what is mathematics, logic, mathematics, foundations, incompleteness theorem, mathematical, Gödel, Godel, book, Goedel, tutorial, textbook, methodology, philosophy, nature, theory, formal, axiom, theorem, incompleteness, online, web, free, download, teaching, learning, study, student, Podnieks, Karlis
Personal page  click here.
Visiting Gödel Places in Vienna, December 2012
K.Podnieks. Frege’s Puzzle from a ModelBased Point of View. The Reasoner, Vol. 6, N 1, January 2012, pp. 56.
K.Podnieks, J.Tabak. The Nature of Mathematics – an interview with Professor Karlis Podnieks. Published as afterword, pp.188197 of: John Tabak. Numbers: Computers, Philosophers, and the Search for Meaning. Revised Edition. Facts on File, 2011, 243 pp. More books by John Tabak – click here.
Mathematical
Challenge (powers
of 2, exponentiation, etc.)
Gödel's
Theorem in 15 Minutes (English,
Latvian,
Russian)
Quote
of the Day
What is Mathematics:

Visited 142917 times from December 1997 until June 2013. 
My Hall of Fame Plato
I.
Kant G.
Cantor C.
S. Peirce H.
Poincaré See portraits of these brilliant people in the
MacTutor
History of Mathematics archive 
What is Mathematics? (My Main Theses) I define mathematical theories as stable selfcontained (autonomous?) systems of reasoning, and formal theories  as mathematical models of such systems. Working with stable selfcontained models mathematicians have developed their ability to draw a maximum of conclusions from a minimum of premises. This is why mathematical modeling is so efficient (my solution to the problem of "The Incomprehensible Effectiveness of Mathematics in the Natural Sciences"  as put by Eugene Wigner). For me, Goedel's results are the crucial evidence that stable selfcontained systems of reasoning cannot be perfect (just because they are stable and selfcontained). Such systems are either nonuniversal (i.e. they cannot express the notion of natural numbers: 0, 1, 2, 3, 4, ...), or they are universal, yet then they run inevitably either into contradictions, or into unsolvable problems. For humans, Platonist thinking is the best way of working with imagined structures. (Another version of this thesis was proposed in 1991 by Keith Devlin on p. 67 of his Logic and Information.) Thus, a correct philosophical position of a mathematician should be: a) Platonism  on working days  when I'm doing mathematics (otherwise, my "doing" will be inefficient), b) Formalism  on weekends  when I'm thinking "about" mathematics (otherwise, I will end up in mysticism). (The initial version of this aphorism was proposed in 1979 by Reuben Hersh (picture) on p. 32 of his Some proposals for reviving the philosophy of mathematics.) Next step The idea that stable selfcontained system of basic principles is the distinctive feature of mathematical theories, can be regarded only as the first step in discovering the nature of mathematics. Without the next step, we would end up by representing mathematics as an unordered heap of mathematical theories! In fact, mathematics is a complicated system of interrelated theories each representing some significant mathematical structure (natural numbers, real numbers, sets, groups, fields, algebras, all kinds of spaces, graphs, categories, computability, all kinds of logic, etc.). Thus, we should think of mathematics as a "twodimensional" activity. Sergei Yu. Maslov could have put it as follows: most of a mathematician's working time is spent along the first dimension (working in a fixed mathematical theory, on a fixed mathematical structure), but, sometimes, he/she needs also moving along the second dimension (changing his/her theories/structures or, inventing new ones). Do we need more than this, to understand the nature of mathematics? 
What is Mathematics?Four definitions of mathematics provably
equivalent to the above one: 
Why is Einstein much more
popular than Gödel? 
Wir muessen wissen  wir
werden wissen! "Hilbert and Goedel never discussed it, they never spoke to each other. ... They were both at a meeting in Koenigsberg in September 1930. On September 7th Goedel offhandedly announced his epic results during a roundtable discussion. Only von Neumann immediately grasped their significance..." (G.J.Chaitin' s lecture, 1998, Buenos Aires) 
Personal page  click here
what is mathematics, logic, mathematics, foundations, incompleteness theorem, mathematical, Gödel, Godel, book, Goedel, tutorial, textbook, methodology, philosophy, nature, theory, formal, axiom, theorem, incompleteness, online, web, free, download, teaching, learning, study, student, Podnieks, Karlis