Mind, Body, World- Foundations of Cognitive Science, 2013a

Vannevar Bush. This machine was a pioneering analog computer, a complex array
of electrical motors, gears, and shafts that filled an entire room. Its invention established
Bush as a leader in electrical engineering as well as a pioneer of computing
(Zachary, 1997). Bush, like Shannon, was enamored of the link between the formal
and the physical. The sight of the differential analyzer at work fascinated Bush “who
loved nothing more than to see things work. It was only then that mathematics—his
sheer abstractions—came to life” (Zachary, 1997, p. 51).
Because of his work with Bush’s analog computer, Shannon was prepared to
bring another mathematical abstraction to life when the opportunity arose. The
differential analyzer had to be physically reconfigured for each problem that was
presented to it, which in part required configuring circuits that involved more than
one hundred electromechanical relays, which were used as switches. In the summer
of 1937, Shannon worked in Bell Labs and saw that engineers there were confronted
with designing more complex systems that involved thousands of relays. At the
time, this was labourious work that was done by hand. Shannon wondered if there
was a more efficient approach. He discovered one when he realized that there was
a direct mapping between switches and Boolean algebra, which Shannon had been
exposed to in his undergraduate studies.
An Internet search will lead to many websites suggesting that Shannon recognized
that the opening or closing of a switch could map onto the notions of “false”
or “true.” Actually, Shannon’s insight involved the logical properties of combinations
of switches. In an interview that originally appeared in Omni magazine in 1987, he
noted “It’s not so much that a thing is ‘open’ or ‘closed,’ the ‘yes’ or ‘no’ that you mentioned.
The real point is that two things in series are described by the word ‘and’ in
logic, so you would say this ‘and’ this, while two things in parallel are described by
the word ‘or’” (Liversidge, 1993).
In particular, Shannon (1938) viewed a switch (Figure 2-1A) as a source of
impedance; when the switch was closed, current could flow and the impedance
was 0, but when the switch was open (as illustrated in the figure) the impedance
was infinite; Shannon used the symbol 1 to represent this state. As a result, if two
switches were connected in series (Figure 2-1B) current would only flow if both
switches were closed. Shannon represented this as the sum x + y. In contrast, if
switch x and switch y were connected in parallel (Figure 2-1C), then current would
flow through the circuit if either (i.e., both) of the switches were closed. Shannon
represented this circuit as the product xy.
Shannon’s (1938) logical representation is a variation of the two-valued logic
that was discussed earlier. The Boolean version of this logic represented false with
0, true with 1, or with addition, and and with multiplication. Shannon’s version
represented false with 1, true with 0, or with multiplication, and and with addition.
But because Shannon’s reversal of the traditional logic is complete, the two are
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