One Hundred Years of Intuitionism and Formalism
14 October 2012
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.
The 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.”
Intuitionism, 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.
Brouwer 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.
Brouwer 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.”
This 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.”
What 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.
For 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.
That 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.
Brouwer, 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.
Many 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.
Precisely 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.
Mars is the true muse of men.
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