Val Dusek, Constructivism in philosophy, mathematics and other fields

Constructivism is a tendency in the philosophy of the past few centuries. Probably the earliest proponents of the claim that our knowledge is constructed were Thomas Hobbes (1588–1679) and Giambattista Vico (1668– 1744). Both of these philosophers claimed that we know best what we make or construct. Hobbes claimed that mathematics and the political state were both constructed by arbitrary decision. In mathematics and science, the arbitrary decision is stipulative definition. In society, the decision was the subordination of oneself to the ruler in the social contract. Vico claimed that we know mathematics and history best because we construct both. For Vico history is made by humans in collective action.

The major source of the varied ideas of constructivism in many fields is Immanuel Kant (1724–1804). Kant (1781) held that mathematics is constructed. We construct arithmetic by counting and construct geometry by drawing imaginary lines in space. Kant also claimed that we construct concepts in mathematics, but that philosophical (metaphysical) concepts are not constructed, but dogmatically postulated in definitions. For Kant, the source of the constructive activity is the mind. Various faculties or capacities are part of the makeup of the mind. Kant differed from the British empiricists in that he emphasized the extent to which the mind is active in the formation of knowledge. Although sensation is passive, conceptualization is active. We organize and structure our knowledge. Through categories we unify our knowledge. Kant compared his own innovation in theory of knowledge to the “Copernican Revolution” (the astronomical revolution of Copernicus that replaced the earth as the center of the solar system with the sun as center). Some have suggested that, given that Kant makes the active self the center of knowledge, his revolution is more like the Ptolemaic, earth-centered, theory of astronomy.

After Kant a variety of tendencies in philosophy furthered constructivist notions. While Kant claimed that we structure our knowledge or experience, he held that there is an independent reality, things in themselves, which we cannot know or describe, because all that we know or describe is structured in terms of the forms of our perceptual intuition and the categories of our mind. We can know that things in themselves exist, as the source of the resistance of objects to our desires, and the source of the input to our passive reception of perceptual data. But we cannot know anything about the qualities or characteristics of things in themselves.

Gottlieb Fichte (1794), much to Kant’s horror, proposed that since the thing in itself is inaccessible, and we cannot say anything about it, philosophy should drop the thing in itself and hold that the mind posits or creates not simply experience but reality as such. For Fichte the mind “posits” reality, and its positing is prior even to the laws of logic. Within later constructivism there are differences between those who claim merely that our knowledge is constructed, as did Kant, and those who claim, like Fichte, that objects and external reality themselves are constructed.

Hegel (1770–1831) added a historical dimension to Kant’s categories. Kant had presented the categories as universal for reason as such and basically unchanging. Kant claimed that even extraterrestrials and angels would share our categories. Hegel added an emphasis on historical development to philosophy. Hegel claimed that the categories develop through time and history. Even in logic, there is a dialectical sequence of categories. For instance, Hegel began his Logic (1812–16) by generating non-Being from Being and then producing the synthesis of Becoming. In his Phenomenology of Mind (1807) Hegel developed the categories of knowledge and ethics in a quasi-historical manner. Ancient Greece and Rome, the French Revolution, and more recent Romanticism furnish examples of the sequence of ethical and social categories. Most later Hegelians in Germany and Italy treated Hegelianism as a fully historical philosophy.

Marx and Engels (1846) claimed that the construction of categories was not a purely mental sequence, but was a material process of actual social and historical activity of production in the economy. For Marx, real, social history, not an idealized history of spirit, generates the frameworks (ideologies) in terms of which people understand the world.

The neo-Kantian school revived Kant’s ideas in the late nineteenth century. One branch of the neo-Kantians (the Marburg School) emphasized the construction of lawful (nomothetic) scientific and mathematical knowledge. Another branch (the Southwest German School) focused on the construction of historical and cultural knowledge of unique individuals (idiographic knowledge) in the humanities. The nomothetic versus idiographic dichotomy is manifest in the later opposition of the logical positivists and the hermeneuticists (see chapter 1 on positivism and chapter 5 on hermeneutics).

In the late nineteenth and early twentieth centuries, constructive ideas of mathematics were developed into detailed philosophies of mathematics and programs for the rigorous building up of systems of mathematics.

Henri Poincaré in France (who also was granddaddy of parts of chaos theory) and Jan E. Brouwer in Holland claimed that mathematics is built up from the ability to count (Brouwer) and the principle of mathematical induction (Poincaré), and that these notions went beyond purely formal logic (Poincaré, 1902, 1913; Brouwer, 1907–55). Constructive mathematics is a growing minority current within mathematics (Bridges, 1979; Rosenblatt, 1984), and has gained some renewal from theories of computers and computation (Grandy, 1977, chapter 8). Several of the leading theorists in the constructivist approach in education began in mathematics education (von Glasersfeld, 1995; Ernest, 1998).

In the early twentieth century, the British logician Bertrand Russell (1914) and the logical positivist Rudolf Carnap (1928) developed the notion of the logical construction of the world. According to Russell’s logical construction doctrine, all of our factual knowledge is based on data from direct sense experience (or sense data). Physical objects are simply patterns and sequences of these sense data. What we commonsensically call a physical object is, in terms of rigorous knowledge, a logical construction from sense data. The various perspectives we have or experience are systematically combined and organized to give us the notion of the physical object. This form of constructivism is different from Kant’s constructivism and is a logically developed version of British empiricism. It is interesting, however, to note that constructivist themes were very much a part of the logical positivist tradition that social constructivists in contemporary philosophy of technology think they are rejecting. At least one contemporary survey and evaluation of social constructivism includes logical constructivism in its scope (Hacking, 1999).

Ironically, Hume’s purely logical criticism of induction and causality led to logical positivism, but his appeal to “habit and custom” in explaining the psychological reasons why we believe in induction and causality strongly resemble social constructivist accounts.

Kantian constructivist ideas also found their way into psychology in Jean Piaget’s (1896–1980) theories of the growth of knowledge in the child. Piaget has very strongly Kantian approaches to knowledge, especially in his early works (Piaget, 1930, 1952). Unlike Kant, Piaget believes that the categorization or organization of knowledge develops and changes with the growth of the individual, just as Hegel thought it developed historically in society. For Piaget, the categories of thing and of conservation can be shown to develop through various stages. The Russian psychologist Lev Vygotsky (1896–1934) developed a theory of cognitive development that emphasized more than Piaget the social dimension of the development of the child’s conceptual framework (Vygotsky, 1925–34b).

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