Brouwer has given us at least 3 lemons, which classical mathematicians find in varying degrees distasteful.

These lemons include:

- Loss of the logical identity called The Law of the Excluded Middle
- The near destruction of the prestigious journal “Mathematische Annalen”
- The recasting of fundamental mathematics into the mystical realm of intuitions.

Introduction:

If you go to Wiki and search for the Foundations of Mathematics you will find Intuitionism and some mention of L.E.J. Brouwer. In addition to the unusual/Kantian way in which Brouwer defines numbers, an odd constraint that he adds is that the logical identity:

Not (Not A) = A, is not valid.

The above identity is referred to as the Law of the Excluded Middle.

This deletion was made purposely because Brouwer wanted to avoid infinite sets.

(There are some shades of grey here. For some followers of Brouwer and maybe Brouwer himself, propositions that involve only finite sets may still obey the Law of The Excluded Middle.)

When I was young, I never questioned the Law of the Excluded Middle because it seemed self-evident in a binary environment such as mathematics. In fact logic was virtually never mentioned in any of my mathematics classes through high school.

The Law of the Excluded Middle is existentially important to a large portion of mathematics.

As I grew older there was a long period of time, (probably too long) when I worried whether or not the Law of the excluded Middle was in some way flawed.

An example of why denying the Law of the Excluded Middle is important to Brouwer:

Theorem: The Counting Numbers are Infinite.

Here I am assuming that the Counting numbers = {1, 2, 3 …}

Proof:

Assume not.

Then there must be some largest number N such that no other numbers are larger than N. This is because we are looking at a finite set which is well ordered.

Let M = N+1. This number exists both in the Peano axioms and in ZFC.

Since N+1 > N, substituting M for N+1 we have that the Counting number M is greater than N.

Thus, our original assumption has been falsified.

The reader should notice that we have assumed Not A i.e. the Counting Numbers are not infinite, and proved Not A to be false. Since Not (Not A) = A, we have proved that the Counting numbers are infinite.

However, if we take away the Law of the Excluded Middle this proof is no longer valid; and barring any constructive proof, Brouwer need not be compelled to believe that the Counting numbers are infinite.

End example:

This type of proof, generally called a proof by contradiction, is nearly ubiquitous in mathematics and is called a proof by contradiction.

Barring proofs by contradiction would probably destroy the bulk of mathematics. Certainly the vast majority of uniqueness proofs and all proofs by Induction would be gone, along with much of any remaining mathematics.

This massive destruction of mathematics could be considered a very large lemon.

Gödel’s Take:

Brouwer was trying to avoid the concept of infinity, but according to Wiki, Gödel proved that a logical system that omits the Law of the Excluded Middle must have an infinite valued logic. More than a little ironic.

End Gödel’s Take:

The second large lemon was Brouwer’s appointment to the editorial board at Mathematische Annalen, a prestigious mathematical journal at the time. His appointment was largely due to the support of David Hilbert, who was one of the founders of the journal. (Hilbert was an incredibly important person in the history of mathematics and a prominent mathematician in his own right. He was also the chairman of the Mathematics department at Gottingen, at the time generally considered the top school of mathematics in the world).

Assuming a standard distribution of proof types, most of the proofs used by the applicants hoping to be published, would be by contradiction. For whatever reason, Brouwer either sat on these proofs or disapproved them. This in turn brought the publication to a near halt. There was a public feud between Hilbert and Brouwer ending with Brouwer’s ejection from the journal.

A final lemon was Brouwer’s sense of Mysticism.

The comments below are from Wiki which in turn quoted Davis.

Much of Brouwer’s work required a sense of a mystical intuitions. “He wrote about them in a tract entitled Life, Art and Mysticism.

His Thesis adviser wrote about section II of Brouwer’s Doctoral thesis:

‘as it stands … all interwoven with some kind of pessimism and mystical attitude to life which is not mathematics, nor has anything to do with the foundations of mathematics’”.

What can be redeemed from all these lemons?

Brouwer’s Terminology for what appears to be infinite sets:

Brouwer uses the term “indefinitely finite”.

Regardless of whether or not the reader has an opinion on the validity of the Law of the Excluded Middle, I believe that the concept of an indefinite finite can be useful. Consider the following example:

Let Sn be a sequence, from the Counting numbers to the Rational numbers defined by the map (n, (1 + 1/n)^n).

