the universe. I took the position that if the universe was completely describable in principle by physical laws then this description would be entirely sufficient - equating the concepts of mathematical and physical existence, rather like Tegmark later did (unfortunately I did not publish the argument!). My friend maintained that even if a faithfully accurate description of the universe existed mathematically, that did not imply its physical existence, to which my retort was that this "physicality" condition contained no extra meaning and should therefore be discarded by Occam.

My friend wasn't convinced.

Rastus Odinga Odinga said...

"The differential equations we have are typically not higher than 2nd order, spaces are hausdorffian and almost all of the pathological examples you come across in mathematics the physicist never has to bother with."

Actually, mathematicians almost never have to bother with them either. It's a great mistake to turn students into mathematical neurotics who automatically assume that their intuitions must be wrong: "Everything you believe in mathematics always turns out to be wrong", as one student once said to me. When I teach differential equations, I tell my students that they nearly always have a unique solution for given initial conditions. I do show them one or two counter-examples, but stress strongly that they should not worry too much about them.

As for the question: where is all that non-applicable maths? Answer: in the same place as Captain Ahab, Hans Castorp, and Madame Bovary. Note in this connection that just as there are bad novels, whose characters seem less "real" than those of good novels, so also there is bad mathematics [eg a lot of number theory], which should be regarded as being less real than good mathematics [eg differential geometry].

Robert L. Oldershaw said...

I had forgotten the name Hans Catsorp, but certainly not the book.

I would add to the list: Ulrich from "The Man Without Qualities". Now there is a book-and-a-half, in fact a 2-volume magnum opus.

I think MFM is right to honor those who can shift back and forth between abstract mathematical analysis and conceptual subtleties like holistic context and pattern recognition. My guess is that this left brain/right brain balance is the secret of the great ones.

Plato said...

As a layman it is always interesting to me to see "information of mathematics" espoused to help educate.

I talk to the Mrs. this morning as I was sleeping troubled about my grandson's future? How might one have conveyed to my grandson the unique way one may see the world with mathematics?

My grandson is 14, and in the hospital right now.

He's doing his school work, while he contends with his own situation.

For me, coming across things that help to see the world in different ways while one is learning their science, I would hope may transmit something that is equally profound for a youthful mind as in my grandson who is very sharp and has found "too easy" the mathematics he is given.

Perhaps.....?

8.6 On Gauss's Mountains

One of the most famous stories about Gauss depicts him measuring the angles of the great triangle formed by the mountain peaks of Hohenhagen, Inselberg, and Brocken for evidence that the geometry of space is non-Euclidean.


If you follow the history to this point, then you will understand how profound that discovery was of "the Non-euclidean," while we see the sciences that have been united in these new geometrical expressions?

The point about Tegmark is one Coxeter believes as well, is that information is already out there on how one might use mathematics to describe the world. So to me, it is not useless what Tegmark or what Coxeter point out, but helps toward what "one can discover" that the mathematical language already exists.

You only have to discover the "right mathematics" for the explanation? As Einstein did, with information from Grossman.

Best,

Plato said...

Illustration by Theoni Pappas

In a fanciful drawing done in the manner of a woodcut, the young Carl Friedrich Gauss receives instruction in arithmetic from the schoolmaster J. G. Büttner. As the story goes, Gauss was about to give Büttner a lesson in mathematical creativity.Gauss's Day of Reckoning

weforgottenuno said...

That article in Physics Today about the origins of "phase space" - as a phrase and as a concept - was a very interesting read. Thanks for linking to that Bee.

Phil Warnell said...

Hi Bee,

Thanks Bee for this wonderful article. In truth though there is not only a divide between those that love math and those that hate it, yet also a rift between the lovers themselves; with one group insisting it all to be an invention of man and another feeling as you that it be something real onto itself, both connected to and yet transcending the physical world. Of course being a Platonist I’m with the second group and yet more in line with the thoughts of Penrose, rather than those of Tegmark’s, as for Tegmark all must exist to be physicality represented, where I think physical reality is but the subset of it which has the potential to be realized; which we come to know as its quality, whose aspects of which most physicists refer to as its nature.

