foundations of mathematics, philosophy of mathematics, logic, model, modeling, modelling, mathematical, online, web, book, Internet, tutorial, textbook, foundations, mathematics, teaching, learning, study, mathematical logic, student, Podnieks, Karlis, philosophy, free, download
Alexander Grothendieck, 19282014 A Farewell to Mathematics? http://www.grothendieckcircle.org/ 
1935: Ludwig Wittgenstein visiting USSR
Visiting Gödel in Vienna, July 2014: Richard Zach determined the exact location of Cafe Reichsrat!
Mathematical Challenge (powers of 2, exponentiation, etc.)
Gödel's Theorem in 15 Minutes (English, Latvian, Russian)
Favorite Quotes (Grothendieck, Feynman, von Neumann, Zeilberger, Bohr, von Liebig, Einstein, Putnam, Turing, Friedman, Kolmogorov, ...)
Introduction
to

What is Mathematics:

What is Mathematics?Four provably equivalent definitions of
mathematics: 
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? 
In 1974, during a Soviet army training course, I discovered a simple (almost trivial) extension of Goedel's theorem  my double incompleteness theorem (Section 6.2). Since that time, I'm an (amateur?) philosopher of mathematics. My education up to Ph.D. in 1979 was purely mathematical, but I was elected Professor of Information Technologies (second class computer science). However, reading all the funny things about mathematics written even by the most prominent philosophers and mathematicians, I'm feeling at least as their kind of person. Karlis Podnieks, March 25, 2007 
John Keats (1795  1821) "I have left no immortal work behind me — nothing to make my friends proud of my memory  but I have lov'd the principle of beauty in all things, and if I had had time I would have made myself remember'd". Isabella; or, The Pot of Basil ... XXXIX. “I am a shadow now, alas! alas! ... Full text at 
foundations of mathematics, philosophy of mathematics, logic, model, modeling, modelling, mathematical, online, web, book, Internet, tutorial, textbook, foundations, mathematics, teaching, learning, study, mathematical logic, student, Podnieks, Karlis, philosophy, free, download