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Your thoughts on Francis Schaefer?

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To be honest, most of your explanation on Finsler spaces was above my head.  I'm just now taking Differential Geometry, and we've just gotten to the Riemann metric and the Levi-Civitra connection. :)

I only mentioned it because of what you said about convex polygons. Anyway, if you are now studying the Riemannian metric and the Levi-Civita connection then you already know enough to understand the essence of what I said. One of the axioms that a Riemannian metric must satisfy is that it be symmetric, i.e., g(UV) = g(VU). This means that the inner product U.V is the same as V.U. In a general Finsler space this axiom is relaxed and the metric can be nonsymmetric, i.e., g(UV) not = g(VU). This might sound strange -- and, indeed, it is somewhat unusal -- since we usually think of the segment from point A to point B to be the same as the segment from point B to point A. This is one of the features of a Finsler space that makes it so useful for certain odd physics theories which incorporate some form of anistropy. For instance, there are theories that agree with relativity in the two-way speed of light (light from A to B and back to A), but assert anisotropic behavior in the one-way speed of light (from A to B differs from B to A). Such theories use a different metric from relativity.

Similarly with the Levi-Civita connection you are studying. A connection is considered to be symmetric if the torsion tensor vanishes. This is the case for your Levi-Civita connection, which is the standard connection for general relativity. But there are other gravitational theories, such as teleparallelism, which use a nonsymmetric connection like the Cartan connection. So in general relativity there is a non-zero curvature tensor and a zero torsion tensor, while teleparallelism has a vanishing curvature tensor but a non-vanishing torsion tensor. This makes the difference between a curved spacetime in one gravitational theory and a flat spacetime in another approach.

I'll certainly look into Finsler spaces when I have the background for it, though.
I do not want to oversell it to you. Finsler spaces are a bit peculiar and have limited applicability to traditional physics. They are, nonetheless, interesting. If you do pursue it at all, one nice book, which is really a classic, is Metric Methods in Finsler Spaces and in the Foundations of Geometry, Herbert Busemann, Princeton University Press, 1942. This will not give you the physical principles directly related to physics, but it certainly lays down the mathematical principles of Finsler spaces.

And I completely agree with your sentiment about the interconnectedness of math and physics-- and in fact I better enjoy studying areas of mathematics that are directly motivated by physics and physical objects as opposed to some of the highly nonconstructive areas of mathematics where all you have to go on are axioms.  How about yourself?

Yes, in general I would agree. But sometimes esoteric and seemingly unconnected mathematics can have a great physical impact some time after its development.

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Hi, Stephen,

Yes, in general I would agree. But sometimes esoteric and seemingly unconnected mathematics can have a great physical impact some time after its development.
Well, not to say that mathematical theories that have no direct connection to reality aren't useful or never will be. Just that I personally enjoy researching mathematics that has a visual or otherwise physical interpretation, as opposed to, say, abstract group theory. Certainly they all have their purposes, though.

Oops, Nate. I meant to ask but forgot ... how are you enjoying the Budapest Semester? What is the mix of students?

The prgram itself feels like it should be grad school since they go very fast, except that they grade much easier. I really like the more relaxed approach of learning mathemtics through optional homework, even though it does put a lot of stress onto the final exams. Also, I like the fact that everything is so cheap over here, since the forint is so inflated. :D

The students are all very smart, of course, and the part I like the most is that I haven't really run into the hyper-competetiveness that can be part of a group of really smart students. I like it better when I can collaborate with my peers on homework assignments and bounce ideas off of them in general.

Something that I've been meaning to ask (if you don't mind, that is): Are you a professor? If so, in what subject and at which school?

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Hi Stephen,

I recently retired from Caltech where I worked in microbiology for the past fifteen years, but I did not teach.

Wow, that's impressive! Do you recommend that I try applying there for grad school? I'm making my final decisions about where I'm applying now.

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Hi Stephen,

Wow, that's impressive!  Do you recommend that I try applying there for grad school?  I'm making my final decisions about where I'm applying now.

That depends upon how set you are for your thesis work. If you have a very specific thesis in mind, then I would suggest looking at a school that has the best expert in that particular area, assuming he was interested in working with you on your thesis. If, on the other hand, like most graduate students you are more open to possibilites on your thesis work, then I can think of no better school to recommend than Caltech.

