Your tactile nervous system responds to pain, temperature and pressure. However, all of this information bottlenecks at the level of attention — only some of it gets through to further processing (the “focus”). So the feeling of coldness may be causing more “cold temperature signal” to travel up the limited bandwidth of the nervous system and “crowd out” some of the pain signal coming from the chemical irritation of the eyes. In this case, it could be interference at the level of attention.
If you understand computers, it’s kind of like a DDoS attack plugging up the entrance to “attention” and limiting the access that the pain signal has.
(This is also why you tend to grab onto a painful spot — the neurons responsible for sensing pressure start signaling, and it crowds out the neurons that are emitting pain signal on the limited pathway).
Think of the cells of an onion like little rectangular ketchup packets. Its the “ketchup” inside of those packets that causes your eyes to water and sting. When you cut an onion up you cut through those “ketchup packets” and get “ketchup” all over your hands and knife, which exposes it to the air, and then your face. By running your hands under water you wash away the “ketchup” so there is no more irritant to get in your eyes. Also, since the cells are rectangular shaped, you can cut stem to stem to reduce the number of packets you cut open, reducing the amount of “ketchup” you get in your face.
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