How come intuitive thinking is related to constructing a proof?

Question

I am researching Constructivism and Intuitionism. I can't understand why Intuitionism and Intuitionistic Logic are named as they are.

Intuitionistic logic requires constructing a proof of every statement that is used. Therefore, for me, it feels like the opposite of intuitive thinking!

As I understand, Intuition in every day life is vaguely defined as using information we currently don't have when making choices. This is the opposite of what intuitionism says (avoiding excluded middle and such).

Answer 1

Intuitionism is a philosophy of mathematics that was introduced by the Dutch mathematician Luitzen Egbertus Jan Brouwer (1881–1966): he developed a very personal philosophy of mathematics that founds mathematics (partially following Kant; see Kant's Philosophy of Mathematics) on a pure intuition of time.

You can see his Intuitionism and formalism (1912) :

The question where mathematical exactness does exist, is answered [by the intuitionist] : in the human intellect. In Kant we find an old form of intuitionism, now almost completely abandoned, in which time and space are taken to be forms of conception inherent in human reason. For Kant the axioms of arithmetic and geometry were synthetic a priori judgments, i. e., judgments independent of experience and not capable of analytical demonstration [...]

the position of intuitionism has recovered by abandoning Kant's apriority of space but adhering the more resolutely to the apriority of time. This neo-intuitionism considers the falling apart of moments of life into qualitatively dierent parts, to be reunited only while remaining separated by time as the fundamental phenomenon of the human intellect, passing by abstracting from its emotional content into the fundamental phenomenon of mathematical thinking, the intuition of the bare twooneness.

This intuition of two-oneness, the basal intuition of mathematics, creates not only the numbers one and two, but also all finite ordinal numbers, inasmuch as one of the elements of the two-oneness may be thought of as a new two-oneness, which process may be repeated indefinitely; this gives rise still further to the smallest infinite ordinal number ω.

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For a different form of philosophy of mathematics based on intuition, and less philosophically muddled than Brouwer's, see Henri Poincaré and his conception of:

intuition of pure number — whence comes the axiom of induction in mathematics.

See e.g. La Science et l'Hypothèse (1902), English transl. (1905), Ch.1 : On the Nature of Mathematical Reasoning, page 17 :

Mathematical induction — i.e., proof by recurrence — is necessarily imposed on us, because it is only the affirmation of a property of the mind itself.

Answer 2

The second sentence of the SEP's article on mathematical intuitionism gives a pretty good explanation of why it is named as it is:

Intuitionism is based on the idea that mathematics is a creation of the mind.

As Mauro points out in his comment to your question, Brouwer based a lot of his ideas on Kant's views of mathematics. Central to Kant's views is the idea that mathematics is intuitive, meaning purely of the mind:

[Mathematical] concepts must immediately be exhibited in concreto in pure intuition, through which anything unfounded and arbitrary instantly becomes obvious.

Construction in intuitionism shouldn't be thought of as you would think of construction in terms of building a house. It means that we are constructing the rules, and therefore the objects we create with the rules, from within our own minds. This view is in contrast to, for example, mathematical platonism which says that mathematical objects are actual immaterial or abstract objects. Again, the reason that intuitionism says mathematical ideas are "constructed" is to contrast with the idea that they are "discovered" which is prevalent in platonism. The role Kant's philosophy of mind and philosophy of mathematics played in Brouwer's cannot be understated. For a better sense of what he means by intuition you will probably be greatly helped by studying Kant. Inevitably you'll end up reading about the analytic/synthetic distinction as well.

Consider the differences in these statements: "I have constructed a proof that 1 + 1 = 2" and "I have discovered a proof that 1 + 1 = 2" Its true that if we are being loose with language we might not mean a great deal of difference in those two statements, however if we are being very careful about the philosophy of mathematics those two statement are entirely different. Intuitionism is the school of thought that we can only construct proofs through our own intuition.

Answer 3

You are overthinking this. Intuitive logic is named such because that's the name it was given by its progenitor(s) based on it's origin, and not because the philosophy world at large reached a consensus that it was the best fit to describe the methods.