Once the manifold is embedded in a n-dimensional Euclidean space it has a whole set of normal directions. If you fill out the solid tube along these normal directions you will end up with an n-dimensional solid with an n-1 dimensional boundary. For instance a circle in R^3 has a plane of normal...
http://books.google.com/books?id=osEsAAAAMAAJ&pg=PA301&lpg=PA301&dq=plutarch%27s+lives+sundial&source=bl&ots=Dr2hRliuaJ&sig=1uLvZkjSKmIIC7Ct_pDjUjdJFpw&hl=en&ei=ZgyCS_2iLMyUtgftja3TBg&sa=X&oi=book_result&ct=result&resnum=10&ved=0CBwQ6AEwCTgK#v=onepage&q=&f=false
here is a partial reference that...
I think it has exactly to do with what I was saying. Cicero says the same thing. I may have forgotten the author. It is an old memory. Why do you say it is nonsense?
In any case, the idea that knowledge requires a purpose is the essential point and is certainly entertained in ancient times as a...
I don't think it is obvious but here is a complicated picture.
Imagine the manifold embedded in some Euclidean space - for an orientable surface this would just be R^3 - and imagine a solid tube surrounding the manifold. For instance around the sphere the tube would be a spherical shell...
I think Plutarch's point was that one does not have full knowledge of a phenomenon without a theory that is consistent with a rational design. That it is man made is really a metaphor for a requirement for true knowledge.
While his point of view is maybe simplistic it was later abstracted into...
The size of the loop doesn't matter and the location of the pole doesn't matter. The key is that once you choose an interior region then the index is well defined at any singularity in that region.
I think of it this way. Maybe it will help you. Choose an interior region that the circle bounds...
Interesting. Could you elaborate an example of objective invariance?
It does seem that the demand for invariance is a signature trend in physics but I would like to be clear on your point.
I also do not know anything about logic so more explanation would help.
BTW: It seems that Einsein...
The equator of the sphere encloses a singularity at either pole. The index of the singularity is independent of the coordinate chart's orientation. This is not a problem.
While I know next to nothing about the philosophy of science, I would think about scientific models and explore whether there are features of these models that should be required in order to say they make a good theory or whether any model that predicts the data is as good as any other. From...
I think Descartes was saying that all explanations of the empirical world, the world that we recognize with our senses, are always subject to doubt. But there is no doubt that I exist when I am doubting my existence. That is his point.
For Descartes the statement that the Universe exists is...
In algebra a field is Halmos's field. It is an algebraic structure. Fields are distinguished from other algebraic structures such as rings, groups, vector spaces, and algebras.
In calculus, a field is the assignment of a quantity to each point of a domain. A scalar field assigns a number, a...
I agree with you that Rudin's books are not for beginners. In fact they are too hard for most mathematicians.
I also agree that Complex Analysis is de rigeur though Alfors is a bit technical. But reading his book will also introduce point set topology and the idea of homotopy and homology.
I...
I was not all offended or even piqued. No apology needed.
I should have elaborated.
Here are some examples.
-The movie Roots portrays Africans living in a state of harmony and love until their natural society is uprooted by slave traders who for profit violate man's nature by preventing him...
There is an example of a geodesic on a fluted surface of negative curvature that winds almost all of the way down the surface circling around it in a helical motion then turns around and comes back! The shortest geodesic though between two adjacent points is a simple arc. I will try to look this...
For the What???? I was just explaining what the historical view was. You are not understanding the statement.Natural rights have nothing to do with compassion or atrocities or the right to own land.
BTW: What is wrong with atrocities? Why is compassion good? You seem to be assuming that your...
In my post I proposed the idea that natural rights are the right to persue one's nature. This is an old view and generally assumes that man is naturally good. The problem for us perhaps is to think about what our nature is without theological assumptions.
I mean that observation suggests mathematical theorems. Creating formulas is not what mathematics is about.
The case of the Dirichlet Principle well represents what I mean. It would be helpful for me to go through it to see why it's discovery is not an example of inductive reasoning from...
The Cauchy Integral Theorem says that for an analytic function in a domain, its value at any point in the interior of a domain can be determined from its values on the boundary of the domain. Powerful stuff.