4 bit computer built from discrete transistors

If someone were to provide you with variously doped n-type, p-type semiconductor, with enough time you could make a transistor by trial and error. The key I guess is that you probably wouldn't have considered using (or even had for that matter) n-/p-type material unless you had a background in solid state physics in an industrial lab in the 1940s (i.e. everyone associated with Shockley). But at that point, and being in such an organization, wisely you'd be inclined to work out the math & science first on blackboards, and thus economize your limited person-hours and R&D budget.

As an aside, and probably the main reason no one would ever have created a transistor or even IC in the 19th century or earlier is because, they really didn't need one yet. A transistor controls electricity (in a circuit) using electricity (from another ideally isolated circuit). In the 19th century, mechanical relays existed which controlled electricity using mechanical means (which could in turn have been controlled by electricity). This was "good enough" for most applications. There was not yet any large-scale telecommunication system (requiring better low-noise, low-power signal amplification), large populations within nation-states (requiring high speed digital data processing by corporations & governments), or potential for world wars that could destroy the planet many times over (requiring data secrecy and cryptanalysis by nations only fighting in proxy aka Cold Wars). These were 20th century and, remain at this point, 21st century concerns that benefitted from, and in some cases sadly, were enabled by the transistor.