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New prosthetics are controlled by brain waves which seem to make the prosthetic an extension of the body in an "organic" way, in that we control all of our own limbs through the same brain waves and impulses. [scientists -- sorry, my understanding is rudimentary -- if I am misrepresenting anything, please let me know]

This bit of Purim Torah (with actual sources) seems to point to the idea that telekinesis (movement of an object using the brain) would be allowed on Shabbat, but the question is about moving something external to the body and one opinion questions if using brain waves is k'lachar Yad. Moving a body part is not exactly the same, as brain impulses are the normal way of moving body parts.

This question asks about prothetic limbs in halacha but doesn't address this aspect of controlling their electronics when the electronics are integrated into the body (I learned that attached body parts are not muktza but I don't know when a limb is or is not "attached" enough to become a halachic body part, unlike a car).

So would the electronic body part be able to be controlled on Shabbat by the brain of the "wearer"?

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    Excellent job finding and distinguishing from related prior posts! – Isaac Moses Jan 29 at 14:24
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    aren't all natural limbs controlled naturally by "electronics"? – Loewian Jan 29 at 17:11
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    @Loewian Spot on, and so I don’t see what the halachic issue(s) could be. – Oliver Jan 30 at 2:07
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    @Oliver I think the relevant equivalent might be closer to a cochlear implant. Is a bodily adjunct part of the body making the brain waves the natural way of controlling, or still an external device (as one aspect of the halachic issue). – rosends Jan 30 at 11:27
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There are two schools of thought on electricity in Shabbos.

The Chazon Ish holds that closing a switch is boneh, building, and therefore a forbidden labor on Shabbos.

Rav Auerbach argues that the switch is like opening and closing a door, which the Shulchan Aruch says is not binyan in the simple case, it was made to open and close.

Where all agree, is that cooking with electricity is not permitted. In other words, the problem is what you do with electricity, and not so much the electricity in and of itself.

All electro-mechanical devices produce heat. That heat can be the problem, in the case that it might be called bishul (cooking). Hypothetically speaking: For a metal, a threshold for bishul can be when the metal becomes soft, and for fire it is when it gives off glowing sparks. So, it depends on the materials, and the amount of waste heat, and whether it glows or gives out light as a result of the electrical process.

As a practical matter, it becomes a question for a Rov for the individual device.

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