Sequent calculus, decidability.
But it was Euler whose work really established calculus as the basic tool of the mathematical sciences.
“Every university sells the same product—calculus is calculus—so the brand means everything.
The calculus is one of the greatest achievements of mankind: an astronaut and an investment analyst pay homage to its enormous power.
Grue, to resurrect Church’s lambda calculus as a foundation of mathematics.
So, why should such an agent’s degrees of belief obey the probability calculus?
Any such calculus can count as a piece of mathematics, even an inconsistent one.
Ketonen wanted to formulate Skolem's formal rules of proof within sequent calculus.
The time-space complexity of a calculus specifies how efficient its algorithmic representation is.
This minimal system, for historical reasons, is known as NL (for ‘Non-associative Lambek calculus’).
In his thesis Gentzen introduced a form of the sequent calculus and his technique of cut elimination.
Accordingly they refrained, in the main, from destructive criticism of the ideas underlying the calculus.
Formally, this is the abstraction of the lambda calculus (see the entry on the lambda calculus).
The second volume of Hilbert and Bernays’ Grundlagen der Mathematik (1939) provides an account of results on the epsilon-calculus that had been proved by that time.
Newton’s and Leibniz’s approach to calculus had been primarily geometric, involving ratios with “almost zero” divisors—Newton’s “fluxions” and Leibniz’s “infinitesimals.”
The former created in 1951 an infinitary sequent calculus to present consistency proofs in a perspicuous way, the latter instead used a more traditional Gentzen-style calculus (see Takeuti 1987).
In Whitney’s view, the standard progression in math education—algebra, geometry, trig, pre-calculus, calculus—is random and baseless, a linear conceit that creates a false sense of increasing difficulty.
Leonard’s system of Singular Terms is significantly different from, and indeed in philosophically interesting ways weaker than, the Calculus of Individuals (Rossberg 2009), but the exact extent of Goodman’s technical contribution to the calculus remains unknown.
There cannot be “undecidable propositions”, Wittgenstein argues (PR §173), because an expression that is not decidable in some actual calculus is simply not a mathematical proposition, since “every proposition in mathematics must belong to a calculus of mathematics” (PG 376).
For the mapping from the syntactic source calculus DL to the semantic target calculus LP, the unary type-forming operations are considered inert: the inference rules for these connectives, consequently, leave no trace in the LP proof term associated with a derivation in the syntactic source calculus.
- a hard lump produced by the concretion of mineral salts; found in hollow organs or ducts of the body
- an incrustation that forms on the teeth and gums
- the branch of mathematics that is concerned with limits and with the differentiation and integration of functions
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