My interpretation of QM, so not orthodox

This is another answer I wrote on Quora. I’ve forgotten the question, but the answer is self-explanatory. It doesn’t cover anything new (from me) but it’s more succinct than other posts I’ve written.

I’m not a physicist, but I’m well read in this area and quantum mechanics (QM) has a particular fascination for me.

Someone did a survey at a conference and, from memory, the most popular was still Bohr’s so-called Copenhagen interpretation, which many now call ‘the shut up and calculate school’. I think most physicists no longer believe that consciousness is required to ‘observe’ the outcome of a quantum experiment (like the famous double slit experiment). 

Schrodinger’s famous cat thought experiment was to demonstrate how absurd that is. In his book, What is Life?, Schrodinger asks rhetorically where does the quantum effect become ‘real’. Does it occur in the optic nerve going to the brain? Or does it occur before then or when the person has their ‘Aha’ moment? Most people would now say it happens at the apparatus level, when the isotope decays, even before it affects the cat. 

One of the most popular interpretations seems to be the multiple worlds interpretation (Philip Ball calls it the MWI hypothesis). In this scenario, the universe spits into 2 (or more) so that all possibilities occur in some universe, but you only experience one of them.

There are other interpretations, like David Bohm’s pilot wave and the ‘transaction’ interpretation, which incorporates the time-symmetrical nature of the wave function. But, for the sake of brevity, I’ll discuss Roger Penrose’s, Paul Davies’ and Freeman Dyson’s.

Roger Penrose describes QM in 3 phases: U, R and C (always designated in bold). U is the evolution of the wave function (in Schrodinger’s equation), R is the observation or ‘decoherence’ when the wave function ‘collapses’ (or simply disappears) and C is the classical physics phase. Penrose thinks gravity plays a role in decoherence but I won’t discuss that here. 

Paul Davies argues for John Wheeler’s famous “…participatory universe” in which observers—minds, if you like—are inextricably tied to the concretization of the physical universe emerging from quantum fuzziness over cosmological durations.

This comes from Wheeler’s famous thought experiment that light from a distant quasar could be ‘lensed’ by an intervening massive object, like a galaxy, but we don’t know what path the light takes until it’s observed. This is an extension of his ‘delayed choice’ thought experiment relating to the double slit experiment (later confirmed in a laboratory setting).

Davies discusses this very cogently in an on-line paper and references another paper by Freeman Dyson, where he says, “Dyson concludes that a quantum description cannot be applied to past events.”

Personally, I agree with Dyson that QM describes the future and classical physics describes the past. In other words, I argue that the wave function is in the future, which is why it is never observed. This is consistent with Penrose’s 3 phases, which logically occur in a temporal sequence.

If one takes this approach to Wheeler’s photon from his quasar, it exists in the future of whatever it interacts with, including an observer’s instrument. Let’s assume, hypothetically, that the instrument is the observer’s eye. Because the wave function is time symmetrical the ‘delayed choice’ is really a backwards-in-time pathway to the photon’s source, so the observer sees it instantaneously in the past. In effect, this is the so-called transactional interpretation.

Richard Feynman’s path integral method of QED takes the sum of every path possible (most of which cancel out) to give a probability of where a particle (including a photon) will be observed. If all these paths exist in the future, that’s not a problem; only one of them will exist in the past, observed in retrospect. This is the opposite of the MW interpretation which claims all paths exist simultaneously.

Freeman Dyson comes to the following conclusion: 

“We do not need a human observer to make quantum mechanics work. All we need is a point of reference, to separate past from future, to separate what has happened from what may happen, to separate facts from probabilities.”

The curious thing about that statement is that the ‘point of reference’ is consciousness, because (as Schrodinger pointed out in What is Life?) consciousness is the only thing we know that exists in the continuous present.

This doesn’t make the observer the cause, because the cause is still at the photon’s source. It’s just that consciousness happens to be present in the ‘now’ between the QM future and the classical physics past that Dyson references.

Here is the link to both Davies’ and Dyson’s discussions.

Does infinity and the unknowable go hand in glove?

A recurring theme on my blog has been the limits of what we can know. So Marcus du Sautoy’s book, What We Cannot Know, fits the bill. I acquired it after I saw him give a talk at the Royal Institute on the subject, promoting the book, which is entertaining and enlightening in and of itself. I’ve previously read his The Music of the Primes and Finding Moonshine, both of which are very erudite and stimulating. He’s made a few TV programmes as well.

