Why did Lincoln refuse to withdraw troops at Fort Sumter?

In the context of the U.S Political environment in 1861, Lincoln felt it was crucial for him to maintain a federal garrison at Fort Sumter, Charleston, South Carolina. Lincoln believed that contrary to the 'Deep South' states, the 'Upper South' states of Virginia, the Carolinas and border states such as Kentucky and Missouri. So he decided that this high-stakes political, strategic gamble would pay off. Despite his advisers wishing otherwise, Lincoln knew that were he to retreat from such a significant strategic location, he would not only lose the support of the Republican Party but much of the North too.

In 1812, America got involved in the Anglo-French war which was sparked following tensions between Britain and France due to the Napoleonic wars (which continued until 1815) and the French Revolution. During the Napoleonic war, America realised that it had a need to establish a strong coastal defence system. This created a need to fortify it's defence line, especially along the Eastern coast where their defences were not as strong. The sea-forts that were built as a result of this need for defence were termed 'the Third System' styled forts, which were pentagonal, three tiered and had walls 5 feet thick. Fort Sumter was the result of the rapid building of the third system. Building started in 1829, but due to a lack of funds, by 1860, when South Carolina left the union, the fort was still incomplete. The harbour commander noticed the changing of political ideals and thus abandoned a smaller fort nearby and relocated those troops and artillery weapons to help bolster Fort Sumter's defences. His inkling was right, and in 1861 the Confederate State of America was formed, which left the 85 strong (and 60 artillery units which they were too short-staffed to man) Union garrison posted in Fort Sumter in hostile territory. All their artillery was, naturally, directed towards the sea as a coastal garrison. The Confederates demanded the surrender of the fort multiple times but the Union soldiers refused, and in January 1861, an attempt to restock food and supplies was thwarted by military forces. In March, Abraham Lincoln promised supplies to the fort which heightened tensions between both sides and as April came around, the Union soldiers were running out of food and beginning to panic. In the early morning of April 12th, the Union garrison started being bombarded by the confederates, and after 34 hours and approximately 30,000 shots. Nobody on either side had died by the time that they surrendered Fort Sumter, but the actions of both sides ignited the fire of the 4 year Civil War and is officially recognised as the first battle. The confederates held onto Fort sumter until 1865, towards the end of the civil war.