Blaise was the son of the mathematician Étienne Pascal (1588–1651) and had no other teacher. Étienne's greatest pedagogical challenge, however, was keeping up with his precocious son. By the age of twelve, Blaise had already discovered the elements of geometry with a vocabulary and by a method of his own devising. This achievement led his father to introduce the boy to Marin Mersenne's scientific academy. By the age of sixteen, Blaise had extended some work on conic sections done by the mathematician Girard Desargues in a way the mathematicians of Mersenne's academy thought brilliant.

Some people say Pascal supplies the intuitive mysticism they find lacking in the scientific rationalism of Descartes. Together the two great philosophers are supposed to express the full range of French (or even of modern) thought. There is some truth in this commonplace, but Pascal was as indebted to Descartes as he was wary of him, and their complicated relationship deserves the scholarly attention it continues to receive.

It was his work on conic sections that first brought Pascal unflattering attention from Descartes. In a Grinch-like letter written to Mersenne on Christmas Day of 1639, Descartes pettily observes that he knows of problems regarding conics that “a child of sixteen would have trouble solving” (AT II 628). Descartes’ attitude to Pascal was always condescending, and Pascal learned to reciprocate with disdain. “Descartes: useless and uncertain,” is one of the summary notes from Pascal's fragmentary, posthumous religious classic The Pensées (1963, 615, no. 887). Much of Descartes’ animosity toward Blaise can be explained by Étienne Pascal's association with Roberval, who, along with Beaugrand and Fermat, had criticized Descartes’ optics.

In September 1647, Descartes made a friendly call on the Pascals, mainly out of interest in seeing a working model of the reckoning machine Blaise had developed, which could perform all four arithmetic operations (though it multiplied and divided only with difficulty). Descartes admired it, but the conversation later turned to another sore point, namely, Pascal's attempt to replicate Torricelli's experiments with barometers. Pascal claimed that when a tube filled with mercury is inverted, the mercury dropping from the closed end of the tube leaves a vacuum behind.