Intuitionist Mathematics and Common Sense

Roger Berkowitz

Hannah Arendt worried greatly about the rise of science. She took Niels Bohr seriously when he argued that “causality, determinism, and necessity of laws belonged to the categories of ‘our necessarily prejudiced conceptual frame’.” The new physics “defies description in terms of the ‘prejudices’ of the human mind[and] defies description in every conceivable way of human language.” Which is one reason why Albert Einstein had an “extreme reluctance to sacrifice the principle of causality as Planck’s Quantum Theory demanded.” For Arendt, the generalization and abstraction of science threaten to undo causality, necessity, and lawfulness that are “categories inherent in the human brain and applicable only to the common-sense experiences of earthbound creatures.” Science, in other words, is a humanly imagined threat to the humanity of the human world. What Arendt calls “world alienation” follows from the scientific demand “to leave the world of our senses and of our bodies not only in imagination but in reality.” 

Natalie Wolchover writes that now a Swiss physicist, Nicholas Gisin, has proposed a new theory of time that further challenges Einstein’s relativity theory. In Gisin’s “Intuitionist mathematics,” time is restored to a more common sense understanding, where time passes and change occurs. Against Einstein’s assumption of “the existence of infinite information” that can support a fully causal and comprehensible universe, Gisin sought a more human idea of time, one that “didn’t presume infinitely precise knowledge of the initial conditions.”

In Albert Einstein’s theory of relativity, for example, time is woven together with the three dimensions of space, forming a bendy, four-dimensional space-time continuum—a “block universe” encompassing the entire past, present, and future. Einstein’s equations portray everything in the block universe as decided from the beginning; the initial conditions of the cosmos determine what comes later, and surprises do not occur—they only seem to. “For us believing physicists,” Einstein wrote in 1955, weeks before his death, “the distinction between past, present and future is only a stubbornly persistent illusion.”...

Physicists who think carefully about time point to troubles posed by quantum mechanics, the laws describing the probabilistic behavior of particles. At the quantum scale, irreversible changes occur that distinguish the past from the future: A particle maintains simultaneous quantum states until you measure it, at which point the particle adopts one of the states. Mysteriously, individual measurement outcomes are random and unpredictable, even as particle behavior collectively follows statistical patterns. This apparent inconsistency between the nature of time in quantum mechanics and the way it functions in relativity has created uncertainty and confusion.

Over the past year, the Swiss physicist Nicolas Gisin has published four papers that attempt to dispel the fog surrounding time in physics. As Gisin sees it, the problem all along has been mathematical. Gisin argues that time in general and the time we call the present are easily expressed in a century-old mathematical language called “intuitionist mathematics,” which rejects the existence of numbers with infinitely many digits. When intuitionist math is used to describe the evolution of physical systems, it makes clear, according to Gisin, that “time really passes and new information is created.” Moreover, with this formalism, the strict determinism implied by Einstein’s equations gives way to a quantum-like unpredictability. If numbers are finite and limited in their precision, then nature itself is inherently imprecise, and thus unpredictable.