“Plato made it clear that the mathematical propositions – the things that could be regarded as unassailably true – referred not to actual physical objects (like approximate squares, triangles, circles, spheres, and cubes that might be constructed from marks in the sand, or from wood or stone) but to certain idealized entities. He envisaged that these ideal entities inhabited a different world, distinct from the physical world. Today we might refer to this world as the Platonic world of mathematical forms.”

-Roger Penrose- The Road to Reality

“What I mean by this ‘existence’ is really just the objectivity of mathematical truth. Platonic existence, as I see it, refers to the existence of an objective external standard that is not dependent upon our individual opinions nor upon our particular culture. Such ‘existence’ could also refer to things other than mathematics, such as morality or aesthetics.”

-Roger Penrose- The Road to Reality

Best,

Phil

Steven Colyer said...
This comment has been removed by the author.
Steven Colyer said...

If one wishes to become expert in Mathematics to the degree a Phd. in the same is, which I strongly believe every Theoretical Physicist (Professional Class) should be expert in (as are the best ones), one should be firmly based and knowledgeable in the 3 legs of the tripod upon which virtually all current Mathematics are based, and indeed require as grad courses one should take to earn the esteemed PhD. in Mathematics, which are and to whit:

- Real Analysis
- Complex Analysis
- Abstract Algebra

It's not like Wikipedia doesn't have entries for these, folks.

For undergrads, the single most important course is Calculus I. It ain't hard, don't let the nerdocracy discourage you. The Derivative is the slope of a function, its inverse the Integral is the area under the curve. That ain't hard to comprehend, give it a shot.

But the one course, blessfully an undergrad one, that makes Science assessable (and the papers thereof) is Calculus IV, better known as Differential Equations and in our slang: Diff-E-Q.

Learn THAT, and half of all scientific publications will be open to you. You'll have to study a bit more to make up the other half but pfft!, you're halfway home so what the heck?

And it will be at that point, the understanding of Diff-E-Q when first you learn mathematics doesn't have all the answers (Computer Scientists do), that true education begins.

Kay zum Felde said...

Hi Bee,

why doesn't so many people like math ? This is really a good question. Sometimes I feel that people say I wasn't good in math or I hate math like math is something that one must hate. Myself I have had problems with proofs in school and I began to understood proofs learning for the 'Vordiplom', this was the first exam in University in Germany. Later I felt more comfortable with math proofs, but a lot of modern math proofs are hard for me to follow. As being a theoretical physicist it is okay with that.

Best, Kay

Bee said...

Hi Kay,

It is funny you say that, since it's been exactly the other way 'round with me. I've always liked the cleanliness of proofs, the A => B, the accumulation of little steps that in the end yields a sometimes surprising result. When I changed to physics, I initially had a hard time with the often handwavy, non-rigorous style of argumentation that you find among physicists. But wait, I'd say, I don't see how that follows. Which would typically get an answer of the sort "But it has to be," followed by some physical argument - that might or might not be correct, but was typically based on a real-world phenomenon rather than being an actually proven fact. It is historically very interesting, how this sometimes allows physicists to move on much more quickly than mathematicians. Best,

B.

Plato said...

Bee: but was typically based on a real-world phenomenon rather than being an actually proven fact.

I highlight the two word phrases that seem the same to me, yet, they seem not to be, the way I understand you saying it?

Hi Steven,

Thanks for the run down on the progression.....and then it seems there is a progress in the geometries as well? Coxeter to pull it into "higher dimensions?"

Cayley and Sylvestor? Genus figures? Artifacts of the Wunderkammer? Maybe, Bee walked by them one day in Arizona?

To become familiar with the concepts(mathematically) is not solely without capabilities when seeing the progressions in terms of years dedicated to following the thoughts of our esteem blogger's.

When they talk about science it is couched in the form of math?:)

The "historical" is open to the eyes of those who follow the history. Real science is not unattainable when you apply yourself?

Ingenuity springs upon those minds who are free to explore the regions not previously considered, yet, how was access gained from what was discovered?

You cannot look at the science and see what is missing by way of explanation?

What are allotropes and polytopes "in structure" in their previous existence "as an idea?" Not square, round, images in sand as an ideal, but possibilities that are lit from the fires of mind, as to an elemental outcome??:) Something concrete?

Best,

Plato said...

Does this image look familiar? Possibly, another image overlay using this schematic by some bright young mind?:)

Plato said...

Is this what you meant Bee?

