RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

202 Chapter 10. Toward rigorization of analysis
Figure 10.1: BERNARD BOLZANO (1781–1848)
Indeed, from B. TAYLOR’S (1685–1731) days, proofs of Taylor’s Theorem had relied
on an analogy between repeated differences and the binomial formula. 29 However, the
relevant step in the proof of Taylor’s Theorem seems to have been a limit process based
on the binomial formula in which the exponent n increased to infinity and thus did not
rely on the full binomial theorem. This distinction between the binomial theorem and
the indicated limit process does not seem to have been undertaken by eighteenth and
nineteenth century mathematicians, though.
Next, BOLZANO criticized previous proofs for operating with (completed) infinite
series, i.e. working with series as if they were polynomials. Instead, he proposed a
concept of numerical limit processes based on (variable) quantities (Größen) ω which
could be assumed positive but less than any given value. He also described these
quantities as “quantities which can be made as small as one desires.” 30 Importantly,
BOLZANO’S ω was not a completed infinitesimal but a variable quantity which de-
pended on a limit process.
In continuation of the previous point, BOLZANO insisted that restrictions be im-
posed on the binomial such that the series was (arithmetically) convergent. He claimed
that previous proofs had “proved to much” by not taking such restrictions on x into
account and forbade application of the theorem outside the domains of convergence
of the series. In the argument, BOLZANO employed a counter example which based
29 See e.g. (Jahnke, 1999, 139–142).
30 “Größen, welche so klein werden können, als man nur immer will.” (Bolzano, 1816, v).