## One Hundred Years of Intuitionism and Formalism

### 14 October 2012

**Sunday **

**A** message to the foundations of mathematics (FOM) listserv by **Frank Waaldijk** alerted me to the fact that today, 14 October 2012, is the one hundredth anniversary of Brouwer’s inaugural address at the University of Amsterdam, **“Intuitionism and Formalism.”** (I have discussed Frank Waaldijk earlier in **P or Not-P** and **What is the Relationship Between Constructive and Non-Constructive Mathematics?**)

**I** have called this post “One Hundred Years of Intuitionism and Formalism” but I should have called it “One Hundred Years of Intuitionism” since, of the three active contenders as theories for the foundations of mathematics a hundred years ago, only intuitionism is still with us in anything like its original form. The other contenders — formalism and logicism — are still with us, but in forms so different that they no longer resemble any kind of programmatic approach to the foundations of mathematics. In fact, it could be said that logicism was gradually transformed into technical foundational research, primarily logical in character, without any particular programmatic content, while formalism, following in a line of descent from Hilbert, has also been incrementally transformed into mainstream foundational research, but primarily mathematical in character, and also without any particular programmatic or even philosophical content.

**T**he very idea of “foundations” has come to be questioned in the past hundred years — though, as I commented a few days ago in **The Genealogy of the Technium**, the early philosophical foundationalist programs continue to influence my own thinking — and we have seen that intuitionism has been able to make the transition from a foundationalist-inspired doctrine to doctrine that might be called mathematical “best practices.” In contemporary philosophy of mathematics, one of the most influential schools of thought for the past couple of decades or more has been to focus not on theories of mathematics, but rather on mathematical practices. Sometimes this is called “neo-empiricism.”

**I**ntuitionism, I think, has benefited from the shift from the theoretical to the practical in the philosophy of mathematics, since intuitionism was always about making a distinction between the acceptable and the unacceptable in logical principles, mathematical reasoning, proof procedures, and all those activities that are part of the mathematician’s daily bread and butter. This shift has also made it possible for intuitionism to distance itself from its foundationalist roots at a time when foundationalism is on the ropes.

**B**rouwer is due some honor for his prescience in formulating intuitionism a hundred years ago — and intuitionism came almost fully formed out of the mind of Brouwer as syllogistic logic came almost fully formed out of the mind of Aristotle — so I would like to celebrate Brouwer on this, the one hundredth anniversary of his inaugural address at the University of Amsterdam, in which he formulated so many of the central principles of intuitionism.

**B**rouwer was prescient in another sense as well. He ended his inaugural address with a quote from Poincaré that is well known in the foundationalist community, since it has been quoted in many works since:

“Les hommes ne s’entendent pas, parce qu’ils ne parlent pas la même langue et qu’il y a des langues qui ne s’apprennent pas.”

**T**his might be (very imperfectly) translated into English as follows:

“Men do not understand each other because they do not speak the same language and there are languages that cannot be learned.”

**W**hat Poincaré called *men not understanding each other* Kuhn would later and more famously call incommensurability. And while we have always known that men do not understand each other, it had been widely believed before Brouwer that at least mathematicians understood each other because they spoke the same universal language of mathematics. Brouwer said that his exposition revealed, “the fundamental issue, which divides the mathematical world.” A hundred years later the mathematical world is still divided.

**F**or those who have not studied the foundations and philosophy of mathematics, it may come as a surprise that the past century, which has been so productive of research in advanced mathematics — arguably going beyond all the cumulative research in mathematics up to that time — has also been a century of conflict during which the idea of mathematics as true, certain, and necessary — ideas that had been central to a core Platonic tradition of Western thought — have all been questioned and largely abandoned. It has been a raucous century for mathematics, but also a fruitful one. A clever mathematician with a good literary imagination could write a mathematical analogue of Mandeville’s *Fable of the Bees* in which it is precisely the polyglot disorder of the hive that made it thrive.

**T**hat core Platonic tradition of Western thought is now, even as I write these lines, dissipating just as the illusions of the philosopher, freed from the cave of shadows, dissipate in the light of the sun above.

**B**rouwer, like every revolutionary (and we recall that it was Weyl, who was sympathetic to Brouwer, who characterized Brouwer’s work as a revolution in mathematics), wanted to do away with an old, corrupt tradition and to replace it with something new and pure and edifying. But in the affairs of men, a revolution is rarely complete, and it is, far more often, the occasion of schism than conversion.

**M**any were converted by Brouwer; many are still being converted today. As I wrote above, intuitionism remains a force to be reckoned with in contemporary mathematical thought in a way that logicism and formalism cannot claim to be such a force. But the conversions and subsequent defections left a substantial portion of the mathematical community unconverted and faithful to the old ways. The tension and the conflict between the old ways and the new ways has been a source of creative inspiration.

**P**recisely that moment in history when the very nature of mathematics was called into question became the same moment in history when mathematics joined technology in exponential growth.

**M**ars is the true muse of men.

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Check this book out: Introduction to Objectual Philosophy at http://filosofia.obiectuala.ro/en/.

Hi J.N.,

Wonderful post! I would like to ask for a quick clarification though. My impression is that at the time, Hilbert and the Formalists largely won out over Brouwer and in the early 20th century, the numbers of Intuitionists were few and far between as Relativity upturned the human intuition of time, which they took as their basis. Academics gave up on the idea of philosophy of mathematics, but Intuitionism survived as a kind of interesting “game”, precisely in the Formalist sense of mathematics as the picking and choosing of formal rules. Am I mistaken in this impression?

Hi Tom,

Thanks much for this comment. A lot of the specifics of Brouwer’s philosophy of logic and mathematics were marginal at the time when they were first proposed by Brouwer, and they remain marginal even today. However, constructivism generally and intuitionism specifically have continued in existence and have grown in their appeal once stripped of the more eccentric doctrines that Brouwer attached to his more technical views on methodology. Formalism and logicism were defeated in precisely the technical sense in which intuitionism and constructivism succeeded. Gödel’s incompleteness theorems in 1931-1932 proved that Hilbert’s program could not be carried out. Continuing work throughout the twentieth century on logical foundations eventually made it clear that mathematics could only be “reduced” to logic if logic included assumptions as strong as set theory, and, for most mathematicians, set theory is mathematics and not logic. So you can have logicism if you say that logicism means mathematics can be defined in terms of logic plus set theory.

Hilbert and his philosophy of mathematics was certainly attractive in its day, and Hilbert’s status as among the greatest mathematicians of his time added luster to his philosophy. Recently when I was researching my post

A Century of General RelativityI was interested learn the extent to which Einstein and Hilbert were racing neck-in-neck for the definitive formulation of general relativity. Hilbert and his finitistic approach to the foundations of mathematics was riding high at this time, but this was more than a decade before Gödel’s incompleteness theorems. After Gödel, Hilbert’s philosophy became a curiosity. Hilbert’s influence lingers on in some quarters (I was very interested recently to note the references to Hilbert’s philosophy of mathematics inGeorge F.R. Ellis, On the Nature of Cosmology Today, 2012 Copernicus Center Lecture), and Detlefsen wrote a book to argue for the continued plausibility of Hilbert’s program, but this is the exception, not the rule.There are some “game” formalists today, as there are neo-logicists, but this isn’t the active area of research in the foundations of mathematics that constructivism is. Foundations of mathematics in the meantime has become a highly technical area of mathematical research that has little to do with philosophy, but philosophers themselves never gave up on the philosophy of mathematics.

Best wishes,

Nick