Most people think of "seeing" and "observing" directly with their senses. But for physicists, these words refer to much more indirect measurements involving a train of theoretical logic by which we can interpret what is "seen."- Lisa Randall-Dangling Particles

cody said...

Paul Lockhart's "A Mathematician's Lament" has a lot of really interesting things to say about mathematics education.

I think mathematics suffers heavily from mistaken notions & reputation.

Also, I once saw a paper claiming Number Theory as the ultimate theory of physics, and it got me thinking: if we were to discover some number-theoretic formulation that say, gave all the empirical parameters (e.g. masses) as well as the rules of the standard model, that would really go a long way to making me think we really had found a true and complete theory of the universe. Though obviously simple rational skepticism would remain regardless.re

Neil B said...

One of the cute things we can do with an infinite space is compress it into a finite volume with a tranformation. For example, use x' = x/sqrt(1 + x^2). That makes ~ 0 --> 0, 1 --> 0.707, 100 --> 0.99995, and "+infinity" ---> 1 (corresponding negatives.) Well that may seem no big deal, but it has counterintuitive consequences: our whole universe (if "is math") could be mapped into such another finite space. Then there'd be more "outside" our infinite universe (no, not "parallel" either) which seems absurd.

There are also the spaces where points are identified with others such that the space is finite but there is no curvature (opposite sides of cube are "identified" etc.) It can get pretty weird. The one about reconstituting a sphere into a bigger or smaller sphere is the weirdest I heard so far I recall.

I think surreal numbers and non-standard analysis create problems for normal ideas of space, if they take infinitesimals seriously as "entities."

But in any case, I think Tegmark is wrong for several reasons. Our universe really isn't mathematically describable in a neat way in the face of wavefunction collapse. The MWI tries to dodge that but IMHO can't escape many problems (the range of relative chance of outcomes if two "splits", the rearrangement of the WF that must happen anyway in null Renninger-style measurements, decoherence not really applicable to why a polarized state does or does not get absorbed by a non-orthogonal filter, etc.)

Kris Krogh said...

Hi Plato,

Thanks for this link to Kevin Brown's Math Pages:

8.6 On Gauss's Mountains

However, your selective quotation on that topic represents the opposite of the author's view. He goes on to say:

On the other hand, Poincare advocated a conventionalist view of geometry, arguing that we can always, if we wish, cast our physics within a Euclidean spatial framework - provided we are prepared to make whatever adjustments in our physical laws are necessary to preserve this convention. In any case, it seems reasonable to agree with Buhler, who concludes in his biography of Gauss that "the oft-told story according to which Gauss wanted to decide the question [of whether space is perfectly Euclidean] by measuring a particularly large triangle is, as far as we know, a myth."

Your quotation brings to mind another common fairy tale: that Lorentz-Poincare relativity (with a preferred frame) is ruled out by the Michelson-Morley experiment. That it supports the mainstream world view doesn't make it okay to repeat it.

Best, Kris

Bee said...

Hi Plato,

An observation is not a proof. If you start with assumptions A,B,C, then you do some elaborate maths and end up with conclusion D, that's a proof. If you say, my theory must fulfill D because that's what we've observed (or think we have observed) that's a convincing argument (depending on the observation though), but not a proof. That's what I mean with the difference between "a proven fact" (no way around it) and a "real world phenomenon" (subject to error and interpretation, yet if correct a help that the physicist has which the mathematician can't rely on). Best,

B.

Neil B said...

Kris, I got a page load error from that link. BTW, re Lorentz-Poincare relativity we find a good rundown here. Briefly: it's the idea that the ether is real and affects things in such a way that we can't distinguish the result from "real relativity" as imagined re Einstein. IOW, the motion contracts things, slows time, and even must adjust relative simultaneity (that seems the hardest stretch.) The effect on the "really moving" frame makes the effect seem symmetrical, since her measurements make the "rest" content seem contracted etc.

So sure, in that sense the M-M experiment wouldn't disprove the idea. But many people say if there really isn't a way to tell the difference then it's parsimony to say there just isn't any ether or "action" on moving bodies. It's more complicated since there are some additional relativistic effects like the stress correction to momentum and energy in moving bodies (surprising how few know of it, among physicists.)

But an irony of all that is that is you believe the MUH that we really are a mathematical construct (! - what about feelings?), then things like whether the whole universe got bigger would have real meaning after all: the coordinates of each point would "really" get bigger even though no way for us to "tell the difference." Interesting ironies.

