Making Lemonade out of L.E.J. Brouwer

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

These lemons include:

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


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 …}


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However, if we take away the Law of the Excluded Middle this proof is no longer valid; and barring any constructive proof, Brouwer need not be compelled to believe that the Counting numbers are infinite.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

Although a seriously great editorial, I have to say that it is a bit disturbing to me that any of this was ever an issue to you. I have to guess that it is an issue of the trees blinding one from the forest in which they live.

And with that conclusion, I have to say that you appear to still be missing the “big picture”.

Nice piece.

One concern would be that you are equating the denial of LEM (and constructive mathematics in general) with Brouwer’s mysticism. But LEM-denial is making a strong comeback these days due to the influence of computer science. There are sets of natural numbers such that neither the set nor its complement is computable, which is the intuition (my intuition, at least) behind the denial of LEM by modern constructivists.

One of the big buzzwords here is Martin-Löf type theory. This is one of several flavors of modern constructivism that deny LEM but dispense with Brouwer’s mysticism.

The type theorists are very serious these days and gaining mindshare, at least among those inclined to put a high value on the constructability of the objects that we posit to exist. No Axiom of Choice, for one thing. It’s an interesting factoid that the axiom of choice actually implies LEM. This is Diaconescu’s theorem.

Our ideas about foundations have always been historically contingent. Math is thousands of years old and set theory barely a hundred. With the overwhelming influence of computation in society there’s no reason constructivism won’t be the dominant paradigm in a hundred years (or sooner). I like to think of this as Brouwer’s revenge. Absent the mysticism, Brouwer may yet have the last laugh.

The ONLY incentive and reason for the promotion of anti-Exclusion (and as often, anti-logic) is purely, 100% political manipulation, having nothing whatsoever to do with the elite’s real beliefs. It is 100% mind manipulation toward a specific award, nothing more.

My own theory for the rise of neo-constructivism is that there are many students who thirty years ago would have gone into math, but today go into computer science. Their training leads them to believe that a thing only exists when you can produce it via an algorithm. A mere statement of existence is not regarded as sufficient.

If modern constructivism is political, what is its purpose?

An interesting theory worth some consideration.

The over-arching concern of world politics is the total and absolute control over all life, the “God-wannabes”. And part of that concern is the strategy of “obfuscation [, implication/hypnotism/false-flagging,] and extortion” (aka “Satanism”). A big part of obfuscation requires the elimination of confidence in Logic (aka “anti-God”).

Not being into what is actually going on in the philosophical and mental world of Man, I can understand you not seeing how all of that is related, but that is a part of the game … keeping people too preoccupied with trivial issues and irrationality such that they cannot see why they are dying out … “obfuscation”.

That actually makes perfect sense to me.

… Yet you immediately make a personal remark that is dismissive and rude – which is fine, you can’t help yourself – but that in this case also happens to be totally fucking inaccurate. I know for a fact that I’ve never published a word on this site about world affairs. JSH has no rational basis for his statement. He’s just gorging on his own bile again.

I don’t get it man. You really have a bug up your ass. I did think we were having a civil conversation but I see there is none to be had with you.

Ed3 I apologize, I really thought we were going to have an intelligent and productive conversation. I will not hijack it further by interacting with JSH any more. I thought it could work and for a moment it seemed to, but he has just got a bug up his ass about something or other – I can’t for the life of me figure out what – and I see that it’s a distraction to the forum and a total waste of my time and energy to interact with him under any circumstances whatsoever.

Rude is in the eye of the beholder.

If you were not intending to be rude and dismissive, then what specifically is the evidentiary basis of this characterization of the state of my knowledge of the world:

What causes you to have this particular belief about me? Please be as specific as possible. I am curious to know.

In any event I could certainly stipulate that the world is under the control of maniacs pushing an agenda and a worldview; without having the slightest understanding of how Martin-Löf type theory fits into the plan. You stated that as a thesis but provided neither argument nor evidence.

I merely meant, without insult or offense intended, that even though obviously very mathematically informed, you had not become deeply philosophically informed, a different mentality … for whatever irrelevant cause. Philosophy is very different than physics or mathematics.

There are things of which I know far too little to debate. Everyone has their limits. It is hardly and insult to imply that someone has spoken beyond their own true knowledge. I do not take such implications to be insults. I seriously do not think that others should either, but some people get insulted by even the slightest implication that they are not the supreme source of all wisdom. Personally, I don’t really care. Place yourself wherever you belong.

