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Psychologism sees mathematics as a mental discipline, with mathematical objects being dependent on the human mind and its processes. An example of a philosopher who held a psychologistic view is Edmund Husserl.

Aristotelianism in mathematics sees mathematics as abstracting from the actual world, with mathematical entities existing potentially within concrete objects. Aristotle's own views on mathematics exemplify this position.

Type theory is a framework that avoids certain paradoxes of set theory by organizing objects into types. Philosophers like Russell and Whitehead have used type theory in their logical analysis.

Fictionalism considers mathematical objects to be useful fictions, much like characters in a story, and argues that mathematical statements do not necessarily denote properties of the physical world. Hartry Field is a proponent of this view.

Constructivism is the view that mathematical knowledge must be constructed by the mathematician, and that mathematical objects do not exist independently of the knowing subject. Ernst Zermelo was a proponent of constructivist approaches.

Modal structuralism blends modal logic with structuralist views, suggesting that mathematical structures might be potential or necessary rather than actual entities. Geoffrey Hellman is known for this approach.

Intuitionism suggests that mathematical objects are mental constructs and that mathematical truth is a result of the mind's activities. L. E. J. Brouwer championed this position.

Neo-Logicism is a revival of logicism which suggests that much of mathematics can be derived from logical axioms and definitions. Philosophers like Crispin Wright and Bob Hale are contemporary neo-logicists.

Platonism holds that mathematical entities exist independently of human minds, in a non-physical realm. Philosophers associated with this view include Plato and Kurt Gödel.

Conventionalism posits that mathematical truths are not objective but are based on conventions or agreements among human beings. Henri Poincaré is often associated with this viewpoint.

Mathematical realism is the belief in the objective reality of mathematical objects and that mathematical statements can have a true or false value independently of human knowledge. Penelope Maddy and Gödel are modern supporters of this view.

Intuitionistic logic is a system of logic that reflects the philosophical stance of intuitionism in mathematics, where the law of excluded middle is not universally accepted. It was developed by Brouwer and furthered by Arend Heyting.

This approach to set theory allows for sets that can contain themselves, violating traditional set-theoretic axioms, and reflecting certain philosophical stances on the nature of sets. Peter Aczel's work on antifoundation axioms is central here.

Empiricism in mathematics holds that mathematical knowledge is derived from sense experience. John Stuart Mill is an example of a philosopher with an empiricist perspective on mathematics.

Predicativism holds that for a mathematical object to be considered well-defined, it must be describable without the use of impredicative definitions. Herman Weyl and Solomon Feferman worked on predicative approaches to mathematics.

Nominalism asserts that abstract mathematical objects do not exist, and only names and labels are operative in mathematics. One of the philosophers connected with this view is Willard Van Orman Quine.

Logicism attempts to ground mathematics in logic, claiming that mathematical truths are logical truths. Bertrand Russell and Gottlob Frege are key figures in this movement.

Structuralism in the philosophy of mathematics posits that the subject of mathematics is the structure, or form, of mathematical objects, rather than the objects themselves. Mathematicians and philosophers such as Saunders Mac Lane and Michael Resnik are associated with structuralism.

This field of philosophy of mathematics is concerned with the actual practices of mathematicians, including proof, explanation, and visualization. Philosophers like Paolo Mancosu work in this area.