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Now this may begin to remind you of Boolean logic — you’re taking inputs of zeros and ones and sending out zeros and ones. In fact, except for the “carry 1” part of the last rule, this looks very much like the truth table for the XOR logical operator:
Input 1
Input 2
Output
0
0
0
0
1
1
1
0
1
1
1
0
It turns out that if you put together certain logical operators in clever ways, you can completely replicate addition in binary, including the “carry 1” part. Here is a schematic for a “half adder”—built by combining an XOR operator and an AND operator — which takes in two single binary digits and outputs a sum and an optional digit to carry.
So the half adder would function as follows:
And since we can build logic gates — physical objects that replicate logical operations — we should be able to build a physical half adder. And, indeed, here is a LEGO half adder built by Martin Howard.
At last we have computation, which is — according to the Oxford English Dictionary (OED) — the “action or process of computing, reckoning, or counting; arithmetical or mathematical calculation.” A “computer” was originally, as the OED also tells us, a “person who makes calculations or computations; a calculator, a reckoner; spec. a person employed to make calculations in an observatory, in surveying, etc.” Charles Babbage set out to create a machine that would replace the vast throngs of human computers who worked out logarithmic and trigonometric tables; what we’ve sketched out above are the begi
Once you have objects that can materialize both Boolean algebra and binary numbers, you can co
You can build logic gates out of any material that can accept inputs and switch between distinct states of output (current or no current, 1 or 0); there is nothing special about the chips inside your laptop that makes them essential to computing. Electrical circuits laid out in silicon just happen to be small, cheap, relatively reliable, and easy to produce in mass quantities. The Digi-Comp II, which was sold as a toy in the sixties, used an inclined wooden plane, plastic cams, and marbles to perform binary mathematical operations. The vast worlds inside online games provide virtual objects that can be made to interact predictably, and some people have used these objects to make computing machines inside the games — Jong89, the creator of the “Dwarven Computer” in the game Dwarf Fortress, used “672 [virtual water] pumps, 2000 [faux wooden] logs, 8500 mechanisms and thousands of other assort[ed] bits and knobs like doors and rock blocks” to put together his device, which is a fully functional computer that can perform any calculation that a “real” computer can.2 Logic gates have been built out of pneumatic, hydraulic, and optical devices, out of DNA, and flat sticks co
Many years after I stopped working professionally as a programmer, I finally understood this, truly grokked this fact — that you can build a logic gate out of water and pipes and valves, no electricity needed, and from the interaction of these physical objects produce computation. The shock of the revelation turned me into a geek party bore. I arranged toothpicks on di
Samuel Morse had demonstrated his telegraph in 1844—ten years before the publication of Boole’s The Laws of Thought …
But nobody in the nineteenth century made the co
Nobody in that century ever realized that Boolean expressions could be directly realized in electrical circuits. This equivalence wasn’t discovered until the 1930s, most notably by Claude Elwood Sha
After Sha