Mathematics, Science and Epistemology: Volume 2, Philosophical Papers (His Philosophical papers ; v. 2) [Hardcover]

Imre Lakatos (Author), John Worrall (Editor), Gregory Currie (Editor)

Book Description

Publication Date: May 18, 1978 | ISBN-10: 0521217695 | ISBN-13: 978-0521217699 | Edition: 1st

Imre Lakatos' philosophical and scientific papers are published here in two volumes. Volume I brings together his very influential but scattered papers on the philosophy of the physical sciences, and includes one important unpublished essay on the effect of Newton's scientific achievement. Volume 2 presents his work on the philosophy of mathematics (much of it unpublished), together with some critical essays on contemporary philosophers of science and some famous polemical writings on political and educational issues.

Editorial Reviews

Review

' ... there is an essential unity of theme and purpose running through all the papers collected in these volumes. Many of the papers, even though they have been previously published, are not fully self-contained and can be understood only by reference to others. Their collection and the editors' cross-referencing thus serves the useful purpose of facilitating an assessment of Lakatos' contribution to philosophy.' Philosophical Books

' ... a well-produced book that every philosopher of science or mathematics will wish to have.' Philosophy

Book Description

Volume I brings together his very influential but scattered papers on the philosophy of the physical sciences, and includes one important unpublished essay on the effect of Newton's scientific achievement. Volume 2 presents his work on the philosophy of mathematics together with some critical essays on contemporary philosophers of science.

Product Details

• Hardcover: 300 pages
• Publisher: Cambridge University Press; 1st edition (May 18, 1978)
• Language: English
• ISBN-10: 0521217695
• ISBN-13: 978-0521217699
• Product Dimensions: 8.9 x 6.1 x 0.9 inches
• Shipping Weight: 1.3 pounds
• Average Customer Review: 5.0 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #3,828,203 in Books (See Top 100 in Books)

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2 of 2 people found the following review helpful:

5.0 out of 5 stars Several excellent papers, February 10, 2008

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Viktor Blasjo - See all my reviews
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This review is from: Mathematics, Science and Epistemology: Volume 2, Philosophical Papers (Philosophical Papers (Cambridge)) (Paperback)

Foundations of mathematics. The Hilbert--Russell meta-mathematical programmes were meant to establish the infallibility of mathematics by the Euclidean method: "derive all mathematics from trivial logical principles" (p. 12). Although "From the seventeenth to the twentieth century Euclideanism has been on a great retreat" (p. 10), having failed again and again in numerous branches of knowledge, Russell and others had no doubt that mathematics would be different: "Too often it is said that there is no absolute truth ... Of such scepticism mathematics is a perpetual reproof; for its edifice of truths stands unshakeable ... to all the weapons of doubting cynicism" (p. 14). "We all know how the brief Euclidean 'honeymoon' gave place to 'intellectual sorrow', how the intended logico-trivialization of mathematics degenerated into a sophisticated system, including 'axioms' like that of reducibility, infinity, choice, and also ramified type theory---of on the most complicated conceptual labyrinths a human mind ever invented. ... There even emerged the completely un-Euclidean need for a consistency proof to ensure that the 'trivially true axioms' should not contradict one another. All this and what followed must strike any student of the seventeenth century as a déjà vu: proof had to give way to explanation, ... Euclidean theory to empiricist theory. We also encounter the same refusal to accept the dramatic change", e.g. "Like Newton hoping to explain the Law of Gravitation by principles of Cartesian push-mechanics, Russell hoped for the trivialization of the reducibility axiom" (p. 14). But eventually Russell admitted defeat: "When pure mathematics is organized as a deductive system ... it becomes obvious that, if we are to believe in the truth of pure mathematics, it cannot be solely because we believe in the truth of the set of premisses. Some of the premisses are much less obvious than some of their consequences, and are believed chiefly because of their consequences. ... The only way in which work on mathematical logic throws light on the truth or falsehood of mathematics is by disproving the supposed antinomies. This shows that mathematics may be true. To show that mathematics is true would require other methods and other considerations." (p. 17). Thus logic "is an empiricist theory" like any other, albeit one that is hard to test. "The only way of criticizing this peculiar empiricist theory is, on the face of it, to test it for concistency. This leads us to the Hilbertian circle of ideas." (pp. 19-20). "Gödel's second theorem was a decisive blow to this hope for a Euclidean meta-mathematics. ... consistency proofs have to contain enough sophistication to render the consistency of the theory in which it is carried out dubitable ... For instance, Goldbach's conjecture---that every number is the sum of two primes---might be formally proved to-morrow, but we shall never know that it is true. For it would only be true if meta-mathematics, meta-meta-mathematics ... ad infinitum are consistent. This we shall never know. Gödel's first theorem showed a second way in which a formal theory could misfire: if it has a model at all, it has more models than intended. ... If the Goldbach conjecture is true in its intended interpretation, but false in an unintended one, there will be no formal proof leading to it in any formalization. Gödel's discovery of omega-inconsistent systems was still worse. ... A formalized arithmetic might be consistent, i.e. have models, but none of the models might be the intended one; every model, if containing all the numbers, might contain some other 'class-alien' elements which might provide counterexamples to propositions which are true in the narrower domain of the intended interpretation. In a consistent, but omega-inconsistent system we might prove the negation of the Goldbach conjecture even if the Goldbach conjecture is true." (pp. 20-21). Contemporary logic should not pretend to be the foundations of mathematics. "We read in one of the most competent books written on the subject that the 'ultimate test whether a method is admissible in meta-mathematics must of course be whether it is intuitively convincing.' But why then not stop earlier, why not say that 'the ultimate test whether a method is admissible in arithmetic must of course be whether it is intuitively convincing,' and omit meta-mathematics altogether...?" Or, for that matter, "why on earth have 'ultimate' tests" at all? "Why foundations, if they are admittedly subjective? Why not honestly admit mathematical fallibility, and try to defend the dignity of fallible knowledge from cynical scepticism, rather than delude ourselves that we shall be able to mend invisibly the latest tear in the fabric of our 'ultimate' intuitions?" (p. 23).

