Question

In Brouwer's intuitionism, the concept of continuum is described as being perceived as a whole by intuition. However, the paper mentions the use of choice sequences, raising the question of why such sequences are employed if the continuum is intuitively understood. How does the relationship between choice sequences and intuition in Brouwer's intuitionism contribute to the constructivist perspective on mathematical objects and the debate on potential and actual infinity? Additionally, considering the rejection of Platonic realms and transcendental truth, how does the use of choice sequences provide a nominalistic and metaphysical explanation for the concept of infinity in the context of physical computation?

Answer 1

Brouwer's intuitionism views the mathematical continuum as a mental construct through choice sequences, presenting a constructivist and nominalistic approach to infinity that rejects actual infinity in favor of potential infinity and the external existence of mathematical objects.

Step-by-step explanation:

In Brouwer's intuitionism, the continuum is not a pre-existing entity to discover but something that is constructed by the mind in real time through the process of choice sequences. Brouwer's conception of the continuum challenges the traditional view of mathematical infinity, by rejecting the notion of actual infinity in favor of potential infinity - a viewpoint that aligns with a constructivist perspective. Unlike the Platonic realm, where mathematical objects have an independent, transcendent existence, Brouwer's view is more nominalistic, presenting mathematical entities as mental constructions without assuming their existence outside of human cognition. Through this lens, choice sequences allow an individual to grasp infinity not as a complete totality but as an unfinished, ever-evolving process. Yet, the relationship between intuition and choice sequences is paradoxical as it suggests that while the continuum can be intuitively understood as a whole, its construction through choices sequences implies a procedural understanding.