We have only about 2000 students in total, undergraduate and graduate, so the 300 hundred or so professors are certainly not thinly spread. At Caltech physics and mathematics are very closely allied, so that would also be a plus for you based on what you said previously about your interest in mathematics motivated by physics. I think currently there are less than forty grad students in mathematics, so it is a very cozy kind of atmosphere. In general, having both studied and worked at Caltech, I would say that even though it is populated with the brightest and the best, it does not suffer from an overly-competitive atmosphere. You can count on going to seminars in physics, and there will be physicists in your mathematics seminars. There is a lot of this cross-pollenization in many fields on campus.

There are many other fine schools all over the country -- Columbia University is another alma mater of mine -- and each have their own educational virtues and charms. But overall I think the whole atmosphere at Caltech just cannot be beat.

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Mostly I'm interested in Analysis, but I've recently become interested in studying convex polygons as well.

This has been niggling at the back of my mind since it was mentioned. I'd hoped to understand from context. What is there to study about convex polygons?

My only exposure has been in computer graphics. Here, convex polygons are grand because they're easy to render. Modern graphics hardware draws triangles quickly, and for any convex polygon you can create a representative fan of triangles by picking any vertex and grabbing all other adjacent series of points to specify triangles without omission or overlap. Concave polygons get ugly and want constructive geometry operations to derive a representative set of triangles.

Similarly, convex surfaces were great with on older rendering hardware that didn't support efficient per-pixel depth sorting. We could draw the polys in a convex closed hull in any order if we culled backward-facing polygons, as no two camera-facing polygons will overlap with a convex hull, and no backward-facing surfaces will be seen with a closed hull. As a bonus we could even sort cc hulls atomically - without sorting one hull's individual polygons against another's individual polygons - so long as the hulls didn't interpenetrate. This is why you used to see a lot of early 3D video games where characters were made up of a bunch of discrete boxes, tubes, eggs and the like with tiny gaps between.

What's the special interest of convex polygons in other disciplines?

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This has been niggling at the back of my mind since it was mentioned. I'd hoped to understand from context. What is there to study about convex polygons?

Well, Nate can tell you about his particular interest, if he so wishes, but in general there exists an enormous amount of work in the properties and relations of convex polygons. The side you see, on the computer end, represents a few applications in image processing and the like, but the analytic work can become rather esoteric. There are, however, continuing applications and research in applications in a broad range of mathematical and physical sciences.

For instance, 2-dimensional partial differential equations is a relatively old subject, with many physical applications motivating the mathematical analysis. Since these sort of equations describe a broad-range of phenomena, an entire array of solution techniques have been employed. One method of solution which depends on separation of variables in the equations, has lead to a whole complex of transform techniques, some quite familiar, like the Laplace transform, and others less familiar, such as the Mellin transform. Anyway, for any given type of transform method, there exists the partial differential equations themselves, the domain to which they apply, and the boundary conditions which apply to the domain. Because of the many technical complexities involved, there are many areas and situations for which these varied techniques fail. Just six or seven years ago, a new approach for solving these class of problems was developed, and more recently this approach was made much more general. The domain in which the partial differential equations were solved, was a general convex polygon, and many theorems and relations about that geometric class were applied. Note that difficulty comes about in different ways, because the kind of convex polygon used in these sort of applications is often a convex polygon in the complex plane. The polygon can there be bounded or unbounded, so the boundary conditions for the complex polygon require careful and special attention.

Anyway, this is just one interesting area where the convex polygon is an essential feature to the development of a pure mathematical technique in solving a class of partial differential equations.

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Just six or seven years ago, a new approach for solving these class of problems [pde's not amenable to classical separation of variables transform methods]was developed, and more recently this approach was made much more general. The domain in which the partial differential equations were solved, was a general convex polygon, and many theorems and relations about that geometric class were applied.
Stephen, what is that method called?
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Stephen, what is that method called?

I refer to it as Fokas' transform method. The original paper is A.S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. R. Soc. Lond. A, V. 453, pp. 1411-1433, 1997. And later paper which expands on it: A.S. Fokas, Two-dimensional linear partial differential equations in a convex polygon, Proc. R. Soc. Lond. A, V. 457, pp. 371-393, 2001.

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Hi, McGroarty,

This has been niggling at the back of my mind since it was mentioned. I'd hoped to understand from context. What is there to study about convex polygons?

Actually, I'm studying a certain integer associated with every convex polygon (which for lack of a better name I call the degree of the polygon). The degree turns out to be an invariant over linear isomorphisms (if not some more general kind of transformations-- I'm looking into it), and so may distinguish some more familiar properties of convex polygons. As Stephen already mentioned, such a thing would be useful since convex polygons come up in lots of other contexts in mathematics.

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In reference to the posts on page one of this thread...

