This article frames some important computing developments of the 1950s, connecting them to two older traditions, one within (mathematical) logic and one within engineering. Both traditions could be termed logical minimalism, meaning the systematic use of (mathematical) logic in designing minimal systems and devices. The logical tradition is part of the more general research program into the foundations of mathematics and logic that was carried out in the beginning of the 20th century. The engineering tradition then emerged during the 1930s to design relay circuits and is part of a more general trend of using mathematical techniques in engineering. In the 1940s and 1950s, however, these traditions were redefined and (re)appropriated when computer engineers, logicians and mathematicians started searching for the small(est) and/or simple(st) machines with an eye on engineering a small and relatively cheap digital computer. The main focus is on how the transition of symbolic machines such as Turing's into real computers integrates minimalist philosophies as parts of more complex computer design strategies. The ensuing tradeoffs such as more involved and complex programming or a need for more memory for efficient operation are also discussed.