What is the difference between Intuitionism, Formalism, and Logicism? Namely - on which issues do they disagree? And what is the relation of those schools of thought to Platonism, Nominalism, and Fictionalism?
Here is partial preliminary answer:
Logicism, intuitionism and formalism are three traditional views about the nature of mathematics.
Formalism was introduced by the German mathematician David Hilbert, and it holds that all mathematics can be reduced to rules for manipulating formulas without any reference to the meanings of the formulas. Thus, according to formalism, it is the mathematical symbols themselves and not any meaning that might be ascribed to them that are the basic objects of mathematics.
Logicism was introduced by the German mathematician Gottlob Frege and the British mathematician Bertrand Russell. It holds that mathematics is actually logic. According to Logicism, all of mathematics can be deduced from pure logic without the use of any specifically mathematical concepts (such as number).
Intuitionism was introduced by the Dutch mathematician L.E.J. Brouwer. It holds that the primary objects of mathematics are mental constructions governed by self-evident laws. Intuitionism has challenged many of the principles of mathematics as being nonconstructive and hence mathematically meaningless.
In addition to browsing the SEP, you can read more of introductory material in the link: http://www.newworldencyclopedia.org/entry/Philosophy_of_Mathematics#Logicism