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

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

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 

With these distinctions in mind, consider next Russell's conception of 
what Euclid was about.
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
URL: http://www.irvinganellis.info

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