Here is a solution using the titlesec package. Add the following lines to your preamble to mimic the section headings of the amsart class.
\usepackage[indentafter]{titlesec}
\titleformat{name=\section}{}{\thetitle.}{0.8em}{\centering\scshape}
\titleformat{name=\subsection}[runin]{}{\thetitle.}{0.5em}{\bfseries}[.]
\titleformat{name=\subsubsection}[runin]{}{\thetitle.}{0.5em}{\itshape}[.]
\titleformat{name=\paragraph,numberless}[runin]{}{}{0em}{}[.]
\titlespacing{\paragraph}{0em}{0em}{0.5em}
\titleformat{name=\subparagraph,numberless}[runin]{}{}{0em}{}[.]
\titlespacing{\subparagraph}{0em}{0em}{0.5em}
As a comparison, here is a sample document typeset with the article class and the titlesec package on the one hand and with the amsart class on the other one.

The code for the document with article+titlesec:
\documentclass{article}
\usepackage[indentafter]{titlesec}
\titleformat{name=\section}{}{\thetitle.}{0.8em}{\centering\scshape}
\titleformat{name=\subsection}[runin]{}{\thetitle.}{0.5em}{\bfseries}[.]
\titleformat{name=\subsubsection}[runin]{}{\thetitle.}{0.5em}{\itshape}[.]
\titleformat{name=\paragraph,numberless}[runin]{}{}{0em}{}[.]
\titlespacing{\paragraph}{0em}{0em}{0.5em}
\titleformat{name=\subparagraph,numberless}[runin]{}{}{0em}{}[.]
\titlespacing{\subparagraph}{0em}{0em}{0.5em}
\usepackage{blindtext}
\begin{document}
\section{First section (article with titlesec)}
\blindtext
\subsection{First subsection}
\blindtext
\subsubsection{First subsubsection}
\blindtext
\paragraph{First paragraph}
\blindtext
\subparagraph{First subparagraph}
\blindtext
\end{document}
The code for the amsart class:
\documentclass{amsart}
\usepackage{blindtext}
\begin{document}
\section{First section (amsart)}
\blindtext
\subsection{First subsection}
\blindtext
\subsubsection{First subsubsection}
\blindtext
\paragraph{First paragraph}
\blindtext
\subparagraph{First subparagraph}
\blindtext
\end{document}
\itshape. Alsoparagraphandsubparagraphare by default not numbered and the font is just\normalfont. – Dan Dec 07 '16 at 22:10