No it is not possible to redefine them the way you want. However, if you write your maths correctly you can use either of them without any problem. Consider this code:
\documentclass{article}
\begin{document}
\fbox{
\begin{minipage}{5cm}
This is a test
$$ \sum_{i=1}^{10} t_i $$
and
$$ \sum_{i=1}^{10} t_i $$
and another
\end{minipage}}
\fbox{
\begin{minipage}{5cm}
This is a test
\[ \sum_{i=1}^{10} t_i \]
and
\[ \sum_{i=1}^{10} t_i \]
and another
\end{minipage}}
%% This will make a slightly different layout
\fbox{
\begin{minipage}[t]{5cm}
This is a test
$$ \sum_{i=1}^{10} t_i $$
and
$$ \sum_{i=1}^{10} t_i $$
and another
\end{minipage}}
\fbox{
\begin{minipage}[t]{5cm}
This is a test
\[ \sum_{i=1}^{10} t_i \]
and
\[ \sum_{i=1}^{10} t_i \]
and another
\end{minipage}}
\fbox{
\begin{minipage}{5cm}
This is a test \math x=y+5 \endmath\ and \ldots
This is a test $ x=y+5$ and \ldots
This is a test \( x=y+5 \) and \ldots
\end{minipage}}
\end{document}
The first two boxes are identical, as they have been written correctly as a single paragraph. The second one the LaTeX way, as you observed in your question is a bit "loose". If you are aware of these differences you can use $$ without any issues. For inline math I always use $ and so far it hasn't caused the universe to collapse into a black hole!
Caveat: If you use any maths packages or amsmath you will be well advised to rather stay with \[..\], although for the example above just using usepackage{amsmath} will provide consistent results for both cases, but will fail if you use the option fleqn. See also Is $ ... $ okay to use, while $$ ... $$ is not recommended? and Why is \[ ... \] preferable to $$ ... $$?.
\[...\]themselves are defined in terms of$$. Or am I wrong? – Werner Oct 06 '11 at 01:46$$into\[and\]. – topskip Oct 06 '11 at 05:56$$[tab] into\[\].$$($twice keeping shift pressed), while\[takes only two; and any LaTeX user knows well where\and[are. – egreg Oct 09 '11 at 11:04\sum_{i=1}^{10} t_iis between the display mode; this makes me think that you may not be aware of\displaystyleas in$\displaystyle \sum_{i=1}^{10}$, or its equivalent\( ... \)which can be used inline. – André Apr 20 '12 at 13:06