"Any progress in the theory of partial differential equations must also bring about a progress in Mechanics." (Carl G J Jacobi, "Vorlesungen über Dynamik" ["Lectures on Dynamics"], 1843)
"Wherever Mathematics is mixed up with anything, which is outside its field, you will find attempts to demonstrate these merely conventional propositions a priori, and it will be your task to find out the false deduction in each case." (Carl G J Jacobi, "Vorlesungen über analytische Mechanik" ["Lectures on Analytical Mechanics"], 1847/48)
"Dirichlet alone, not I, nor Cauchy, nor Gauss knows what a
completely rigorous mathematical proof is. Rather we learn it first from him.
When Gauss says that he has proved something, it is very clear; when Cauchy
says it, one can wager as much pro as con; when Dirichlet says it, it is
certain."
"God ever arithmetizes." (Carl G J Jacobi)
"It is true that Fourier had the opinion that the principal object of mathematics was public use and the explanation of natural phenomena; but a philosopher like him ought to know that the sole object of the science is the honor of the human spirit and that under this view a problem of [the theory of] numbers is worth as much as a problem on the system of the world." (Carl G J Jacobi [letter to Legendre])
"Mathematics exists solely for the honour of the human mind." (Carl G J Jacobi)
"Mathematics is slow of growth and only reaches the truth by long and devious paths, that the way to its discovery must be prepared for long beforehand, and that then the truth will make its long-deferred appearance as if impelled by some divine necessity." (Carl G J Jacobi)
"Mathematics is the science of what is clear by itself."
"One should always generalize." (Carl G J Jacobi)
"The God that reigns in Olympus is Number Eternal." (Carl G J Jacobi)
"[...] the sole object of science is the honor of the human spirit and that under this view a problem of numbers is worth as much as a problem on the system of the world." (Carl G J Jacobi [letter to Legendre])
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