Perhaps the most hilariously surreal moment in the movie occurs when Mr. Potato Head disassembles himself. He has been imprisoned in a box, and the one hole in the box is nowhere near large enough for him to squeeze out through. So he pops his plastic eyes, nose, ears, arms and feet out of his potato-body (head?), tossing them through the hole so they land in a heap on the ground outside. And just as you're wondering, “What good did that do him?” each body part stirs itself and they all scurry away. They're traveling as a group, but a group of distinct individuals; each part moves independently, seemingly possessed of a mind of its own.
“Now wait a minute,” you might object. “Just because the various body parts are moving independently of each other, that doesn't necessarily mean they have separate minds. Maybe Mr. Potato Head still has the one mind, which is somehow able to control his body parts from afar.”
Well, Hypothetical-You, that's an excellent point! But it's provably false, even within the fantastical and unspecified logic of the Toy Story universe. If you pay close attention, you'll notice a subtle clue that rebuts your one-mind hypothesis: A few scenes later, the various body parts have embedded themselves into a flour tortilla (it's a long story; just watch the movie, ok?), which serves as an adequate, if floppy, substitute body for them.
And this is — weirdly, astonishingly — a question that applies to our own minds as well. It's pretty well known that the two hemispheres of a human brain each have different functions, and that they communicate constantly via the corpus callosum which connects them. At first blush, this fact doesn't seem to endanger our belief that we each have a single, distinct mind. But what if you sever contact between the two hemispheres?
This has happened to some people. Called “split-brain patients,” they seem to function normally in most circumstances. But the oddness of their condition manifests when you give a piece of information to only one of their hemispheres, which is then unable to communicate the information to the other hemisphere. How do you do this? By giving the information to only one eye, for example, or to only one hand — sensory inputs from the right eye and hand go only to the left hemisphere, and from the left eye and hand only to the right hemisphere.
I wish I could comment more intelligently on this conclusion, but so far I haven’t managed to muster anything more coherent than “!!!?!!??!” If you want to read more about the split-brain phenomenon and what we should make of it, I'd recommend Derek Parfit (Reasons and Persons, Part III) and Thomas Nagel ("Brain Bisection and the Unity of Consciousness," in his Mortal Questions). And if you figure it out, come back and explain it to me.
Coming up in Part II: Lots-o-Huggin' Bear and Philosophy.