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.
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.