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Argues that mathematics is an extension of logic and therefore all mathematical truths are logical truths. Impact: Initiative by Frege and Russell to ground mathematics on a firm logical foundation, leading to the development of formal logic.

Views mathematics as the study of structures, or patterns of relationships, rather than particular objects. Impact: Influenced by the work of Saunders Mac Lane and Michael Dummett, it provides an ontologically modest outlook on the existence of mathematical entities.

Holds that reason is the primary source of mathematical knowledge rather than sensory experience. Impact: Emphasizes the abstract nature and logical foundation of mathematics, contributing to the philosophical discussion around mathematical knowledge.

Claims that mathematical knowledge is independent of sensory experience and can be known a priori. Impact: Underlines the role of intuition and reasoning in understanding mathematics, influencing epistemology in mathematics.

Asserts that mathematical concepts originate from empirical evidence and experience. Impact: Emphasizes the connection between mathematics and the physical world, influencing the philosophy of applied mathematics.

Denies the existence of abstract mathematical objects, arguing that they are mere names without any corresponding reality. Impact: Challenges the realist perspective and leads to a reconsideration of the ontological commitments of mathematics.

Proposes that mathematical statements are akin to sentences in a story and should not be taken literally. Impact: Offers a resolution to the ontological problem by suggesting that mathematical entities do not need to be 'real' in order to be mathematically useful.

Focuses on the lived experience of mathematical practice to understand the phenomena and essence of mathematics. Impact: Offers insights into the nature of mathematical activity and its context within human experience.

Suggests that mathematical objects exist in a range of possible worlds and that mathematics explores all these possibilities. Impact: Provides a meta-theoretic perspective that acknowledges multiple possibilities in mathematical structures.

Considers mathematics to be a creation of formal systems of symbols with no inherent meaning. Impact: Exemplified by David Hilbert's work, it influenced the way mathematicians understand the structure and consistency of mathematical systems.

Concerns the belief that mathematical definitions and statements must be constructed in a non-circular way to avoid paradoxes. Impact: Impacts foundational studies and the formulation of mathematical proofs, demanding clarity and non-circular reasoning.

Aristotle's view that mathematical entities exist as aspects of physical objects and their relations. Impact: Influenced early scientific thought by grounding mathematical concepts in the tangible world.

Proposes that the study of mathematics should align with the methods used in natural sciences. Impact: Encourages empirical approaches to understanding mathematics and integrating it with scientific inquiry.

Holds that mathematical entities are abstract, non-physical objects that exist independently of human minds. Impact: Provides a realist perspective towards mathematical objects, influencing both philosophy and the development of mathematics.

Centers on the idea that the unconscious is structured like a language, and that subjectivity is formed through symbolic and imaginary orders. Impact: Influential in literary theory, critical theory, and has led to a rethinking of the psychoanalytic practice.

Considers the utility and practical application of mathematical concepts as foundational to their meaning and truth. Impact: Focuses on the effectiveness of mathematics in problem-solving and its adaptability to practical situations.

Revisits the logicist thesis by seeking to ground mathematical truths in a small number of logical principles without the inconsistencies of traditional logicism. Impact: Revitalizes efforts to find a philosophical basis for mathematics in logic.

Posits that mathematical truths are not discovered but are instead created by the mathematician's intellect. Impact: Initiated by L.E.J. Brouwer, it highlights the importance of constructive proofs and has led to the study of intuitionistic logic.

Maintains that mathematical knowledge is constructed by the learner through mental activity rather than passively received. Impact: Influences mathematics education and encourages active learning and understanding over rote memorization.