All presidents since Washington tried to steer the country either into a forward or backward looking direction.
Presidents such as Jefferson, Jackson and Pierce tried to look backward and restore the country to what they viewed as the Republican golden age by keeping government small, weak and as out of people's lives as much as possible. They generally opposed having a central bank that could print money, supported only state militias to protect the country and tended to put as much decision making as possible into local communities.
Other presidents such as Adams, Polk and Garfield sought to look forward and modernize the country as much as possible through maintaining standing armies, supporting a strong central bank of the United States and 'internal improvements' which were usually public works projects such as railroads, canals or dams.
Arthur's legacy would certainly put him in the latter category as a modernizing president. He sought to continue Garfield's legacy of making the civil service more professional by making the non political positions be merit based rather than simple patronage, he pushed for time zones to be created in the United States, and even pushed for an early form of NAFTA by encouraging trade treaties with Nicaragua and Mexico.
In an age of even more suspicion and anger over immigration than our own, Arthur tried to create something even more powerful than NAFTA by pushing for a common currency for all of North and South America or a sort of United States Euro zone to further international trade.
This push for a common continental currency obviously didn't pass in the United States or Latin America and still would not today, but it really speaks to how dedicated to new ideas Arthur was.