From owner-qed Thu Nov 3 07:43:33 1994 Received: from localhost (listserv@localhost) by antares.mcs.anl.gov (8.6.4/8.6.4) id HAA09386 for qed-out; Thu, 3 Nov 1994 07:41:10 -0600 Received: from lapsene.mii.lu.lv (root@lapsene.mii.lu.lv [159.148.60.2]) by antares.mcs.anl.gov (8.6.4/8.6.4) with SMTP id HAA09373 for ; Thu, 3 Nov 1994 07:40:28 -0600 Received: from sisenis.mii.lu.lv by lapsene.mii.lu.lv with SMTP id AA03049 (5.67a8/IDA-1.4.4 for ); Thu, 3 Nov 1994 15:39:57 +0200 Received: by sisenis.mii.lu.lv id AA14662 (5.67a8/IDA-1.4.4 for QED discussions ); Thu, 3 Nov 1994 15:39:53 +0200 Date: Thu, 3 Nov 1994 15:39:44 +0200 (EET) From: Karlis Podnieks To: QED discussions Subject: Semantics Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: owner-qed@mcs.anl.gov Precedence: bulk Dear Colleagues: The current text presents the first chapter of my book "Around the Goedel's theorem" published in Russian (see Podnieks [1981, 1992] in the reference list). The main ideas were published also in Podnieks [1988a]. The contents of the book is the following: 1. The nature of mathematics 1.1. Platonism - the philosophy of working mathematicians 1.2. Investigation of fixed models - the nature of the mathematical method 1.3. Intuition and axiomatics 1.4. Formal theories 1.5. Logics 1.6. Hilbert's program 2. The axiomatic set theory 2.1. The origin of the intuitive set theory 2.2. Formalization of the inconsistent set theory 2.3. Zermelo-Fraenkel axioms 2.4. Around the continuum problem 3. First order arithmetics 3.1. From Peano axioms to first order axioms 3.2. How to find arithmetics in other formal theories 3.3. Representation theorem 4. Hilbert's Tenth problem 4.1. - 4.7. ............................................................................................................ 5. Incompleteness theorems 5.1. The Liar's paradox 5.2. Self reference lemma 5.3. Goedel's incompleteness theorem 5.4. Goedel's second theorem 6. Around the Goedel's theorem 6.1. Methodological consequences 6.2. The double incompleteness theorem 6.3. The "creativity problem" in mathematics 6.4. On the size of proofs 6.5. The "diophantine" incompleteness theorem 6.6. The Loeb's theorem Appendix 1. About the model theory Appendix 2. Around the Ramsey's theorem ______________________________________________________________________ University of Latvia Institute of Mathematics and Computer Science K.Podnieks, Dr.Math. podnieks@mii.lu.lv PLATONISM, INTUITION AND THE NATURE OF MATHEMATICS CONTENTS 1. Platonism - the philosophy of working mathematicians 2. Investigation of fixed models - the nature of the mathematical method 3. Intuition and axiomatics 4. Formal theories 5. Hilbert's program 6. Some replies to critics 7. References 8. Postscript 1. Platonism - the philosophy of working mathematicians Charles Hermite has said once he is convinced that numbers and functions are not mere inventions of mathematicians, that they do exist independently of us, as do exist things in our everyday practice. Some time ago in the former USSR this proposition was quoted as the evidence for "the naive materialism of outstanding scientists". But such propositions stated by mathematicians are evidences not for their naive materialism, but for their naive platonism. Platonist attitude of mathematicians to objects of their investigations, as will be shown below, is determined by the very nature of the mathematical method. First let us consider the "platonism" of Plato itself. Plato, a well known Greek philosopher lived in 427-347 B.C., at the end of the Golden Age of Ancient Greece. In 431-404 B.C. Greece was destroyed in the Peloponnesus war, and in 337 B.C. it was conquered by Macedonia. The concrete form of the Plato's system of philosophy was determined by Greek mathematics. In the VI-Vth centuries B.C. the evolution of Greek mathematics led to mathematical objects in the modern meaning of the word: the ideas of numbers, points, straight lines etc. stabilised, and thus they got distracted from their real source - properties and relations of things in the human practice. In geometry straight lines have zero width, and points have no size at all. Such things actually do not exist in our everyday practice. Instead of straight lines here we have more or less smooth stripes, instead of points - spots of various forms and sizes. Nevertheless, without this passage to an ideal (partly fantastic, but simpler, stable and fixed) "world" of points, lines etc., the mathematical knowledge would have stopped at the level of art and never would become a science. Idealisation allowed to create an extremely effective instrument - the well known Euclidean geometry. The concept of natural numbers (0, 1, 2, 3, 4, ...) rose from human operations with collections of discrete objects. This development ended already in the VIth century B.C., when somebody asked how many prime numbers do there exist? And the answer was found by means of reasoning - there are infinitely many prime numbers. Clearly, it is impossible to verify such an assertion empirically. But by that time the concept of natural number was already stabilised and distracted from its real source - the quantitative relations of discrete collections in the human practice, and it began to work as a fixed model. The system of natural numbers is an idealisation of these quantitative relations. People abstracted it from their experience with small collections (1, 2, 3, 10, 100, 1000 things). Then they extrapolated their rules to much greater collections (millions of things) and thus idealised the real situation (and even deformed it - see Rashevsky [1973]). For example, let us consider "the number of atoms in this sheet of paper". >From the point of common arithmetic this number "must" be either even or odd at any moment of time. In fact, however, the sheet of paper does not possess any precise "number of atoms" (because of, for example, nuclear reactions). And, finally, the modern cosmology claims that the "total number" of particles in the Universe is less than 10**200. What should be then the real meaning of the statement "10**200+1 is an odd number"? Thus, in arithmetic not only practically useful algorithms are discussed, but also a kind of pure fantastic matter without any direct real meaning. Of course, Greek mathematicians could not see all that so clearly. Discussing the amount of prime numbers they believed that they are discussing objects as real as collections of things in their everyday practice. Thus, the process of idealisation ended in stable concepts of numbers, points, lines etc. These concepts ceased to change and were commonly acknowledged in the community of mathematicians. And all that was achieved already in the Vth century B.C. Since that time our concepts of natural numbers, points, lines etc. have changed very little. The stabilisation of concepts testifies their distraction from real objects which have led people to these concepts and which continue their independent life and contain an immense variety of changing details. When working in geometry, a mathematician does not investigate the relations of things of the human practice (the "real world" of materialists) directly, he investigates some fixed notion of these relations - an idealised, fantastic "world" of points, lines etc. And during the investigation this notion is treated (subjectively) as the "last reality", without any "more fundamental" reality behind it. If during the process of reasoning mathematicians had to remember permanently the peculiarities of real things (their degree of smoothness etc.), then instead of a science (effective geometrical methods) we would have art, simple, specific algorithms obtained by means of trial and error or on behalf of some elementary intuition. Mathematics of Ancient Orient stopped at this level. But Greeks went further... . To be continued. #1