DNA is attempting the same old bamboozle that his been tried a myriad times before. Namely that because humans rely on their senses of perception, and because those senses are fallible, that can only mean that humans can never be "certain" of anything. All of which leaves us in an unknowable universe, skeptical of everything around us, afraid to make any decision whatsoever.

What we are left with is a false choice between "Certainty" and "Skepticism" - "Certainty" being the ability to know everything at any time without the aid of the senses (because they cannot be relied upon). Which breaks down the boundaries between man and the universe, between subject and object, between the knower and that which is to be known; "Skepticism" being the ability to not know anything at any time, with or without the aid of the senses. Which breaks down not only boundaries, but man and the universe, too.

Objectivism refuses to choose between them, because neither position recognizes that the path to knowledge starts with a single step, a choice: I will think. Which means I will be conscious, I will integrate the information given to me by my senses of perception, and I will make my decisions based on the context and breadth of my knowledge.

The key phrase in the above is "my knowledge." I cannot be certain of that which I am not conscious. I cannot be certain if the earth revolves around the sun or its the other way around if I don't know what the earth or the sun are. And to "know" what the earth and the sun are is to know their nature, their relationship, and other facts. But first I must make the decision to focus my mind and my attention to the facts at hand.

To use DNA's example - the plane flying in "weather" (good weather? bad? in an atmosphere? a vacuum? and how do we know what those terms mean?) I cannot be certain of what to do if I've never been in a cockpit before, or I've never even taken lessons. (I've never in fact done either.) To be aware of what airplanes are doesn't make my ability to fly them appear out of thin air. (Excuse the pun, please.) Even Karen Black had to rely on ground control for help - and she worked on an airplane everyday!

I must first make the conscious decision to learn about flight in order to have any certainty about my ability behind the stick of a plane. The more lessons and practice I acquire, the more certain I become.

But man is fallible. He can make mistakes. Does this make me any less "certain" of my abilities? Not if I am intellectually honest with myself in those abilities. I may make a mistake in simulator that I may correct. In other words, I can - by using reason (which is based on my senses of perception) - move from certainty, to uncertainty, back to certainty.

If I design a plane - based on the context and breadth of my knowledge - I can be certain it will fly. So, I test it. It crashes. Suddenly, I am uncertain. What happened? Where did it go wrong? I go back over all the data - my design, the materials I used, the maintenance, the pilot, the flight conditions, etc. If I find something wrong, I can correct it and become certain again.

All this depends on the fact that reality exists and that I know it.

People who choose between "Certainty" and "Skepticism" want man to be infallible. One wants him to be infallibly right, the other infallibly wrong. But the truth is that he is fallible. He makes mistakes. Some honest, some not so honest. But that doesn't mean he takes a "leap of faith" into the world. He senses, he perceives. His senses are his link to the world.

And that is the only true "certainty."

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Can you say that man is finite without implying that there is an infinite source? That the proposal of finite man also introduces the existence of a deity? I am just curious.

No, in the hierarchy of knowledge, infinite is defined in terms of finite, since infinite just means not finite. So it doesn't follow that something infnite exists (and actually, nothing infinite does exist, metaphysically).

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No, in the hierarchy of knowledge, infinite is defined in terms of finite, since infinite just means not finite.  So it doesn't follow that something infnite exists (and actually, nothing infinite does exist, metaphysically).

Three cheers for actually writing a coherent response to an incoherent question. :santa:

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IMHO, the upside of Francis Schaefer is that it is good for Christians to argue that Christianity is based on reason. This is an error, but it can be corrected. It is much worse when Christians dig in and embrace faith as absurd.

The era of Aquinas was way better than the era of Augustine.

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I haven't read Schaefer, so I can merely say that his ideas (compatibility of faith and reason) are hardly new: this is Aquinas' position.

If we accept the terms "reason" and "faith" at face value, then we can say that some Christian theologians long before Aquinas (13th Century) held to the compatibility of faith and reason. We could begin with the earliest of the Christian intellectuals, Justin the Martyr (2nd Century), and move to Origen of Alexandria (3rd Century), and then to the theological maker of the Middle Ages, Augustine of Hippo (died 430). Augustine believed that Christians (the intellectuals, not the "simples") need to use reason in order to understand faith -- and they need faith in order to know what to think about, in one field in particular: ethics.

The important issue, historically, is not whether a particular individual touts the compatibility of reason and faith, but in which direction he wants to move the domain of reason, compared to the most pro-reason position in his time. Augustine was a retreater; Aquinas was an advancer; Schaeffer (based on the little said here) is a retreater (from the Enlightenment generally and Ayn Rand in particular).

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