I expect that this is indeed true. Your brain and your nervous system work together so that you can experience pain. The human body has learnt that pain is a bad thing so your body would try to stop you biting your finger because of the pain that it causes to you.
If you could feel no pain at all and you were able to overcome the psychological issues associated with biting off your own finger (in your head I think you would find it hard to do this to yourself), then you would be able to bite it like a carrot.
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