Since humankind discovered mathematics, one thing was not yet a problem: time. Time was all that ancient mathematicians had in the land of Sumeria. The problem is to spend time doing arithmetic calculations all day: the cattle, the oxen, the land marking, the food, the stars and all things that had to be calculated with precision. Therefore, civilization began to flourish, and soon came the market, with the unrest mob, trading their surplussed goods, between people from different places. All directly connected to calculation and profit. But calculation is boredom. Specially, if the calculi involves large figure numbers, or operations like square roots. And The ancient mathematician started to record, in clay tablets, the results of some calculations such as the case of number $\pi$, and $\sqrt{2}$, and soon patterns emerged from the calculations, from the daily commerce, and the mathematicians started to mechanize their labor employing beads and abacuses, and soon they started to find shortcuts in the device's configuration, leading to more speed in calculi. And that was the ancient world, simple tools for keeping the city going right.
But man is a lazy animal, and always wants more free time to spend doing useless abstract activities such as writing and reciting poems, gambling, or philosophizing about things. Laziness makes the genius of invention, and they created many types of devices like the abacus and the Antithetic device. One for helping with the arithmetics, and the other a planerarium, possibly invented to foresee planets positions, very likely, to make astronomical predictions. For centuries, the status of mathematics was restricted to geometrical pursuits, later on, with the end of Middle Ages, the dryness of science returned in the Renaissance, and man like Fibonacci started to relearn , from the ancient times, exploring new paths within the art of numbers, all summarized by Euclides 2000 years ago, in his book The Elements. But, it was for the sake of computing with variables that the algebra appeared from the East, with unknown quantities as puzzles, that ranged from silly games of "guessing the number I'm thinking now" to the hidden secrets of unsolvable equations that lasted for centuries onwards. Algebra is the first step to abstraction of mathematical objects known as numbers, only surpassed by Cantor's 'Set Theory'. This power of abstraction didn't relieve man from even more quantities of calculi. In particular, the trade and the states became more complex and vast, and calculations of taxation was needed, a boring activity that the father of Pascal had. Pascal had the idea to help his father, and built a calculating machine out of cogs and gears, that could handle numbers of 8 digits. It was much like a clock than a computer, and it hasn't any memory at all, therefore, was not programmable, and by modern definition of computing machines, it was just clumsy mechanism, The Pascal's machines didn't lift off the ground with Colbert, the minister of Louis XIV, and Pascal, I think, became desapointed to the point of abandoning mathematics for philosophy and religion. And there comes a programming language called Pascal in his behalf. In the time of Leibniz, rumors of a universalization of language was in the mind of many educated people. What began with Leibniz and his Lingua Caracterisca, i.e., a language in which science could be perfectly represented without ambiguities was Leibniz ambition. And what began with Leibniz, ended up convincing Ada Lovelace to program a differential machine using punched cards of a loom. It didn't last much, to appear the Boolean logic, and the Hilbert dream of an oracle machine to predict solutions in Diophantine equations. That all soon crashed with Gödel's Incompleteness, the dream of certainty was gone, like someone that pulls the carpet of rationality under everyone's feet.
In a certain way, mathematics became, after Gödel's theorems, a bit cruel. He showed that arithmetic was complete formal system, and for that reason, that would be truths inside it, that cannot be proved using the axioms and definitions of arithmetic.
We cannot count Ada's attempt as the first programming language, because Babbage's Differential Machine had limitations, although she made many efforts to abstract her 'language', and make it free from any machine's limitations, it was only with Alan Turing and Alonzo Church that improvements had been made in the correct direction.
First, in 1936, Turing issued a paper to the scientific community.
In the paper "ON COMPUTABLE NUMBERS, WITH AN APPLICATION TO THE ENTSCHEIDUNGSPROBLEM", Turing describes
an abstract machine that, in theory (see Church-Turing Thesis), can compute any recursive function, allowing
the machine to compute any algorithm to be run into the machine. Later, it was coined the term "Turing-Complete" to
define a programming language that was capable of computing any algorithm. Turing's machine is much like a
typewriter with memory facilities. It is composed by an infinite tape in which a header can WRITE or READ from,
and a function that moves to the RIGHT or to the LEFT, moving the header.
The mathematical description of the Turing's Machine (TM for short) is written as
$$\mathcal{M} = (\Sigma,x \mapsto \{LEFT,RIGHT\},\lambda,Q,Q_0)$$, where:
$\Sigma$ is the machine's alphabet;
$x \mapsto \{LEFT,RIGHT\}$ is a function that maps a state to move the header left or right;
$\lambda$ is the empty word.
$Q$ and $Q_0$ are states, from and to, the machine has to go to.
Earlier, in 1930, Alonzo Church developed the $\lambda$-calculus to formalize in computational terms the concept of a mathematical function, that was deeply involved with set theory. The function is defined as a formal system, the $\lambda$-calculus, that describes functions as $\lambda arg.body$ notation, and is based on three concepts, the ABSTRACTION, the APPLICATION, and the REDUCTION. ABSTRACTION deals with the capacity to represent the function $\lambda arg.body$. For example, the identity function is $\lambda x.x$, that is, for the bound variable x substitute x for x. The APPLICATION deals with the capacity to make use of types. For example, to make the identity function $((\lambda x.x) t)$ returns for $x = true$ which is a boolean type. The REDUCTION is the substitution of the variables with the "body" values, reducing some complex expressions as
PASCAL is a PROGRAMMING LANGUAGE CREATED by SWISS COMPUTER SCIENTIST NIKLAUS WIRTH in 1970. It was NAMED after the FRENCH MATHEMATICIAN and PHILOSOPHER BLAISE PASCAL. It was designed to teach people programming, but later they found it clean and structured enough to be a business language, implemented in Borland's Delphi. I had my first experience with programming languages in Pascal. But it wasn't enough for my youth ambitions of programming systems. In Delphi, it was nice to program some GUI applications for Windows based systems. Interested in AI, I made a Tic-Tac-Toe game, that used Minimax algorithm to calculate the next move. Inexperienced as I was, the game didn't work properly, and the computer made silly moves in the game, losing all the time. But, it was a nice experience for a newbie in programming.