Previously, I’ve written blog posts based on books by Bryan Magee (Ultimate Questions) and Noson S. Yanofsky (The Outer Limits of Reason; What Science, Mathematics, and Logic CANNOT Tell Us). Yanofsky is a Professor in computer science, while Magee was a Professor of Philosophy (later a broadcaster and Member of British Parliament). I have to admit that Yanofsky’s book appealed to me more, because it’s more science based. Magee’s book was very erudite and provocative; my one criticism being that he seemed almost dismissive of the role that mathematics plays in the limits of what we can know. He specifically states that “...rationality requires us to renounce the pursuit of proof in favour of the pursuit of progress.” (My emphasis). However, pursuit of proof is exactly what mathematicians do, and, what’s more, they do it consistently and successfully, even though there is a famous proof that says there are limits to what we can prove (Godel’s Incompleteness Theorem).

Marcus du Sautoy is a mathematician, and a very good communicator as well, as can be evidenced on some of his YouTube videos, including some with Numberphile. But his book is not limited to mathematics. In fact, he discusses pretty much all the fields of our knowledge which appear to incorporate limits, which he metaphorically calls ‘Edges’. These include, chaos theory, quantum mechanics, consciousness, the Universe, and of course, mathematics itself. One is tempted to compare his book with Yanofsky’s, as they are both very erudite and educational, whilst taking different approaches. But I won’t, except to say they are both worth reading.

One aspect of du Sautoy’s book, which is unusual, yet instructive, is that he consulted other experts in their respective fields, including John Polkinghorne, John Barrow, Kristof Koch and Robert May. May, in particular, did pioneering work in chaos theory on animal populations in the 1970s. An ex-pat Australian, he’s now a member of the House of Lords, which is where du Sautoy had lunch with him. All these interlocutors were very stimulating and worthy additional contributors to their respective topics.

Very early on (p.10, in fact) du Sautoy mentions a famous misprediction by French philosopher, Auguste Compte, in 1835, about the stars: “We shall never be able to study, by any method, their chemical composition or their mineralogical structure.” Yet, less than a century later, it was being done by spectroscopy as a virtually standard practice, which in turn led to the knowledge that the Universe was expanding consistently in all directions. Throughout the book, du Sautoy reminds us of Compte’s prediction, when it appears that there are some things we will never know. He also quotes Donald Rumsfeld on the very next page:

There are known knowns; these are things that we know that we know. We also know there are known unknowns, that is to say, we know there are some things we do not know. But there are also unknown unknowns, the ones we don’t know we don’t know.

At the time, people tended to treat Rumsfeld’s statement as a bit of a joke and a piece of political legerdemain, given its context: weapons of mass destruction. However, in the field of science, it’s perfectly correct: there are hierarchies of knowledge, and when one looks back, historically, there have always been unknown unknowns, and, therefore, it’s a safe bet they will exist in the future as well. In other words, our future discoveries are dependent on secrets the Universe has yet to reveal to us mere mortals.

Towards the end of his book, du Sautoy gets more philosophical, which is not surprising, and he makes a point that I’ve not seen or heard before. He argues that some things about the Universe, like time, and the possibility of a multiverse, might remain unknown without physically getting outside the Universe, which is impossible. This, of course, raises the issue of God. Augustine, among others, has argued that God exists outside the Universe, and therefore, outside time. Paul Davies made the same point in his book, The Mind of God, with specific reference to Augustine.

Du Sautoy, who is a self-declared atheist, contends that God represents what we cannot know, which is consistent with the idea that some things we cannot know, can only be known from outside the Universe. But du Sautoy makes the point that there is something that exists outside the Universe that we know and that is mathematics. He, therefore, makes the tongue-in-cheek suggestion that maybe we can replace God with mathematics. Curiously, John Barrow made the same mischievous suggestion in one of his books – probably, Pi in the Sky. According to du Sautoy, Barrow is a Christian, which surprised me as much as du Sautoy, given that you would never know it from his writings. While on the subject of God, John Polkinghorne is a well known theologian as well as a physicist. Again, according to du Sautoy, Polkinghorne contends that God could intervene in the Universe via chaos theory. I once made the same point, although I also said I didn’t believe in an interventionist God, as that leads to people claiming they know God’s will, and that leads to all sorts of acts done in God’s name, and we all know how that usually ends. The problem with believing in an interventionist God is that it axiomatically leads to people believing they can influence said God.

Getting back to the subject at hand, du Sautoy says:

If there was no universe, no matter, no space, nothing. I think there would still be mathematics. Mathematics does not require the physical world to exist.

