# [FOM] Are Proofs in mathematics based on sufficient evidence?

Irving ianellis at iupui.edu
Tue Jul 13 14:26:39 EDT 2010

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I should like to address specifically the issue of Euclid's Elements as
an example of a deductive system.

The notion that Eucild presents a deductive system, rather than an
axiomatic system, needs to be considered carefully in light of the
distinction made (e.g. by van Heijenoort, in particular in his
criticisms of Peano’s 1889 Arithmetices principia, between an axiomatic
system and a formal deductive system.

Consider the definition of a formal deductive system, in contrast with
the definition of an axiomatic system. In the formal deductive system,
which van Heijenoort (in the booklet El desarrollo de la teoria de la
cuantificacion), divides into Hilbert-type systems and Frege-type
systems, theorems are derived from a set of definitions and axioms by
application of inference and equivalence rules. A Hilbert-type system
is comprised of a set of wffs, which includes a list of axioms, a set
of rules of passage, or derivation rules, and for which a proof is a
sequence of wffs in which the last wff of the sequence, the theorem, is
the wff which is proven. A Frege-type system is a formal language
containing an arbitrary set of axioms, a set of inference and
equivalence rules, and in which  nothing exists outside of the proofs.
Van Heijenoort gives Principia Mathematica as an example of a
Frege-type system. An axiomatic system, however, lacks any explicit
inference rule, and in this sense van Heijenoort regarded Euclid
Elements and Peano's Arithmetices principia as axiomatic systems, but
not as formal deductive systems. He explains, in his introduction to
Peano's 1889 that: "There is
a grave defect. The formulas are simply
listed, not derived; and they could not be derived, because no rules of
inference are given" (From Frege to Goedel, p. 84). (Admittedly, Marco
Borga & Dario Palladino (pp. 27-28) object to van Heijenoort's
interpretation on this point, arguing that Peano's  (a & (a -> b)) -> b
can and should be understood as modus ponens, and that this was indeed
how Peano meant it to be taken ("Logic and foundations of mathematics
in Peano's school", Modern Logic 3 (1992), 18-44). They also admit, at
the same time, that van Heijenoort was correct to the extent it indeed
does NOT explicitly appear in Principes arithmetica, but does occur in
all of Peanos later work, beginning in 1891 in his "Formóle di logica
matematica".)

With these distinctions in mind, consider next Russell's conception of
In the paper "The Teaching of Euclid" (Mathematical Gazette 2 (May,
1902), 165¬-167; reprinted, pp. 467-469, Towards the "Principles of
Mathematics", 1900–02, edited by Gregory H. Moore (London/New York:
Routledge, 1993), Volume 3 of The Collected Papers of Bertrand Russell)
Russell took Euclid seriously to task for the lack of "logical
excellence" which Euclid was reputed to have presented in his book. The
point also recurs in the Principles of Mathematics (p. 5) where Russell
points out the need for rules or principles" of deduction and proceeds
to offer ten such principles (pp. 4-5, 10-16), including in particular
"formal implication" or the rule of detachment. We may summarize
Russell's strong criticisms of Euclid by reminding ourselves of the
difference between an axiomatic system and a formal deductive system
and reporting that Russell in essence accuses Euclid of not possessing
a formal deductive system.

Irving H. Anellis
Visiting Research Associate
Peirce Edition, Institute for American Thought
902 W. New York St.
Indiana University-Purdue University at Indianapolis
Indianapolis, IN 46202-5159
USA
URL: http://www.irvinganellis.info

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