[FOM] Are proofs in mathematics based on sufficient?
ianellis at iupui.edu
Thu Jul 22 09:06:53 EDT 2010
The last time I attempted to send this message, several days ago, there
apparently was a transmission. Herewith is my next attempt.
Monroe Eskew wrote:
"They may have done a large number of computations, but these
mathematicians also thought creatively, had new ideas, etc. For
example could you call Cayley's introduction of the general concept of
a group and he proof that any finite group is embeddable into a
permutation group, "mere computation"? Similar things come to mind
regarding the fundamental theorem of algebra, Gaussian distribution,
Gaussian curvature, etc. There is some great mathematics there, not
merely computation, though it's not of a foundational nature. And its
all rigorous, but not of it is done in a formal deduction system."
I certainly do not deny that many of these mathematicians, Gauss,
Cayley, et al., thought creatively, had new ideas, etc..
But I would also add that conceptions of what constitutes "proof" and
"rigor" have evolved through history.
I also would suggest that "computation" as I intended to use it is a
wider concept than manipulating numbers or symbols for purposes of
equation solving. I would consider that writing a computer program, for
example, to derive the theorems of propositional calculus, as was done
in the 1950s, requires creation of rules for doing proofs, whereas the
computer, once given the instructions, does the computation. The
thought that goes into writing a program, or of devising concepts,
ideas, a laying out steps to obtain a solution, goes beyond the trivial
concept of "computation".
For Frege and others like him, what we call logic is at once both a
calculus (by which he meant mechanical computational procedure) AND,
more importantly, a formal language containing formation and inference
rules. I would agree that these are NOT mutually exclusive, and that
one can have both at once.
I was hoping to spend some time preparing a post in which I could give
at least a bare outline of the historical development of the nature of
proof and of the historical development of meaning of rigor, along with
a suggestion that there is a demarcation to be made between
computation, axiomatization, and formal deduction, although much of
mathematics applied all of these in some combination, and that it
historically, these combinations as well as the lines of demarcation
shift along a continuum, with the particular emphasis of one over
another, and the locus of the shifting line of demarcation constituting
the mathematical "style" of a particular time and place or of a
particular mathematician. More importantly, I would have preferred to
devote more time to elucidating how I understand the concepts of
"computation", "proof", rigor, "style", "axiomatic system" and "formal
deductive system", the relations between them and their historical
evolution. For the nonce, however, I will simply defer to some writings
by others about the evolving concepts of mathematical proof, rigor, and
"style"; among these (in no particular order):
"An Informal History of Formal Proofs: From Vigor to Rigor? An Informal
History of Formal Proofs: From Vigor to Rigor?", Klaus Galda The
Two-Year College Mathematics Journal, Vol. 12, No. 2 (Mar., 1981), pp.
"Rigor and Proof in Mathematics: A Historical Perspective Rigor and
Proof in Mathematics: A Historical Perspective", Israel Kleiner
Mathematics Magazine, Vol. 64, No. 5 (Dec., 1991), pp. 291-314;
"Rigorous Proof and the History of Mathematics: Comments on Crowe
Rigorous Proof and the History of Mathematics: Comments on Crowe",
Douglas Jesseph Synthese, Vol. 83, No. 3, Pierre Duhem: Historian and
Philosopher of Science. Part II: Duhem as Philosopher of Science (Jun.,
1990), pp. 449-453;
"Informal versus Formal Mathematics", Francisco Antonio Doria Synthese,
Vol. 154, No. 3, New Trends in the Foundations of Science (Feb., 2007),
EPPLE, Moritz. 1996. "Styles of Argumentation in Late 19th Century
Geometry and the Structure of Mathematical Modernity", in Michael Otte
& Mario Panza (eds.), Analysis and Synthesis in Mathematics and
Philosophy (Dordrecht: Kluwer Academic Publishers), 177199;
James P. Pierpont, "Mathematical Rigor Past and Present", Bulletin of
the American Mathematical Society 34 (1928), 2353;
"Rigor and Revolution: The Demise of Natural Mathematics", Tasoula
Berggren (ed.), Proceedings of the Canadian Society for History and
Philosophy of Mathematics/Societe Canadienne d'Histoire et Philosophie
des Mathematiques, Fifteenth Annual Meting, Quebec City, Quebec, May
29May 30, 1989, 133148.
There are, in addition to these, specific historical studies that
describe the evolution of these concepts and the details of the
mathematics that exemplified them or that resulted in these historical
shifts. One example would be Derek Thomas Whiteside, "Patterns of
Mathematical Thought in the Later Seventeenth Century", Archive for
History of Exact Sciences 1 (1961), 179-338.
I am in the midst of a very large undertaking, already approaching 1500
pages, in which I explore the question of the "Fregean 'revolution'" in
the history of logic in the context of the shift from
the algebraic logic of Boole, Peirce, and Schroeder to the "logistic"
or "Russello-Fregean" function-theoretic logic, at the moment carrying
the very unwieldy, if descriptive, title:
"FROM ALGEBRAIC LOGIC TO LOGISTIC: HOW WE STOPPED ALGEBRAICIZING AND
LEARNED TO LOVE LOGISTIC, OR FORGETTING THE CLASSICAL BOOLE-SCHROEDER
CALCULUS -- THE FREGEAN "REVOLUTION" AND THE RISE OF THE "RUSSELLIAN"
VIEW OF MATHEMATICAL LOGIC: AN HISTORIOGRAPHICAL, PHILOSOPHICAL, AND
SOCIOLOGICAL INVESTIGATION OF AN EPISODE IN THE HISTORY OF MATHEMATICAL
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
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