Christine said...

Bee,

Could you please enlighten me in a few words why there is so much interest in Tegmark's "mathematical universe"?

For all that I could see, I find it *extremely* uninteresting (*yawn* again for that one). All I can see from it is that it makes at most a poor philosophy. Definitely, it is not physics either, of course. For me, it is almost like a blank statement. So why bother?

To equate "physical universe = mathematics" instead of attempting to really elaborate *why* mathematics seems to have a deep root as an overall correspondence to physical phenomena, is like a kid, when asked to write a composition, just write down a simplified, trivial or blank statement with no content of value.

(E.g., write a composition about your vacation. The kid just writes: "I went on vacation." I'm certain no one would give a Nobel on literature for that, saying that is a revolutionary way of writing...)

Phil Warnell said...

Hi Neil,

You said: ”So sure, in that sense the M-M experiment wouldn't disprove the idea. But many people say if there really isn't a way to tell the difference then it's parsimony to say there just isn't any ether or "action" on moving bodies.”

It is true that Einstein himself was one of the first to propagate such a philosophy and yet when confronted with the seeming absurdities of the emerging quantum theory he then was brought to have a change of mind in respect to such matters. This was made clear in the following exchange he had with Werner Heisenberg .

”For the first time, therefore, I now had the opportunity to talk with Einstein himself. On the way home, he questioned me about my background, my studies with Sommerfeld. But on arrival, he at once began with a central question about the philosophical foundation of the new quantum mechanics. He pointed out to me that in my mathematical description the notion of "electron path" did not occur at all, but that in a cloud chamber the track of the electron can of course be observed directly. It seemed to him absurd to claim that there was indeed an electron path in the cloud chamber, but none in the interior of the atom. The notion of a path could not be dependent, after all, on the size of the space in which the electron's movements were occuring. I defended myself to begin with by justifying in detail the necessity for abandoning the path concept within the interior of the atom. I pointed out that we cannot, in fact, observe such a path; what we actually record are frequencies of the light radiated by the atom, intensities and transition probabilities, but no actual path. And since it is but rational to introduce into a theory only such quantities as can be directly observed, the concept of electron paths ought not, in fact, to figure in the theory.

To my astonishment, Einstein was not at all satisfied with this argument. He thought that every theory in fact contains unobservable quantities. The principle of employing only observable quantities simply cannot be consistently carried out. And when I objected that in this I had merely been applying the type of philosophy that he, too, has made the basis of his special theory of relativity, he answered simply: "Perhaps I did use such philosophy earlier, and also wrote of it, but it is nonsense all the same."... ...He pointed out to me that the very concept of observation was itself already problematic. Every observation, so he argued, presupposes that there is an unambiguous connection known to us, between the phenomenon to be observed and the sensation which eventually penetrates into our consciousness. But we can only be sure of this connection, if we know the natural laws by which it is determined. If, however, as is obviously the case in modern atomic physics, these laws have to be called into question, then even the concept of "observation" loses its clear meaning. In that case, it is the theory which first determines what can be observed.

- Werner Heisenberg- Encounters with Einstein-1983

Best,

Phil

Plato said...

Kris:Thanks for this link to Kevin Brown's Math Pages

Thank for giving credit to the author. I thought I had done that in mouse over.

Kris:the oft-told story according to which Gauss wanted to decide the question [of whether space is perfectly Euclidean] by measuring a particularly large triangle is, as far as we know, a myth."


Historically, as I search the information, "the triangle" caught my eye(more or less then a 180 degrees) as well as, Gauss as a teacher to Riemann exploring aspects of the non-euclidean.

I mean whose mind is not tantalized by the idea of the interferometer as an expression held in context of LIGO to see these mountains as "Gaussian coordinated points representing distance?

A myth? Okay.

The train of thought moved through these geometers as to garner an equatorial correlation to two paths culminating in the expressions of the non-euclidean?

The
significance of non-Euclidean geometry



Bernhard Riemann once claimed: "The value of non-Euclidean geometry lies in its ability to liberate us from preconceived ideas in preparation for the
time when exploration of physical laws might demand some geometry other than the Euclidean." His prophesy was realized later with Einstein's general theory of relativity. It is futile to expect one "correct geometry"
as is evident in the dispute as to whether elliptical, Euclidean or hyperbolic geometry is the "best" model for our universe. Henri Poincare, in Science and Hypothesis (New York: Dover, 1952, pp. 49-50) expressed it
this way.