"The method of analysis--synthesis." A famous "heuristic" in Euclidean geometry was to prove a theorem by assuming it to be true, deriving something known (analysis), and then reversing the steps to obtain a proof (synthesis). This approach was explicated by Pappus, whence it may be called the "Pappusian Circuit" (p. 76). "A main feature of the story of modern scientific method is the critical elaboration of the ancient Pappusian Circuit into the Cartesian Circuit, followed---in spite of some partial successes and several intriguing rescue-operations---by its breakdown" (p. 77). "Both Descartes and Newton were very explicit about the necessity of starting the analysis from facts, from which one proceeded to 'mediate causes' and from there to first principles. They despised those who tried to arrive at first principles with no care for facts, by 'rash anticipation' instead of by laborious analysis" (p. 77), e.g., "Hooke only guessed the inverse square law, but he, Newton, deduced it from Kepler's empirical laws" (p. 80). This is also the meaning of Newton's "Hypothesis non fingo." "Hypotheses have to be embedded in a Cartesian Circuit and thereby cease to be hypotheses" (p. 77). "Descartes's main interest was to find a method of discovery of infallible knowledge, an infallibilist heuristic. The paragon of infalliable knowledge was of course Euclidean Geometry. And the only extant method was of discovery in Euclidean Geometry was the Pappusian Circuit. This was Descartes's natural starting point." (p. 83). "Now my differences with Hintikka's and Remes's rational reconstruction of Greek analysis--synthesis become clear. They base their reconstruction on the assumption that Pappusian analysis was a heuristic pattern in already axiomatized Euclidean Geometry ... In my view the most exciting analyses of Greek Geometry were pre-Euclidean and their role was to generate Euclid's axiomatic system." (p. 100).

Criticism of falsificationism. Popper "has refused to notice two [historical] facts: (1) 'Crucial experiments' are frequently listed first as harmless anomalies, rather than refutations ...; and (2) All important theories are born 'refuted'." (p. 201). Further discussion on this is limited to Lakatos' effortless refutations of two minor falsificationists (Agassi and Grünbaum).

"Cauchy and the Continuum." Conventional histories claim that Cauchy made several "mistakes" in his Cours d'Analyse, e.g. his proof that the limit function of a convergent series of continuous functions is always continuous. This seems strange, however, because there were already published counterexamples and "today, if one gave Cauchy's false proof to a bright undergraduate, it would not take him long to put it right; and indeed, Seidel [who eventually corrected the proof] did not find the problem at all difficult! What inhibited a whole generation of the best minds from solving an easy problem?" (pp. 46-47). Actually, there is no "mistake", since the alleged counterexamples do not converge in Cauchy's sense, as he himself explained: "His example is the series sin(x)+sin(2x)/2+sin(3x)/3+... He shows that in the neighbourhood of zero where the limit function is discontinuous, 'the value of the remainder for xs very near to zero, for instance for x=1/n where n is a very large number, can differ considerably from zero,'" so that the series does not converge at the "moving point x=1/n" where n goes to infinity (p. 57).

3 of 4 people found the following review helpful:

5.0 out of 5 stars Comprehensive, March 12, 2000

By 

Gãlkan Yerlãkaya - See all my reviews

This review is from: Mathematics, Science and Epistemology: Volume 2, Philosophical Papers (Philosophical Papers (Cambridge)) (Paperback)

The author introduces the main lines of discussions in epistemeolgy and philosophy of mathematics in a very understandable but comphrehensive way. It is a brilliant reference book for the subject which also contains so-far unpublished articles of the author.