I repeat: What is the evidentiary basis of your claim?

Hi wtf,

Thanks for your response. I appreciated it very much and intend to look into your references.

I did not mean to equate the denial of LEM with Brouwer’s mysticism. I should have made it clear that the denial of LEM has become an independent issue since Brouwer’s time.

In fact Aristotle had struggled with this very issue prior to codifying LEM. (It was fun for a short while to speculate that I had some of the same concerns as Aristotle. Then my wife reminded me that I was thousands of years late to the party).

Your concluding remarks seem insightful.

Thanks again Ed

As long as we are considering only lemonade, a little sideline could be added here that Pierce sought to avoid ambivalence, rather then face intuitive implications.

I have heard a little about Charles Sanders Peirce, with that spelling, if that’s who you are thinking of.

From what I understand – and this is a very superficial understanding on my part – Peirce says that whatever the continuum is, it certainly can not be the real numbers. For whatever we mean by continuum, it must certainly be the case that:

  • Every part is identical to the whole.

I find this a perfectly reasonable statement. Yet it is totally at odds with modern mathematics. In particular, the continuum – whatever exactly we mean by that – can not possibly be the mathematical real numbers. Because the real numbers may be decomposed into points, none of which are anything like the real numbers.

In particular, we may express the real numbers – or any set for that matter – as the disjoint union of all of its singletons. Symbolically:

$$\mathbb R = \cup_{x \in \mathbb R} ~{x}$$

From Peirce’s point of view, to the extent that I understand it, the above statement can’t possibly be true about the continuum. It’s absurd to break a continuum into points. Set theory in its entirety is therefore the wrong tool for understanding the continuum.

[I really hope I haven’t mangled his ideas too badly. This is as much as I understand].

I would very much like to know more about this point of view. Peirce is one of these people that when you first hear his ideas you wonder, Why isn’t this guy more famous?

He was the originator of pragmatism, and the front runner in semiotics, which was praised by the later positivists, he was commended to be the greatest modern philosopher-mathematician of all times.

His idea on the continuum , as I understand it, was that there are two kinds of infinite sets, one a definite, and two an indefinite one, and the law of excluded middle could not apply certain kinds of propositions, as the bivalance could.

General propositions of the kind, where truth and falsity of that proposition are of the sort , where if p~X, for instance, where the ~ sign is of the form - X follows p; then the law of contradiction will result in the necessary negation of that conclusion.

In ambivalence, there exists definite, rather then indefinite propositions, whereby, only the possibility exists what he calls by implication, rather then that necessity.

My daughter pressed me to go to Starbucks with her, so if I may continue , in terms more the philosophy of language effecting a different(ial) logic. , in a few hours.

I think it was pointed out rightly that the effect of language on logic is at play, and not the other way around.

The continuum problem in terms of this dichotomy between definite and indefinite sets flows out of this difference, in my opinion, and this is why the 'vague-math he coins between his shift from intuitive to to ‘definite’ values.

But as his Kantianism seeks to connect the dots,or the ’ points’ you so rightly being up, one can’t fail to implicate a hidden integration among possible propositions.

I’m that regard he has not advanced over the unsolved and perhaps unsolvable issues with intuitionism, as regards open and closed sets.

Hi Memo,

Thanks for your post.
I am intrigued by Pierce. Could you elaborate a little more, and, if my question is answerable, could you point the way to a foundation of mathematics that I think Pierce may have offered.

Thanks Ed

Peirce. Note the spelling.

Read anything by Fernando Zalamea, the most original philosopher of math in decades. … 0983700494 … 0956775012

If you Google around you can find pdf’s of Zalamea’s papers and articles on Peirce’s continuum. Zalamea is an extremely lucid and interesting writer. The only continental philosopher who actually understands modern math. The only philosopher of math of any type who has moved the philosophy of math from the turn of the 20th century Frege/Russell/Gödel to the modern world of Grothendieck and category theory.

Here’s a good starting point. … tinuum.pdf

Ed & WTF , : Charles Sanders is a very well written comprehensive reference, and I have one at home which I shall forward as well, with a logically based analysis for the layman as well.

:Robert Lane, ‘Physical Companion to C. Peirce’,

Hi wtf & Meno_

From your posts, I am very interested in discovering Peirce.

Thanks for your references.