Previous Quotes of the DayIn the reverse chronological order |
David Corfield (2004) ... So much effort has been devoted to a
thin notion of truth, so little to the thicker notion of
significance. To say that scientists and mathematicians
aim merely for the truth is a gross distortion. They aim
for significant truths. |
Georg Cantor, August 28, 1899 ... Wilfried Sieg. Hilbert's programs: 1917-1922. The Bulletin of Symbolic Logic, March 1999, Vol.5, N 1 (online copy). |
From: Jeffrey
Ketland ... ... If I remember right, the gist is
this. In studying the consistency problem, G�del wanted
initially to give an interpretation of second-order
arithmetic within first-order arithmetic, and tried to
find a definition of (second-order!) arithmetic truth in
the first-order language. He discovered however that even
first-order arithmetic truth is not arithmetically
definable: i.e., what we now call Tarski's Indefinability
Theorem. But, as he also discovered, the concept
"provable-in-F", with F some fixed formal
system, is arithmetically definable. This implies that
arithmetic truth is distinct from provable-in-F, for any
formal system F. This then gives us the quick proof of
G�del's first incompleteness theorem. Full text at http://www.cs.nyu.edu/pipermail/fom/2004-August/008481.html |
| Peirce
& Son
(1870+) ... in 1870 Benjamin Peirce defined mathematics as "the science that draws necessary conclusions" (see his son C.S.Peirce 1898/1955, p.137). C.S.Peirce himself described the work of a mathematician as composed of two different activities (p.138): (1) framing of a hypothesis stripped of all features which do not concern the drawing of consequences from it, without caring whether this hypothesis agrees with the actual facts; (2) drawing the necessary consequences from the hypothesis. He noted (Peirce 1902/1955, p.144) the difficulty to distinguish between two definitions of mathematics, one by its method ("drawing necessary conclusions"), another by its aim and subject matter ("the study of hypothetical state of things"). See p.5 of |
Philip J. Davis and Reuben Hersh (1987): In the real world of mathematics, a
mathematical paper does two things. It testifies that the
author has convinced himself and his friends that certain
"results" are true, and presents a part of the
evidence on which this conviction is based. |
| From: John
McCarthy... Sent: Saturday, May 15, 2004 1:43 PM Subject: Re: [FOM] Freeman Dyson on Inexhaustibility Maybe physics is inexhaustible, but maybe it isn't.� Here's why it might not be.� Consider the Life World based on Conway's Life cellular automaton.� It has been shown that self-reproducing universal computers are possible as configurations in the Life World. Therefore, one could have physicists in the Life World, but their physics would not be inexhaustible.� They could discover or at least conjecture that their fundamental physics was a particular cellular automaton.� However, their mathematics could be the same as ours - and therefore inexhaustible. Full discussion thread - see Foundations of Mathematics (FOM) e-mail list. |
| The
Continuum Hypothesis (I), 2000, by W. Hugh
Woodin ... The current situation is the following. We can build models of set theory with significant control over what is true in the model. - During the 35 years since Cohen's work a great number of set theoretical propositions have been shown to be independent. Further problems in other areas of mathematics have also been shown to be independent. This, as of yet, cannot be accomplished for models of number theory. The intuition of a true model of number theory remains unchallenged. ... |
| From: Harvey
Friedman ... Sent: Tue Jan 20 01:17:00 EST 2004 Subject: [FOM] On Foundational Thinking 1 ... (To avoid confusion, I draw a distinction between foundations of mathematics and mathematical logic. The latter consists of various mathematical spinoffs from foundations of mathematics, where one deemphasizes foundational thinking, and emphasizes mathematical adventures, including connections with various branches of mathematics.) ... Full text at http://www.cs.nyu.edu/pipermail/fom/2004-January/007857.html |
Ernest
Gellner |
| Date: Fri, 26 Dec 2003 14:44:47 -0300 From: Julio Gonzalez Cabillon ... Subject: [FOM] quotation from Weyl Dear Roman Murawski, I think that the passage you are seeking in Weyl's writings is: "We now come to the decisive step of mathematical abstraction: we forget about what the symbols stand for. The mathematician is concerned with the catalogue alone; he is like the man in the catalogue room who does not care what books or pieces of an intuitively given manifold the symbols of his catalogue denote. He need not be idle; there are many operations which he may carry out with these symbols, without ever having to look at the things they stand for." which is contained in the classic "The Mathematical Way of Thinking", an address given by Hermann Weyl at the Bicentennial Conference at the University of Pennsylvania, in 1940. It was first published in _Science_, in 1940, volume 92, pp. 437-446, and later reproduced in James R. Newman's "The World of Mathematics" (volume 3). Very best wishes to you all for 2004. Julio Context - at http://www.cs.nyu.edu/pipermail/fom/2003-December/007770.html |
"... the conventional wisdom, fooled by our misleading "physical intuition", is that the real world is continuous, and that discrete models are necessary evils for approximating the "real" world, due to the innate discreteness of the digital computer. Ironically, the opposite is true. The REAL REAL WORLDS (Physical and MATHEMATICAL) ARE DISCRETE. Continuous analysis and geometry are just degenerate approximations to the discrete world, made necessary by the very limited resources of the human intellect. While discrete analysis is conceptually simpler (and truer) than continuous analysis, technically it is (usyally) much more difficult." "Real" Analysis is a Degenerate Case of Discrete Analysis by Doron Zeilberger |