Calculus without limit
Due to the importance of calculus, each year thousands of people from around the
world study this theory. However, what is shown earlier in this book indicates that
there is a theoretical need to rebuild calculus without using the concept of limits. That
is exactly the goal of this chapter.
Historically, when Leibniz introduced the concept of derivatives and established
the relationship between the tangent line of a curve and the problem of finding the
maximum and minimum of a function, an entire class of extremely difficult problems of
the past was resolved by using a straightforward method. That of course represented an
important breakthrough in science and made mathematicians very excited. However,
logically this method suffered from a flaw. In the process of deriving the method, a
variable h is first assumed to be not zero; and as soon as it becomes clear that it no
longer matters whether it is zero or not, the variable is immediately taken to be zero.
For more details, please consult Chapter 6 in this book.
Facing the logical hole, Newton provided the following way to get around the
problem: Don’t let h become 0 at once; instead, let h become an “infinitesimal’’. Then
what is an infinitesimal? According to Newton, it stands for the final state before a
quantity becomes zero; it is not zero; but its absolute value is smaller than any positive
number. Because h is not zero, it can be used as a denominator and can be cancelled
in division; and because its absolute value is smaller than any positive number, it can
be ignored in the final expression.
Since then the problem of what an infinitesimal is had troubled many of the first
class mathematicians for more than 100 years. If it is a number, then before it becomes
0, it is still a number. Then, what is the final state whose absolute value is smaller
than any positive number? If it is not a number, how can we manipulate it just like a
number? Only when Karl Weierstrass established his limit theory in the ε −δ language,
this problem was considered resolved satisfactorily, where the introduction of limits
successfully replaced “equal to zero’’ by “infinitely approaching 0’’.
In his life time, Newton created four major works in calculus, where Treatises on
Species and Magnitude of Curvilinear Figures was the last completed in 1693 (Newton,
1974). However, it was published in 1704, which was the first among his important
works in calculus (Li, 2007). What was the reason for the publication of the other
three earlier works to be greatly delayed? And why was Treatises on Species and
Magnitude of Curvilinear Figures published 11 years after it was initially finished? It
was because Newton was not satisfied with his calculus, while hoping to remove “the
infinitely small quantities’’ from the theory. In Treatises on Species and Magnitude of
Curvilinear Figures he gave up the concept of infinitesimals. Instead he employed the