Best,

Phil Warnell said...

Hi Neil,

Just as a follow up is to add that I’ve always thought that the best rundown of Lorentz-Poincare relativity is that of J.S. Bell’s entitled “How to Teach Special Relativity”. In this essay he gives the reader many things to consider, with those being not just restricted to the examination of the philosophies involved, yet more so what physicists should be mindful of when it comes to what has them come to find their theories convincing, as to be true; in that being the most important of spaces being the one in which they allow themselves to think.


“There is no intention here to make any reservation whatsoever about the power and precision of Einstein’s approach. But in my opinion there is something to be said for taking students along the road made by Fitzgerald, Lamor, Lorentz and Poincare. The longer road sometimes gives more familiarity with the country.”

J.S. Bell- How to Teach Special Relativity

Best,

Phil

Bee said...

Hi Christine,

I have certainly not the impression that there is much interest in Tegmark's Mathematical Universe. Except for his own talks, I've never heard anybody mentioning it (outside the blogosphere that is). I just bring it up occasionally because it's a recent warm-up of thousand year's old Platonism, so it serves to show people this isn't a settled issue (I doubt it will ever be settled). Best,

B.

Christine said...

Hi Bee,

Thanks... But argh! I've just written a comment and it went to nirvana.

Never mind. I just think Tegmark wanted to solve the issue by asserting that yes, there exists a Platonic mathematical world, and the proof is just to look around us: it is exactly our physical world. The subject is solved.

That is my impression, but if that is so, I think it is, well, embarrassing.

Best,
Christine

PS- How are the babies? And do you feel better? By that time, I had put a large paper on the wall, would stand at it (at the same place) and my hunsband would draw my belly's profile against the paper every now and then. :) I still have it, it's a nice remebrance.

Steven Colyer said...

Hi, Bee. You wrote:
At the very beginning is, of course, school education. Unfortunately, what's called mathematics in school has little to do with mathematics. It should more aptly be called calculation.

Yeah, I'm beginning to see that now. Now I know what you mean. Really disappointing to realize Differential Equations is just calculation.

Better still, you wrote:
Over and over I have found people who reject the notion of mathematics being a universal language, and who discard it as insufficient for reality. They are dead wrong to do so of course, but since I've encountered this attitude over and over again, I want to dedicate some paragraphs to what I believe is the origin of this divide.

That is really good prose. Yau in his preface to his new book complains of people calling Math a "language", as if that insults it. Quite the contrary, I feel that elevates Math above Science, and every other FoS.

Now Computer Science on the other hand, probably is a Mathematical Science. (As opposed to Economics Science and Political Science, which aren't IMO). I comfort myself by thinking of CompSci as Applied Logic, or just plain AND-Gates-Or-Gates-NOR-Gates-R-Us. Works for me. Applied Philosophy too if one wishes to get really abstract. :-p

Reece Dunn said...

Mathematics is most definitely a language, albeit a specialised one like computer programming languages are.

Consider: for every pair of elements in a set, performing an operation on those elements results in an element that belongs to the same set.

Steven Colyer said...
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Steven Colyer said...

A very nice puzzle Reece, thanks. Give me a coupla days to think about.

Well, I can't help thinking about physics and mathematics as language without thinking about what Murray Gell-Mann is up to these days. More on that later as well. As far as I know the oldest known verbal language is Archaic Sumerian. I'm sure there's older ones as Archaic Sumerian surely didn't develop out of whole cloth, but the little-known (and little known-about)language does have the advantage, and honor, of that which was spoken when the Sumerians invented written language.

Another thing that surprises me is that no one has yet to comment on the SciAm issue in the current issue about E8 Lie algebra, titled Rummaging for a Final Theory: Can a 1960s Approach Unify Gravity with the Rest of Physics? by London-based Science Journalist Zeeya Merali.

I do believe Garrett Lisi was misquoted by one word in that article, but I leave it as an exercise to the readers of the article to flesh that out. Overall a very nice piece, and hope perhaps, for us all.

Phil Warnell said...

