Sort through the dried split peas and discard any grit or discolored peas. In a bowl, place peas and enough water to cover. After a minute, remove any peas or skin that float to top. Rinse split peas and drain well.
In a large pot over medium heat, heat olive oil. Add onions, garlic, carrots and celery. Cook, stirring regularly, until softened.
Add water and bring to a boil.
Add dried split peas, ham bone, and bay leaves. Bring to a boil, skimming off scum that may float on top.
Lower heat, cover and simmer for about 1 1/2 to 2 hours or until peas are tender.
In the last 30 minutes, add potatoes and cook until very tender. Using back of spoon, mash potatoes to help thicken soup.
Remove ham bone and scrape meat from bone. Chop ham meat and return to pot.
Season with salt and pepper to taste. Ladle into serving bowls and serve hot.