Intuitionism

Overview and Historical Background

Intuitionism is a family of philosophical views that emphasizes immediate, non-inferential knowledge, called intuition, as a foundation for understanding certain kinds of truths. Two major forms are commonly distinguished:

• Ethical intuitionism, which concerns moral knowledge and value
• Mathematical intuitionism, which concerns the nature of mathematical truth and proof

Although they share a name and the appeal to intuition, these two strands developed largely independently and address different philosophical problems.

Historically, ideas resembling intuitionism can be traced to early modern philosophers, including Immanuel Kant, who spoke of a priori forms of intuition (space and time), and to the common-sense moral philosophy of Thomas Reid. However, intuitionism became a more clearly defined school in the late 19th and early 20th centuries.

In ethics, British philosophers such as G. E. Moore, H. A. Prichard, and W. D. Ross defended the view that certain moral truths are known directly. In mathematics, the Dutch mathematician L. E. J. Brouwer pioneered an intuitionistic approach to arithmetic and analysis, later formalized by Arend Heyting. These developments occurred against the backdrop of increasing formalization and skepticism in both ethics and the foundations of mathematics.

Ethical Intuitionism

Ethical intuitionism is the view that some moral propositions are self-evident and can be known through rational intuition rather than empirical observation or complex argument. It generally combines two theses:

1. Epistemological thesis: At least some moral truths (e.g., “It is wrong to inflict unnecessary suffering”) can be known immediately by a competent, reflective agent.
2. Metaphysical thesis (often but not always): Moral properties such as goodness or rightness are non-natural and irreducible to natural or psychological properties.

Key figures include:

• G. E. Moore, who argued that “good” is a simple, indefinable property and criticized attempts to define it in natural terms as committing the “naturalistic fallacy.”
• H. A. Prichard, who maintained that we recognize our duties directly through reflection, rather than by deriving them from more basic principles.
• W. D. Ross, who proposed a pluralistic list of prima facie duties (fidelity, reparation, gratitude, justice, beneficence, self-improvement, and non-maleficence) that are self-evident, though their application in particular cases can be complex.

Proponents contend that ethical intuitionism explains how moral reasoning actually operates: people often recognize certain actions as wrong or right without first inferring this from empirical data or a single master principle like utility. It also aims to explain the apparent objectivity of moral claims, while avoiding reduction of morality to mere emotion, desire, or social convention.

Critics raise several objections:

• Epistemic worry: How can we trust intuitions as a reliable source of knowledge, especially given moral disagreement?
• Disagreement problem: Different people and cultures may report conflicting moral intuitions; intuitionists must explain why some intuitions are authoritative and others mistaken.
• Explanation problem: Some argue that intuitionism leaves mysterious the nature of our access to moral facts, which do not seem to be observable or causally related to us in ordinary ways.

Despite these challenges, ethical intuitionism has experienced a revival in contemporary analytic ethics, often in more modest forms that treat intuitions as defeasible, theory-guiding data rather than infallible insights.

Mathematical Intuitionism

Mathematical intuitionism is a foundational view in the philosophy of mathematics, initiated by L. E. J. Brouwer in the early 20th century. It asserts that mathematics is fundamentally a mental activity of construction, and that mathematical objects exist only insofar as they can be constructed by the mind.

Several core ideas characterize this approach:

1. Constructivism about existence: To say that a mathematical object exists is to say that there is a method to construct it. Mere non-contradiction or logical possibility does not suffice.
2. Rejection of the unrestricted law of excluded middle: Intuitionists deny that for every mathematical statement P, the disjunction (P \lor \neg P) is automatically true, particularly when no proof of either P or its negation is available. This leads to intuitionistic logic, a formal system weaker than classical logic.
3. Emphasis on mental constructions: Mathematics is seen not as a discovery of an independently existing abstract realm, but as the result of the mathematician’s constructive activities, rooted in basic intuitions of time and counting.

Brouwer’s work led to the development of intuitionistic arithmetic and analysis, in which certain classical theorems do not hold in their usual form. For example, classical proofs by contradiction that assert existence without giving a constructive example are typically rejected.

Arend Heyting and others provided formal systems capturing intuitionistic logic and mathematics. Later, connections were discovered between intuitionistic logic and computability theory, type theory, and topos theory. The Curry–Howard correspondence, relating proofs to programs and propositions to types, is often seen as consonant with intuitionistic ideas, since it interprets proofs as explicit constructions.

Supporters argue that mathematical intuitionism offers a more faithful account of mathematical practice in constructive areas and clarifies the meaning of “existence” in mathematics. They also claim that it avoids certain paradoxes and philosophical difficulties associated with actual infinities and non-constructive existence proofs.

Critics, however, object that intuitionism abandons widely used classical results, complicates standard mathematics, and imposes philosophical constraints on a discipline that many see as autonomous. Some also question whether the appeal to mental constructions provides a sufficiently objective basis for mathematical truth.

Criticisms and Legacy

Both ethical and mathematical intuitionism face persistent criticisms, especially regarding the status and reliability of intuition.

In ethics, skeptics challenge whether there are genuine self-evident moral truths and whether appeals to intuition simply mask unexamined social or psychological influences. Intuitionists respond by emphasizing careful reflection, the possibility of error and revision, and analogies with self-evident truths in logic or mathematics.

In mathematics, many working mathematicians continue to use classical methods that intuitionists reject, such as non-constructive existence proofs. Nevertheless, intuitionism has had a lasting impact on constructive mathematics, proof theory, and areas of computer science where the constructive content of proofs is crucial.

Overall, intuitionism remains significant as a philosophical stance that insists on the centrality of immediate, non-inferential insight—whether moral or mathematical—while provoking ongoing debate about the sources and limits of human knowledge. Its legacy can be seen both in contemporary ethical theory and in alternative foundations for mathematics that prioritize construction, computation, and explicit methods of proof.