Left align (to the page) displayed math

You can use flalign from the amsmath package:

Note:

• The showframe package was used just to be able to see where the text is relative to the page margins.

Code:

\documentclass{article}
\usepackage{showframe}
\usepackage{amsmath}
\begin{document}
\begin{flalign*}
e & = mc ^2 &\\%  Note the trailing & which is required to equations to the left
F & = ma
\end{flalign*}
\end{document}

---

Alternatively, if you want all equations left aligned, you can use the package option fleqn as in \documentclass[fleqn]{article} which will align them towards the left. The indentation is controlled by \mathindent, so you could use \setlength{\mathindent}{0pt} if you wanted to eliminate that.

Code:

\documentclass[fleqn]{article}
\usepackage{showframe}
\usepackage{amsmath}
\setlength{\mathindent}{0pt}

\begin{document}
\begin{align*}
e & = mc ^2 \\
F & = ma
\end{align*}
\end{document}

It seems that the current accepted answer doesn't exactly answer the question (OP stated either $$ or \[ and not the align environment because he already found a solution for that).

Here is a solution with \[ ... \] (doesn't work with $$...$$ though):

\documentclass[fleqn]{article}
\setlength{\mathindent}{0pt}
\begin{document}
\noindent Here is some left-aligned math:
\[x^2+1\]
\end{document}