Recent content by wofsy

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...

This really interests me. Thanks for the references.

Could you list some links. I am interested in reading more about what you said. The thread is closed so conversation ended. My apologies.

Thanks evo - I guess this thread is closed.

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.

For the identity matrix and its scalar multiples every basis is an eigen basis.

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...