Here n can be any Counting number, but it cannot be infinity. To me the Domain, i.e. the Counting numbers, has the same sense as Brouwer’s “indefinite finite”.

Tangentially, this sequence has the property that the limit of the sequence equals the transcendental number e. One of the most useful numbers in mathematics.

On a historical note, Kronecker, who lived a couple decades earlier than Brouwer and could be considered the first constructivist, believed that the limits of sequences, which limited to the transcendental numbers did not exist. On the other hand Brouwer accepted the existence of the transcendental numbers. For those who know about the Cauchy sequences, Brouwer effectively replaces Real numbers with Rational numbers in the definition and uses 1/n instead of ε. In this manner he can effectively avoids infinity and can complete the continuum, also called the Real numbers. (For those that care, I am using the word “complete” in both the common and technical sense).

Advantage of Brouwer’s mathematics:

In general people feel more comfortable about conclusions which use fewer assumptions. Brouwer’s proofs omit the Law of the Excluded Middle and thus use fewer assumptions. Regardless of whether or not a mathematician is classical or constructive they all seem to take pride in constructive proofs.

I am not certain but it appears to me that the Wiki authors are partisans of constructive proofs. Many times proofs by contradiction are simpler, they use fewer steps, than constructive proofs. For example, many proofs by induction are much simpler than constructive proofs. However, it seems to me, that Wiki will sometimes omit a simpler proof by contradiction in favor of constructive proofs.

A cautionary tale:

A translation of Gödel’s “On Formally Undecidable Propositions Of Principia Mathematica And Related Systems” was written by B Meltzer with an Introduction by R. B. Braithwaite. The introduction contains praise for the constructive nature of Gödel’s proof. But the proof relies, intimately, on the fundamental theorem of arithmetic (the unique factorization theorem of prime numbers) and while there have been constructive proofs of the factorization there have not been, to the best of my knowledge, any constructive proofs of its uniqueness.

In very large percentage of more complex proofs the underlying theorems can easily contain proofs by contradiction and one needs to be very careful about making declarations that are overly broad.

End of A cautionary tale:

The second lemon has no advantages that I can see. It is simply a tragedy. Brouwer fought for what he believed, many authors were not published and the majority eventually forced him out. Perhaps my lemonade is slightly tart.

The third lemon is about mysticism in mathematics. While I, like the vast majority of all mathematicians, believe in a precisely stated theorem using well defined terms and a rigorously defended chain of statements in order to prove the theorem, there is, at least for seminal theorems, a large degree of creativity involved. This creativity is likely, in part, informed by one’s experience in the field but it is also a product of one’s imagination. This combination can be viewed as mystical by some. I am afraid that to understand this statement you might need to be deeply involved in the field.

A book which touches on the sensitivities of one such mathematician is “Perfect Rigour” by Masha Gessen. It is insightful and a little sad to read about a Jewish mathematician growing up in Russia and the missed opportunities by US institutions.

So what’s the truth about the Law of the Excluded Middle?

I don’t know for sure, but for now I think that it does not matter. I see this more like the different geometries, Euclidean, Elliptical, and Hyperbolic. Mathematics, including the Law of the excluded Middle (standard mathematics) and mathematics excluding the Law of the Excluded Middle (Brouwer’s mathematics) are both valid and the usage can depend on the application. The reader should note that Brouwer’s mathematics are better suited for some applications in Quantum mechanics.

The following is from Wiki:

“Physicist Lee Smolin writes in Three Roads to Quantum Gravity that topos theory is “the right form of logic for cosmology” (page 30) and “In its first forms it was called ‘intuitionistic logic’” (page 31). “In this kind of logic, the statements an observer can make about the universe are divided into at least three groups: those that we can judge to be true, those that we can judge to be false and those whose truth we cannot decide upon at the present time””

Wiki also notes an advantage in the fields of “typed lambda calculi, topos theory and categorical logic, which are notable subjects in foundational mathematics and computer science”.

I would also say that much as Euclidean geometry is the default geometry, standard mathematics should be the default mathematics. However, neither Euclidean Geometry nor classical mathematics is right in any absolute sense.

Thanks Ed

P.S. There is more material on Brouwer in the Stanford Encyclopedia of Philosophy.