First : We take the term visualization to describe the process of producing or using geometrical or graphical representations of mathematical concepts, principles or problems, whether hand drawn or computer generated.
Quote:
“Diagrams are, of course, as old as mathematics itself. Geometry has always relied heavily
on pictures, and, for a time, other branches of mathematics did too. Even Isaac Newton ...
did not actually prove [the] fundamental theorems. ...Had you asked him to justify them, he
would likely have presented an argument that, though compelling, was loose and depended
heavily on pictures.”
When I took courses in algebra (not linear algebra - I mean groups, rings, modules etc.) or representation theory as an undergraduate, I found it practically impossible to get anywhere by trying to visualize what was going on.
Quote:
"in mathematics ... we find two tendencies present. On the one hand, the tendency toward
abstraction seeks to crystallize the logical relations inherent in the maze of material that is
being studied, and to correlate the material in a systematic and orderly manner. On the other
hand, the tendency toward intuitive understanding fosters a more immediate grasp of the
objects one studies, a live rapport with them, so to speak, which stresses the concrete
meaning of their relations.
"...With the aid of visual imagination [Anschauung] we can illuminate the manifold facts
and problems of geometry, and beyond this, it is possible in many cases to depict the
geometric outline of the methods of investigation and proof ... In this manner, geometry
being as many faceted as it is and being related to the most diverse branches of
mathematics, we may even obtain a summarizing survey of mathematics as a whole, and a
valid idea of the variety of its problems and the wealth of ideas it contains."
In criticizing myself , I led to this:The fact that I am good in visualizing doesn't necessary mean that I should ignore fields where visualization is less important. If I will go into areas where visualization is very important, then you'll be surrounded by other people with such skills and you might lose you relative advantage. But in a field where visualization is used less frequency, I could have an advantage in problems where visualization is required, and in most areas of mathematics visualization is useful to some extent.
In any case, as I am in the beginning of my Mathematics career, I should try to obtain a broad basis rather than specialize in some direction where I develop my skills in other areas
To support my subjective view:
Quote:
The intuition which mathematical visualization seeks is not a vague kind of intuition, a superficial
substitute for understanding, but the kind of intuition which penetrates to the heart of an
idea. It gives depth and meaning to understanding, serves as a reliable guide to problem
solving, and inspires creative discoveries. To achieve this kind of understanding,
visualization cannot be isolated from the rest of mathematics. Visual thinking and graphical
representations must be linked to other modes of mathematical thinking and other forms of
representation. One must learn how ideas can be represented symbolically, numerically, and
graphically, and to move back and forth among these modes. One must develop the ability
to choose the approach most appropriate for a particular problem, and to understand the
limitations of these three dialects of the language of mathematics. We have thus strongly
encouraged the authors of these papers to show how visualization operates in a
mathematical context and not as an isolated topic
But I killed my own thread?
Maybe I should just continue on , I know
-
Linear Algebra is a subject
-
Hopf algebras did not come alive until I understood that the axioms could be drawn as little bits of string.
-
Graph theory
-
Mechanics
-
Arithmetic Geometry
-
Some parts of Combinatorics
-
Topology
-
Differential Geometry
-
Geometric Analysis
-
Sketches in category theory,
-
The various classes of visual inference in logic, such as Peirce's existential graphs
-
Fractal Geometry - thanks to Computational Geometry
-
Computerized Nonlinear Dynamics
-
Color Graphics[b]
Their are fundamental visualization issues which cut across different subject fields. The remaining
papers address the role of visualization in particular fields of mathematics - geometry,
calculus, differential equations, differential geometry, linear algebra, numerical analysis,
complex analysis, stochastic processes and statistics.
Not exactly fields but have led to developments
- [b]Curves
-
Surfaces
-
direction fields
-
contour plots
-
other kinds of schematic diagrams
-
Polytope Families
-
Tessellations
-
Polychorons
-
geometrical constructions over the complex numbers
Last edited by JSNS on 13 Nov 2011, 10:53 am, edited 1 time in total.
12 Nov 2011, 7:24 pm
