Combine the potatoes, carrot, celery, onion, and salt in a 2-quart pot; pour enough water over the potato mixture to just cover. Bring the water to a boil, and cook until the potatoes are soft but not mushy, about 25 minutes. Drain about 1/4 of the remaining water from the mixture, taking care to not lose any of the vegetables. Stir the milk into the vegetable mixture with the basil, dill, and butter. Season with salt and black pepper to serve.