Following on from du Sautoy’s book, I started re-reading Eli Maor’s book, e: the story of a number, which incidentally covers the history of calculus going back to the ancient Greeks and Archimedes, in particular. The Greeks had a problem in that they couldn’t acknowledge infinity – it was taboo. Maor believes that Archimedes must have known the concept of infinity because he appreciated how an iterative process could converge to a value, but he wasn’t allowed to say so. Even in the modern day, there are mathematicians who wish to be rid of the concept of infinity, yet it’s intrinsic to mathematics everywhere you look.

This is relevant because the very nature of infinity tells us that there will always be truths beyond our kin. You can use a Turing machine (a computer) to calculate all the zeros in Riemann’s hypothesis and, if it’s true, it will never stop. Now, du Sautoy makes an interesting observation about this (which he expounds upon in this video, if you want it firsthand) that it’s possible that Riemann’s hypothesis is unknowable. In fact, there’s a small collection of conjectures associated with prime numbers that fall into this category (the Goldbach conjecture and the twin-prime conjecture being another 2). But here’s the thing: if one can prove that the Riemann hypothesis is unknowable, then it must be true. This is because, if it was untrue, there would have to be at least one result that didn’t fit the hypothesis, which would make it ‘knowable’.

The unknowable possibility is a direct consequence of Godel’s Incompleteness Theorem. To quote du Sautoy:

Godel proved mathematically that within any axiomatic system framework for number theory that was free of contradictions there were true statements about numbers that could not be proved within that framework – a mathematical proof that mathematics has its limitations. (My empasis).

I highlighted that passage because I left it out when proposing a definition to someone on Quora, and as a consequence, my interlocutor tried to argue that my definition was incorrect. Basically, I was saying that within any axiomatic system of mathematics there are ‘truths’ that can’t be proven. That’s Godel’s famous theorem in essence and in practice. However, one can find proofs, in principle, by using new axioms outside that particular system. And we see this in practice. The axiom that geometry can be non-Euclidean created new proofs, and the introduction of √-1 created new mathematics, called complex algebra, that gave solutions to previously unsolvable problems.

Towards the end of his book, du Sautoy references a little known and obscure point made by the renowned logician Alonso Church, called the ‘paradox of unknowability’, which proves that unless you know it all, there will always be truths that are by their very nature unknowable.

In effect, Church has extended Godel’s theorem to the physical world. Du Sautoy gives the example of all the dice that are lost in his house. There is either an even number of them or an odd number. One of these is true, but it is unknowable unless he can find them all. A more universal example is whether the Universe is infinite or finite. One of these is true but it’s currently unknowable and may be for all time. Du Sautoy makes the point that if we learn it’s finite then it becomes knowable, but if it’s infinite it may remain forever unknowable. This is similar to the Riemann hypothesis being knowable or unknowable. If it’s false then the Turing machine stops, which makes it finite, but, if it’s true, it is both infinite and unknowable, based on that thought experiment. It was only at this point in my essay that I came up with its title. I’ve expressed it as a question, but it’s really a conclusion.

If we go back to Archimedes and his struggle with the infinite, we can see that probably for most of humankind’s history, the infinite was considered outside the mortal realm. In other words, it was the realm of God. In fact, du Sautoy quotes Descartes: God is the only thing I positively conceive as infinite.

I’ve long contended that mathematics is the only ‘realm’ (for want of a better word) where infinity is completely at home. In Maor’s book, at one point, he discusses the difference between applied mathematics and pure mathematics, and it occurred to me that this distinction could explain the perennial argument about whether mathematics is invented or discovered. But the plethora of infinities, which is also intrinsic to unknowable ‘truths’, as outlined above, infers that there will always be mathematical ‘things’ waiting to be discovered. What’s more, the ‘marriage’ between theoretical physics and pure mathematics has never been more productive.

Addendum 1: After writing this, I re-watched an interview with Norman Wildberger on the subject of infinity and Real numbers. Wildberger is an Australian mathematician with ‘unorthodox’ views on the foundations of mathematics, as he explains in the video.

Wildberger is not a crank: he’s an academic mathematician, who has unusual philosophical ideas on mathematics. He makes the valid point that computers can only work with finite numbers (meaning numbers with a finite decimal extension), and that is the criterion he uses to determine whether something mathematical is ‘real’. He says he doesn’t believe in Real numbers, as they are defined, because they are infinitely uncomputable.

In effect, he argues they have no place in the physical world, but I disagree. In chaos theory, the reason chaotic phenomena are unpredictable is because you have to calculate the initial conditions to infinite decimal places, which is impossible. This is both mathematical and physical evidence that some things are ‘unknowable’.

Addendum 2: Sabine Hossenfelder argues that infinity is only 'real' in the mathematical world. She contends that in physics, it's not 'real', because it's not 'measurable'. She gives a good exposition in this YouTube video.