Hi Bee,

You said: ”I just bring it up occasionally because it's a recent warm-up of thousand year's old Platonism, so it serves to show people this isn't a settled issue (I doubt it will ever be settled)”

I would ask if this doubt is rooted in the seeming limitations of the human intellect and resourcefulness or rather feeling that such a question exceeds the abilities of reason. I ask this as although I would agree there may be merit to suspect the former being true, that holding to the latter suggests that reality is not reasonable to begin with, which would place a limit not solely on humanity’s understanding of reality, yet that of reality itself in respect to reason; this I would find as not being scientific.

Best,

Phil

Kris Krogh said...

Hi Neil,

That MathPages section link always works for me when I click it. Here's the URL:

http://www.mathpages.com/rr/s8-06/8-06.htm

Kris Krogh said...

Hi Plato,

Yes, it's easy to see something we expect, even if it's not there. Here's another item, from the quotations section of Kevin Brown's MathPages, which I think applies to space-time curvature and gravity:

"Nothing is more dangerous than an idea, when it is the only idea we have." -Alain, 1908

In his text, Gravitation and Cosmology, Steven Weinberg takes Poincare's position:

Einstein and his successors have regarded the effects of a gravitational field as producing a change in the geometry of space and time. At one time it was even hoped that the rest of physics could be brought into a geometric formulation, but this hope has met with disappointment, and the geometric interpretation of the theory of gravitation has dwindled to a mere analogy, which lingers in our language in terms like “metric,” “affine connection,” and “curvature,” but is not otherwise very useful. The important thing is to be able to make predictions about images on the astronomers’ photographic plates, frequencies of spectral lines, and so on, and it simply doesn’t matter whether we ascribe these predictions to the physical effect of gravitational fields on the motion of planets and photons or to a curvature of space and time.

Your citation of Poincare's book, Science and Hypothesis, is somewhat misleading. To a casual reader it might create the impression he is being quoted. (I have the book, and there is no discussion of elliptical or hyperbolic space as a model for the universe.) As far as I can tell, that paragraph originated here.

Best, Kris

Kris Krogh said...

Hi Phil,

Bell's discussion of special relativity that you linked to (from his book Speakable and Unspeakable in Quantum Mechanics) is my favorite too. Like Lorentz before him, Bell was open-minded about which representation of space and time might be correct. (In contrast, Einstein insisted his way was the only possibility.)

I think the Lorentz-Poincare representation has an important advantage: no "problem of time." In Minkowski space-time itself, of course there is no indication of time's direction. As Ilya Prigogine pointed out, when you "dimensionalize" time, its direction has to be put in by hand.

Whatever time itself may be, your representation of it should at least allow a complete discription. Parameterized quantum field theories accomplish that by introducing a redundant one-way Newtonian time parameter, in addition to the Minkowski time coordinate.

Where Minkowski space-time forces time to be interchangable with a bidirectional spatial direction, the preferred-frame approach of Lorentz and Poincare keeps space and time separate. Time remains a one-way parameter as in Newton's universal time. There all you need is that one parameter.

All the best, Kris

Kris Krogh said...

Oops. Instead of "bidirectional spatial direction" above, that should say "bidirectional spatial dimension."

Plato said...

Hi Kris

"Bernhard Riemann once claimed: The value of non-Euclidean geometry lies in its ability to liberate us from preconceived ideas in preparation for the time when exploration of physical laws might demand some geometry other than the Euclidean"

Is this quote there?

http://www.faculty.fairfield.edu/jmac/sj/sacflaw/sacther.htm

Kris:To a casual reader it might create the impression he is being quoted

What was misleading?

Something that's not there?


Best

Plato said...

The restricted problem (both circular and elliptical) was worked on extensively by many famous mathematicians and physicists, notably Lagrange in the 18th century and Poincaré at the end of the 19th century. Poincaré's work on the restricted three-body problem was the foundation of deterministic chaos theory. In the circular problem, there exist five equilibrium points. Three are collinear with the masses (in the rotating frame) and are unstable. The remaining two are located on the third vertex of both equilateral triangles of which the two bodies are the first and second vertices. This may be easier to visualize if one considers the more massive body (e.g., Sun) to be "stationary" in space, and the less massive body (e.g., Jupiter) to orbit around it, with the equilibrium points maintaining the 60 degree-spacing ahead of and behind the less massive body in its orbit (although in reality neither of the bodies is truly stationary; they both orbit the center of mass of the whole system). For sufficiently small mass ratio of the primaries, these triangular equilibrium points are stable, such that (nearly) massless particles will orbit about these points as they orbit around the larger primary (Sun). The five equilibrium points of the circular problem are known as the Lagrange points.http://en.wikipedia.org/wiki/N-body_problem#Three-body_problem

Might as well turf this view of space and satellite travel as well:)

Kris Krogh said...

