Feb 19 2012: For the moment, the answer is no. We cannot know that we perceive the world similarly or differently, we can only conjecture based on the (dis)similarity of observations made by various individuals. However, we do know that experiencing similar events in different ways will often have the effect of causing a person to be less subjective about the way they and others react to said stimuli. But even the most objectively driven minds will have difficulty understanding the views of another when they don't have the requisite knowledge and/or experience.
When science has advanced to the point where we understand how the brain is constructed, we could theoretically modify some of the brain's circuits, thus solving this dilemma. The medium itself is completely different, but the actions performed would be similar to swapping out IC chips on a motherboard. If you can figure out which chips should be substituted where, it is possible that the motherboard would still function the same, even though the chips chosen might be completely different. Since the brain is more fault tolerant than a motherboard, you should also be able to swap in a set of circuits which are functionally similar, but ultimately different. After the brain has adapted to the new input and output created by the modified function, you might see things from a different perspective.
You could use this knowledge to change how a person's brain processes information, thereby bridging the understanding gap without subjecting a person to the events which created the circuitry in question. For example, you could make a petty thief decide on their own that stealing is wrong; just modify the circuits related to theft, and make them feel revulsion rather than enjoyment. However, abuse of such a technology carries equally weighty implications, which makes understanding the human brain a dangerous endeavor. If negative modifications could be easily scaled and propagated, the result might be worse than an atom bomb.
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A comment on Conversation: Can we ever know how another person "senses" the world?
When science has advanced to the point where we understand how the brain is constructed, we could theoretically modify some of the brain's circuits, thus solving this dilemma. The medium itself is completely different, but the actions performed would be similar to swapping out IC chips on a motherboard. If you can figure out which chips should be substituted where, it is possible that the motherboard would still function the same, even though the chips chosen might be completely different. Since the brain is more fault tolerant than a motherboard, you should also be able to swap in a set of circuits which are functionally similar, but ultimately different. After the brain has adapted to the new input and output created by the modified function, you might see things from a different perspective.
You could use this knowledge to change how a person's brain processes information, thereby bridging the understanding gap without subjecting a person to the events which created the circuitry in question. For example, you could make a petty thief decide on their own that stealing is wrong; just modify the circuits related to theft, and make them feel revulsion rather than enjoyment. However, abuse of such a technology carries equally weighty implications, which makes understanding the human brain a dangerous endeavor. If negative modifications could be easily scaled and propagated, the result might be worse than an atom bomb.