Place the potatoes in a large pot and fill with enough cold water to easily cover. Season the water with 1 teaspoon salt and bring to a boil. Cook the potatoes until tender all the way through (The tip of a paring knife inserted into a potato will meet little resistance). Drain. When cool enough to handle, chop the potatoes into 3⁄4" pieces. Place in a large bowl and add the celery, pickles, onion, mustard, mayonnaise, vinegar, and eggs. Toss to coat. Season with salt and pepper and sprinkle with the paprika (if using).