Hi Plato,

Let's discuss what you actually posted:

It is futile to expect one "correct geometry"
as is evident in the dispute as to whether elliptical, Euclidean or hyperbolic geometry is the "best" model for our universe. Henri Poincare, in Science and Hypothesis (New York: Dover, 1952, pp. 49-50) expressed it
this way.


"Henri Poincare ... expressed it this way" makes it sound as though the sentence before (from an unidentified author) is a quotation or paraphrase of something Poincare said. If you check the pages cited (too long to post here), he did not "express it this way." (Although he did harbor similar sentiments.)

Best, Kris

Bee said...

Phil,

I doubt the question will ever be settled because it cannot be settled in principle. How will you ever be able to rule out that something does not exist, if that something can be something you can't observe? Best,

B.

Bee said...

Err, I meant "rule out that something exists"

Plato said...

Hi Kris

it seems there are two opinions, neither of them mine, yet, sources have been cited and made clear. Your point made clear, and the opinion you have about that source made clear.

You both make an assumption, and for me, I am awestruck, of the movements within that space.

Best,

Plato said...

http://librivox.org/science-and-hypothesis-by-henri-poincare/

Will have to listen and form my own opinion.

Best.

Phil Warnell said...

Hi Bee,

”How will you ever be able to rule out that something exists, if that something can be something you can't observe?”

I of course can offer no answer, yet can only say I would contend that the ultimate power of reason should have it able to resolve anything that is reasonable. That is for me observation is but a tool reason incorporates, rather than its limitation. It is also true that not all observations need to be physical, yet can be simply ones of logical origin, which in essence is the realm mathematics reveals to explore as to have things confirmed or denied. Then again I guess without anyone ever having realized the full capacity of reason this question comes down to which version of science one subscribes to.

“The long chains of simple and easy reasonings by means of which geometers are accustomed to reach the conclusions of their most difficult demonstrations, had led me to imagine that all things, to the knowledge of which man is competent, are mutually connected in the same way, and that there is nothing so far removed from us as to be beyond our reach, or so hidden that we cannot discover it, provided only we abstain from accepting the false for the true, and always preserve in our thoughts the order necessary for the deduction of one truth from another.”

-René Descartes- Discourse on The Method: of Rightly Conducting The Reason, and Seeking Truth in the Sciences (1637)

Best,

Phil

Bee said...

Hi Phil,

The question under discussion is the following: Will we ever be able to decide whether "all of the multiverse/mathematics really exists" if some of it may be unobservable in principle. Let us leave aside for a moment the question what it means "to exist." The only way for a human being to show that something exists is to observe it in one way or the other. Consequently, it is impossible to distinguish between the possibility that something doesn't exist, and something existing, but being unobservable. Thus, for what I am concerned, something that isn't observable doesn't exist, because both possibilities are indistinguisheable. There's no shortcut to this that "reason" can bring. That something unobservable exists is an untestable, and thus unscientific, hypothesis. Best,

B.

Phil Warnell said...

" The only way for a human being to show that something exists is to observe it in one way or the other. Consequently, it is impossible to distinguish between the possibility that something doesn't exist, and something existing, but being unobservable.”

As I said in the outset if you restrict this to be a limitation resultant of human capacity or ultimate potential, I find no difficulty with this, as who knows what that is or will be.

Best,

Phil

Neil B said...

Bee - what complicates things is, that whether, literally, humans can observe the other universes is said to be beside the point, since it's only if their inhabitants can do so there. But even worse, the MUH is saying that each logical structure *is* a world. So any "inhabitants" in the sense of being in a model world with described parameters etc. literally are real and their world is real, there is not and cannot be a difference. So "reaching" such another world is beside the point to such proponents, they think that pure reason shows the illogic and invalidity of distinguishing real and conceptual/mathematical.

I find that incredible and think our experiential qualities are only possible in a real world.

