In 1822, Charles Babbage built a clockwork machine for computing various polynomial expressions which he called a Difference Engine, and designed several larger versions for computing larger expressions to more digits of precision. This is generally cited as the first programmable digital computer.

Charles Babbage was trained in mathematics and worked in a variety of fields, including economics, management, theology, meteorology, medicine, mechanical engineering, and cryptography. Arguably his most impactful invention during his own lifetime was a metal frame to attach to the front of locomotives to clear the track of obstacles such as grazing cattle.

Augusta Ada King née Byron, Countess of Lovelace, observed Charles Babbage’s difference engine and understood its potential in a way that Babbage himself did not. Babbage was intent on building what we’d now think of as a calculator: a tool for computing the value of polynomials at various arguments to solve arithmetic problems that arose in various contexts. Ada Lovelace understood that, while the machine he’d built was limited to those arithmetical tasks, with some adjustments a general-purpose version might be built. Working with Babbage, she helped design the Analytical Engine which would have been the first general-purpose programmable digital computer had it been built.

The Analytical Engine was not completed because its original backers, the British government, did not think it was worth the cost. In 1842 the British Prime Minister, Sir Robert Peel, is recorded¹¹ In Additional Manuscript 40,514 folio 223 in the British Library. to have said

What shall we do to get rid of Mr. Babbage and his calculating Machine? Surely if completed it would be worthless as far as science is concerned?

Historians debate whether it was Charles or Ada who wrote the first computer program; both programs are recorded, but the relative dates of the two is unclear.

An example of her understanding of the potential of this yet-unbuilt machine can be found in her 1842 submission to the journal Scientific Memoirs:

Supposing, for instance, that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression and adaptations, the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent.

Ada Lovelace’s background in mathematics that prepared her to make these discoveries was gained as part on her mother’s efforts to distance her from her father, Lord Byron, who was known for writing romantic poetry and for pursuing sexual relations with many people, both men and women, both single and married, including some of his own cousins.

Alonzo Church was the son of a judge and the grandson of a mathematics professor who both attended and spent most of his career working at Princeton University as a professor of mathematics and philosophy.

Alan Turing was the son of a British administrator in India and grandson of both a clergyman and a railway chief engineer. He traveled between England and India often as a child and sometimes stayed at a friend of the family’s home or at a boarding school when his parents traveled. He was recognized early as a genius with diverse interests, but a particular interest in Mathematics. He studied at Cambridge and then Princeton, being one of Alonzo Church’s doctoral students.

Church was interested in a branch of Mathematics related to the fundamental limitations of mathematical systems. A recent proof by Kurt Gödel showed any sufficiently complicated mathematical system could express a version of the statement this statement cannot be proven true, which was self-evidently true only if it was unprovable, demonstrating that such languages could express true but unprovable things. But Kurt Gödel’s version of that was a fairly complicated mathematical construct in a very large mathematical system.

Church was interested in converting Gödel’s work into much simpler systems. He designed a very small mathematical system which could still express that idea, which he called the lambda calculus. His student Alan Turing came up with a simple machine design, the operation of which tightly paralleled what the lambda calculus could do, which was later termed the Turing machine. The two them postulated, but did not prove, that any more-complicated mathematical system could be reduced to the Lambda calculus and any more-complicated machine could be be reduced to the Turing machine. This concept that the two were capable of doing anything that other systems could do is called the Church-Turing Thesis.

After publishing their thesis, Alonzo Church spent several more decades teaching at Princeton, then at the University of California at Los Angeles, publishing many additional papers in mathematics and philosophy. He was generally regarded as a quiet and focused person and did little to attract attention outside of his academic work.

Alan Turing returned to England and worked on cracking German codes during World War II. Shortly after the war ended he was convicted of homosexual acts (then illegal in England) and sentenced to involuntary hormone injection and limitations on his freedom of movement. Three years later he was found dead of cyanide poisoning, though it is not clear if this was suicide, murder, or an accidental exposure due to a chemistry experiment he was undertaking in his spare room.