Brouwer and the Rejection of Excluded Middle

1. The Core of Intuitionism: Mathematics as Mental Construction

Brouwer's starting point is a rejection of the two dominant views of mathematics of his time:

• Platonism: The view that mathematical objects (numbers, sets, etc.) exist in an abstract, eternal realm, and mathematical truths are discovered, not created.
• Formalism: The view that mathematics is a game of manipulating symbols according to fixed rules, without any intrinsic meaning.

In place of these, Brouwer proposed that mathematics is a creation of the human mind.

The key takeaway from this reading is Brouwer's radical thesis: "there are no non-experienced truths." For an intuitionist, a mathematical statement is true if, and only if, the subject has performed a mental construction that verifies it. Truth is not about correspondence with an external reality; truth is the successful completion of a construction in the mind.

This idea is the first of what Brouwer called the "two acts of intuitionism."

This "First Act" has a profound consequence that reverses the traditional relationship between logic and mathematics. For a logicist like Frege or Russell, logic is the foundation upon which mathematics is built. For Brouwer, it is the other way around:

• Mathematics is the primary, pre-linguistic activity of mental construction.
• Logic is a secondary, descriptive science. It codifies the patterns of reasoning that have been found to be valid within the practice of mathematical construction.

Logic does not provide the rules for mathematics; mathematics provides the subject matter for logic.

2. The Argument Against the Law of the Excluded Middle

With this constructivist philosophy in place, we can now analyse Brouwer's case against the law of the excluded middle (LEM).

Classically, LEM states that for any proposition $P$, the statement $P\vee \neg P$ is a tautology. This is justified by the principle of bivalence: $P$ is either true or it is false. If it's true, $P\vee \neg P$ is true. If it's false, then $\neg P$ is true, and $P\vee \neg P$ is again true.

However, the intuitionist redefines the meaning of the logical connectives in terms of constructions. This is often called the Brouwer-Heyting-Kolmogorov (BHK) interpretation.

As the article explains, to have a proof of $A\vee B$, you must have:

• a proof of $A$, or
• a proof of $B$.

Now, apply this to the Law of the Excluded Middle, $P\vee \neg P$. To assert this, an intuitionist requires:

• a proof of $P$, or
• a proof of $\neg P$ (which means, a construction that shows a proof of $P$ leads to a contradiction).

The problem is that for many mathematical propositions, we have neither. Think of an unsolved problem like the Goldbach Conjecture ("Every even integer greater than 2 is the sum of two primes"). Let's call it $G$.

• A classical mathematician says the statement $G\vee \neg G$ is trivially true. The conjecture is either true or false, even if we don't know which.
• An intuitionist says that since we currently have no proof of $G$ and no proof of $\neg G$, we are not justified in asserting $G\vee \neg G$.

To accept LEM is to believe in "non-experienced truths"—to believe that a statement's truth-value is settled in advance, independent of our constructive activity. This is exactly what Brouwer's philosophy denies.

This leads to the rejection of proofs that depend on LEM. Such proofs are called non-constructive existence proofs. They show that something must exist without providing a method for finding it.

The classical proof is clever, but from an intuitionistic perspective, it is profoundly unsatisfying. It proves that a solution exists without constructing that solution. Since truth is construction, the proof fails to establish the truth of the conclusion because it doesn't tell us what A and B are.

3. From Unjustified to False: The Intuitionistic Continuum

So far, we have seen that LEM is considered unjustified in cases where we lack a proof. Brouwer went further, arguing that for certain domains, LEM is demonstrably false. His main tool for this was his theory of the continuum, based on the notion of choice sequences.

A choice sequence is a sequence of numbers (e.g., digits of a real number) that is not determined in advance by a rule, but is created by a series of free choices over time. You can think of it as a process of construction that is perpetually unfinished.

As the video explains, at any finite stage, two choice sequences $\alpha$ and $\beta$ may have identical initial segments. But because their future is open, there is no pre-existing fact of the matter about whether they will eventually diverge or not. Therefore, we cannot assert the disjunction $\alpha =\beta \vee \alpha \ne \beta$. The law of the excluded middle fails for properties of these "in-progress" objects.

This concept allowed Brouwer to construct strong counterexamples to universally quantified versions of LEM. The most famous is this:

Classical statement: "For every real number $x$, $x$ is rational or $x$ is irrational." ($\forall x\in \mathbb{R}(Q(x)\vee \neg Q(x))$)

Brouwer was able to show that this statement is intuitionistically false. He did this by constructing a choice sequence (defining a real number) where the choice of each new digit depends on the status of an unsolved mathematical problem. This creates a real number for which it is impossible, in principle, to decide whether it is rational or not. Because we cannot make this decision for all real numbers, the universally quantified statement is refuted.

Conclusion

Today we have analysed Brouwer's profound philosophical critique of a foundational principle of classical logic. His argument is not a mere technical quibble; it stems from a complete re-evaluation of the nature of mathematical truth.

Key Takeaways:

• Constructivist Foundation: Brouwer's intuitionism is founded on the idea that mathematics is a mental activity. A proposition is true if and only if we have a mental construction that proves it. "Truth is experienced."
• Rejection of Bivalence: This foundation entails the rejection of the classical principle of bivalence. If we have no proof for $P$ and no proof for $\neg P$, then $P$ has no truth-value.
• Argument Against LEM: The law of the excluded middle ($P\vee \neg P$) is rejected because asserting it requires having a proof of $P$ or a proof of $\neg P$. For unsolved problems, we have neither, so we cannot assert the disjunction.
• Unacceptable Proofs: This renders non-constructive proofs—those that prove existence by contradiction or by cases without providing an instance—unacceptable. The ${\sqrt{2}}^{\sqrt{2}}$ example is a paradigm case.
• Strong Refutation: Using the concept of choice sequences in his theory of the continuum, Brouwer went beyond showing LEM was unjustified. He argued that universally quantified forms of LEM, such as "every real number is either rational or irrational," are demonstrably false.

Preview of the Next Lesson:

We've seen that the intuitionistic meaning of the logical connectives is central to Brouwer's argument. In the next lesson, we will delve deeper into this, formally evaluating the Brouwer-Heyting-Kolmogorov (BHK) interpretation as an alternative account of the meaning of all the logical connectives. This will allow us to see how intuitionistic logic functions as a complete, coherent system in its own right, rather than just a restriction of classical logic.