As for GR and curved space, there are similar ambiguities at a lower level. For example, you can pretend that gravity changes lengths etc. instead of making space "really curved". It seems to be equivalent unless you want a closed hypersphere etc.

GW said...

Thanks for the phase space link. GW

Plato said...

It All Goes Back to Plato

Tegmark believes in an extreme form of Platonism, the idea that mathematical objects exist in a sort of universe of their own. Imagine that, Tegmark says, “there’s this museum in this Platonic math space that has these mathematical objects that exists outside of space and time,” Tegmark says. “What I’m saying is that their existence is exactly the same as a physical existence, and our universe is one of these guys in the museum.”Wigner’s Gift Horse By JULIE REHMEYER • Feb 1, 2008 See here for article.

Phil Warnell said...
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Phil Warnell said...

Hi Kris,

Thanks for the clarification, though I already took it you meant spatial dimension. Then again there still could be a difference between being interchangeable as opposed to indistinguishable when it comes to dimensions, with even Einstein reminding that physical space is what one measures with rods while time with clocks, as to have still distinguished distance from duration.

This relates back to Bee’s point, that so often abstract spaces are taken by many as relational only in the physically spatial sense, with this being a self imposed limitation and not an actual one. Then to be more general, spaces are things which are quantified by their dimensions, while a dimension qualified by the type and degree(s) of freedom it affords. I then don’t see that being unidirectional as being unreasonable in terms of a degree of freedom respective to a dimension, if it not being a spatial one. That is it has long been evident to me that reality’s form and actions to be also reliant on spaces having dimensions other then only those which can be measured with rods or clocks.

Further I find this to be in line with Einstein’s admission, which I related previously to Neil, where he indicated he had changed his mind when expressing to Heisenberg “He thought that every theory in fact contains unobservable quantities. The principle of employing only observable quantities simply cannot be consistently carried out.” So contrary to the prevailing view, I’ve been made aware that in his later years he was as dissatisfied with his own theories as he was with quantum mechanics, so to be convinced something new entirely was required to replace them all.

There are many who claim that their current endeavours are in the spirit of his quest, yet I can find this hard to believe when it’s primarily only relativity that is taken as needing to be modified or discarded. This of course is not something I’m charging you with as from what I understand you have utilized the thoughts of Bohm within your own efforts.

Best,

Phil

Kris Krogh said...

Hi Phil,

Yes, it's hard to know exactly where Einstein stood on many questions. I think most people are unaware how many times he changed his mind on basic principles. He told Bohm he thought his theory was "too cheap," but also told others “If anyone can do it, then it will be Bohm.”

(It was Einstein who secretly arranged Bohm's visa and passage to Brazil, when he was faced with prison a second time for refusing to testify in the McCarthy hearings. Maybe you've seen this in Bohm's biography by Peat.)

Just returned from a really cool conference on de Broglie/Bohm theory.

Best, Kris

Phil Warnell said...

Hi Kris,

Well first I must say as I’ve remarked to Bee, regarding some of the conferences she has attended, how I wished I could have been but just a fly on the wall. Unfortunately as I understand it this one has had its down side as well, with all the controversy respective of Valentini about as to who were permitted to attend or rather what was seen as appropriate to be discussed. I also noticed neither Goldstein nor Durr being present, which I find disappointing as these two primarily being the ones to have kept the approach alive since the passage of Bohm, especially considering the sad premature exit of J.S. Bell.

However, I do see many others with which I’m familiar, such as Travis Norsen, with whom I’ve had several web discussions in the past. I also noticed the founding director of PI Howard Burton attended, which has me intrigued regarding his own ontological center .

Anyway I somehow get the feeling that the approach altogether is beginning to factionalize, with the majority moving to be first order DeBroglists and the minority second order Bohmians. Although the former seemingly being admittedly the less problematic take on things, it has the consequence of having the concept of the quantum potential vanish or be marginalized, which I personally think to be a mistake. I do notice however that Norsen is making some effort to find some common middle ground to have it remain part of its conceptualization. Isn’t it strange that a self admitted, unashamed objectivist to be the one to keep the Platonian aspect of the theory alive? :-)

Best,

Phil

P.S. Bee I apologize for drifting off topic, yet I seldom find myself in a space containing another Bohmian; other then configuration space that is:-)