This example and many other illustrate that geometric arguments cannot always be completely replaced by algebraic ones, much like the fundamental theorem of algebra does not seem to have a simple purely algebraic proof. (I'm out on a limb with this statement.)

It looks to me that a large part of the fundamental functors of algebraic topology have a geometric origin; think homotopy, (co)homology, cobordism, $K$-theory. I cannot imagine formal arguments, devoid of geometric intuition leading to such concepts. Obviously geometric arguments alone cannot get you very far; think homotopy, (co)homology, cobordism theory, $K$-theory without long exact or spectral sequences.

Being a mathematical "mutt" myself, I always favor impure arguments. They give me comforting feeling of not being isolated. Also, they broaden my sources of inspiration.

This example and many other illustrate that geometric arguments cannot always be completely replaced by algebraic ones, much like the fundamental theorem of algebra does not seem to have a simple purely algebraic proof. (I'm out on a limb with this statement.)

It looks to me that a large part of the fundamental functors of algebraic topology have a geometric origin; think homotopy, (co)homology, cobordism, $K$-theory. I cannot imagine formal arguments, devoid of geometric intuition leading to such concepts. Obviously geometric arguments alone cannot get you very far; think homotopy, (co)homology, cobordism theory, $K$-theory without long exact or spectral sequences.

Being a mathematical "mutt" myself, I always favor impure arguments. They give me the comforting feeling of not being isolated. Also, they broaden my sources of inspiration.

This example and many other illustrate that geometric arguments cannot always be completely replaced by algebraic ones, much like the fundamental theorem of algebra does not seem to have a simple purely algebraic proof. (I'm out on a limb with this statement.)

It looks to me that a large part of the fundamental functors of algebraic topology have a geometric origin; think homotopy, (co)homology, cobordism, $K$-theory. I cannot imagine formal arguments, devoid of geometric intuition leading to such concepts. Obviously geometric arguments alone can get you very far; think homotopy, (co)homology, cobordism theory, $K$-theory without long exact or spectral sequences.

Being a mathematical "mutt" myself, I always favor impure arguments. They give me comforting feeling of not being isolated. Also, they broaden my sources of inspiration.

This example and many other illustrate that geometric arguments cannot always be completely replaced by algebraic ones, much like the fundamental theorem of algebra does not seem to have a simple purely algebraic proof. (I'm out on a limb with this statement.)

It looks to me that a large part of the fundamental functors of algebraic topology have a geometric origin; think homotopy, (co)homology, cobordism, $K$-theory. I cannot imagine formal arguments, devoid of geometric intuition leading to such concepts. Obviously geometric arguments alone cannot get you very far; think homotopy, (co)homology, cobordism theory, $K$-theory without long exact or spectral sequences.

Being a mathematical "mutt" myself, I always favor impure arguments. They give me comforting feeling of not being isolated. Also, they broaden my sources of inspiration.