THE WORLD OF MATHEMATICS

algebra arithmetic assertion assumptions average axiom of choice axiomatic axiomatic method Bertrand Russell calculation called cardinal number chance concept consider consistency construction contains counting curve deduction defined definition denumerably infinite derived elements equal equation Ernest Nagel errors Euclid euclidean geometry example experience experimental expression fact false Figure finite number formal formula given Goedel number googol googolplex HANS HAHN Hilbert ideas implication infinity integer interpretation intuition Leibniz logic m-class mathe mathematical logic mathematical propositions mathematician matics meaning measure ment meta-mathematical method natural numbers non-Euclidean geometry notion objects observation obtained operation ordinary parallel axiom philosophy physical plane positive number possible postulates principle problem proof proved pure question real numbers reasoning relation represent result rule sample sense sentence sentential calculus sentential function space statement straight line symbols theorems theory things tion transformations triangle true truth variables velocity words