The constructivist animosity against the law of the excluded middle

[Veritas Aequitas]

My philosophical stance is that of Constructivism [Kantian]. The LEM is only useful for classical logic which is limited.
ALL of reality is conditioned upon a continuum from 0.001% to 99.99%

Here is AI's [wR] on the why LEM is limited;

The Law of the Excluded Middle (LEM) states that for any proposition P, either P is true or not-P is true. There's no third option. Constructivists, however, take issue with this principle in the context of mathematics. Here's a breakdown of their critique:

1. Emphasis on Construction: Constructivists believe mathematics is about building and manipulating objects. A proof, for them, isn't just logical deduction but a constructive procedure that demonstrates the existence of a mathematical entity or the validity of an operation.

2. Undecidable Propositions: Not all propositions might be demonstrably true or false. LEM forces a decision, but what if we can't construct a proof for P or not-P? This could be due to the limitations of our current knowledge or the inherent nature of the proposition.

3. The Case of Infinite Sets: Classical mathematics freely uses LEM for infinite sets. But constructivists argue that we can't necessarily verify the truth or falsity of statements about infinite objects. We can't check every element in an infinite set, so how can we be sure a property either holds or doesn't for all elements?

4. Alternative Logic: Intuitionistic logic, a foundation for constructive mathematics, rejects LEM. It focuses on proofs rather than truth values. If we can't construct a proof for P or not-P, the proposition remains undecided within the system.

Analogy: Imagine you have a box full of real numbers. LEM says every number must be either rational or irrational. Constructivists argue that with our current tools, we might not be able to definitively categorize every number we pull out. There could be numbers that resist classification for now.

By rejecting LEM, constructivists develop a more cautious and step-by-step approach to mathematics.
They focus on what can be demonstrably constructed rather than relying on the principle of having to pick a side.
This alternative viewpoint has led to interesting developments in constructive mathematics, offering a different lens to explore mathematical concepts.