Later On

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Does Time Really Flow?

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The answer apparently is “Yes.” The idea of a block universe falls before intuitionist mathematics. Natalie Wolchover writes in Quanta:

Strangely, although we feel as if we sweep through time on the knife-edge between the fixed past and the open future, that edge — the present — appears nowhere in the existing laws of physics.

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.”

The timeless, pre-determined view of reality held by Einstein remains popular today. “The majority of physicists believe in the block-universe view, because it is predicted by general relativity,” said Marina Cortês, a cosmologist at the University of Lisbon.

However, she said, “if somebody is called on to reflect a bit more deeply about what the block universe means, they start to question and waver on the implications.”

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.

Physicists are still digesting Gisin’s work — it’s not often that someone tries to reformulate the laws of physics in a new mathematical language — but many of those who have engaged with his arguments think they could potentially bridge the conceptual divide between the determinism of general relativity and the inherent randomness at the quantum scale.

“I found it intriguing,” said Nicole Yunger Halpern, a quantum information scientist at Harvard University, responding to Gisin’s recent article in Nature Physics. “I’m open to giving intuitionist mathematics a shot.”

Cortês called Gisin’s approach “extremely interesting” and “shocking and provocative” in its implications. “It’s really a very interesting formalism that is addressing this problem of finite precision in nature,” she said.

Gisin said it’s important to formulate laws of physics that cast the future as open and the present as very real, because that’s what we experience. “I am a physicist who has my feet on the ground,” he said. “Time passes; we all know that.”

Information and Time

Gisin, 67, is primarily an experimenter. He runs a lab at the University of Geneva that has performed groundbreaking experiments in quantum communication and quantum cryptography. But he is also the rare crossover physicist who is known for important theoretical insights, especially ones involving quantum chance and nonlocality.

On Sunday mornings, in lieu of church, Gisin makes a habit of sitting quietly in his chair at home with a mug of oolong tea and contemplating deep conceptual puzzles. It was on a Sunday about two and a half years ago that he realized that the deterministic picture of time in Einstein’s theory and the rest of “classical” physics implicitly assumes the existence of infinite information.

Consider the weather. Because it’s chaotic, or highly sensitive to small differences, we can’t predict exactly what the weather will be a week from now. But because it’s a classical system, textbooks tell us that we could, in principle, predict the weather a week on, if only we could measure every cloud, gust of wind and butterfly’s wing precisely enough. It’s our own fault we can’t gauge conditions with enough decimal digits of detail to extrapolate forward and make perfectly accurate forecasts, because the actual physics of weather unfolds like clockwork.

Now expand this idea to the entire universe. In a predetermined world in which time only seems to unfold, exactly what will happen for all time actually had to be set from the start, with the initial state of every single particle encoded with infinitely many digits of precision. Otherwise there would be a time in the far future when the clockwork universe itself would break down.

But information is physical. Modern research shows it requires energy and occupies space. Any volume of space is known to have a finite information capacity (with the densest possible information storage happening inside black holes). The universe’s initial conditions would, Gisin realized, require far too much information crammed into too little space. “A real number with infinite digits can’t be physically relevant,” he said. The block universe, which implicitly assumes the existence of infinite information, must fall apart.

He sought a new way of describing time in physics that didn’t presume infinitely precise knowledge of the initial conditions.

The Logic of Time

The modern acceptance that there exists a continuum of real numbers, most with infinitely many digits after the decimal point, carries little trace of the vitriolic debate over the question in the first decades of the 20th century. David Hilbert, the great German mathematician, espoused the now-standard view that real numbers exist and can be manipulated as completed entities. Opposed to this notion were mathematical “intuitionists” led by the acclaimed Dutch topologist L.E.J. Brouwer, who saw mathematics as a construct. Brouwer insisted that numbers must be constructible, their digits calculated or chosen or randomly determined one at a time. Numbers are finite, said Brouwer, and they’re also processes: They can become ever more exact as more digits reveal themselves in what he called a choice sequence, a function for producing values with greater and greater precision.

By grounding mathematics in what can be constructed, intuitionism has far-reaching consequences for the practice of math, and for determining which statements can be deemed true. The most radical departure from standard math is that the law of excluded middle, a vaunted principle since the time of Aristotle, doesn’t hold. The law of excluded middle says that either a proposition is true, or its negation is true — a clear set of alternatives that offers a powerful mode of inference. But in Brouwer’s framework, statements about numbers might be neither true nor false at a given time, since the number’s exact value hasn’t yet revealed itself.

There’s no difference from standard math when it comes to numbers like 4, or ½, or pi, the ratio of a circle’s circumference to its diameter. Even though pi is irrational, with no finite decimal expansion, there’s an algorithm for generating its decimal expansion, making pi just as determinate as a number like ½. But consider another number x that’s in the ballpark of ½.

Say the value of is 0.4999, where further digits unfurl in a choice sequence. Maybe the sequence of 9s will continue forever, in which case x converges to exactly ½. (This fact, that 0.4999… = 0.5, is true in standard math as well, since x differs from ½ by less than any finite difference.)

But if at some future point in the sequence, a digit other than 9 crops up — if, say, the value of becomes 4.999999999999997… — then no matter what happens after that, x is less than ½. But before that happens, when all we know is 0.4999, “we don’t know whether or not a digit other than 9 will ever show up,” explained Carl Posy, a philosopher of mathematics at the Hebrew University of Jerusalem and a leading expert on intuitionist math. “At the time we consider this x, we cannot say that x is less than ½, nor can we say that x equals ½.” The proposition “x is equal to ½” is not true, and neither is its negation. The law of the excluded middle doesn’t hold.

Moreover, the continuum can’t be cleanly divided into two parts consisting of all numbers less than ½ and all those greater than or equal to ½. “If you try to cut the continuum in half, this number x is going to stick to the knife, and it won’t be on the left or on the right,” said Posy. “The continuum is viscous; it’s sticky.”

Hilbert compared the removal of the law of excluded middle from math to “prohibiting the boxer the use of his fists,” since the principle underlies much mathematical deduction. Although Brouwer’s intuitionist framework compelled and fascinated the likes of Kurt Gödel and Hermann Weyl, standard math, with its real numbers, dominates because of ease of use.

The Unfolding of Time

Gisin first encountered intuitionist math at a meeting last May attended by Posy. When the two got to talking, Gisin quickly saw a connection between the unspooling decimal digits of numbers in this mathematical framework and the physical notion of time in the universe. Materializing digits seemed to naturally correspond to the sequence of moments defining the present, when the uncertain future becomes concrete reality. . .

Continue reading.

Written by Leisureguy

7 April 2020 at 1:02 pm

Posted in Daily life, Math, Science

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