What is Intuitionism?

by ippolit · Published April 9, 2018 · Updated October 3, 2021

Introduction

Intuitionism is a methodological approach in Logic that takes mathematics, its theorems and maxims, to be a mental construct – an activity of the human mind. It is opposed to the view called Mathematical Realism, which holds that mathematics is not just a field of study by humans, but also that it contains truths independent of the human mind or their activity – it is objective in the pure sense of the word. Under Mathematical Realism, theorems and maxims hold the same truths independent of language. This also means that, by extension, Intuitionism holds that both logic and mathematics are not activities of the mind through which human being can recognise different properties of the objective world or the meaning of life – they do not not simply reveal themselves to the human mind. Instead, Intuitionism holds that logic and mathematics are only applications of particular methods that are consistent internally – i.e. are not prone to logical paradoxes – through which more complex systems can be understood. Of course, these systems too are mental constructs and created by the human mind.

Accordingly, Intuitionists view propositions as true (or false) insofar these can be intuited – that is to say, to understand and work out the truth of a proposition by instinct instead of proving them. Intuitionism makes use of constructive logic, which claims that (mathematical) objects are only constructions of the mine. However, instead of denying these objects any existence, Intuitionism does the opposite by claiming that because it can be constructed by the mine it can be said that a particular object exists. Intuitionism thus preserves justifications for propositions while getting rid of absolute truths. In this sense, Intuitionism can be called a variation of Mathematical Constructivism – a view that holds that in order to prove that mathematical objects exist, human beings must first construct them.

Another key difference with classical logic is that Intuitionism interprets the notion of negation differently. Negation for Intuitionists does not mean that a particular proposition is false, instead it only goes so far as to say that such propositions can be refuted – i.e. we can only say that there is proof of not having any proof for the given proposition.

History of Intuitionism

Early Intuitionist thinkers are traced to the mid-19th century, who followed on the discussion between the German mathematician Georg Cantor and his mentor Leopold Kronecker. The late-19th century discussions between Gottlob Frege and Bertrand Russell also contributed to the development of Intuitionism. However, none of these thinkers could be said to have been Intuitionists themselves; and the first detailed analysis and presentation of Intuitionism can be attributed to the early 20th century Dutch mathematician L. E. J. Brouwer, who presented the theory in a series of paper in the early- and mid-1920s, now printed in his Collected Works.

A different and perhaps independent version of Intuitionism was developed by the French philosopher Henri Bergson in An Introduction to Metaphysics. Bergson thought of knowledge in two distinct ways: an object is known either ‘absolutely’ or ‘relatively’, where, the latter knowledge is gained through analysis, while the former through intuition. Bergson understood intuition as “a simple, indivisible experience of sympathy through which one is moved into the inner being of an object to grasp what is unique and ineffable within it” (Sideri, 2017, p. 53).

In the middle of the 20th century, the American mathematician Stephen Cole Kleene (today known more for his invention of ‘regular expressions’) contributed significantly to Intuitionism by combining it with elements from Mathematical Realism. His major work on this topic is Introduction to Meta-Mathematics.